A Computability Perspective on (Verified) Machine Learning
Tonicha Crook
Jay Morgan
Arno Pauly
Markus Roggenbach
There is a strong consensus that combining the versatility of machine learning with the assurances given by formal verification is highly desirable. It is much less clear what verified machine learning should mean exactly. We consider this question from the (unexpected?) perspective of computable analysis. This allows us to define the computational tasks underlying verified ML in a model-agnostic way, and show that they are in principle computable.
perspective_article.tex

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\title{A Computability Perspective on (Verified) Machine Learning}

\titlerunning{A Computability Perspective on (Verified) Machine Learning} %TODO optional, please use if title is longer than one line

\author{Tonicha Crook}{Department of Computer Science, Swansea University, UK}{t.m.crook15@outlook.com}{https://orcid.org/0000-0002-4882-9999}{This author is supported by the UKRI AIMLAC CDT, cdt-aimlac.org, grant no. EP/S023992/1.}

\author{Jay Morgan}{Department of Computer Science, Swansea University, UK}{j.p.morgan@swansea.ac.uk}{https://orcid.org/0000-0003-3719-362X}{}

\author{Arno Pauly}{Department of Computer Science, Swansea University, UK \and \url{https://www.cs.swan.ac.uk/~cspauly/}}{arno.m.pauly@gmail.com}{https://orcid.org/0000-0002-0173-3295}{}

\author{Markus Roggenbach}{Department of Computer Science, Swansea University, UK}{m.roggenbach@swansea.ac.uk}{https://orcid.org/0000-0002-3819-2787}{}

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\begin{document}

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%TODO mandatory: add short abstract of the document
\begin{abstract}
There is a strong consensus that combining the versatility of machine learning with the assurances given by formal verification is highly desirable. It is much less clear what verified machine learning should mean exactly. We consider this question from the (unexpected?) perspective of computable analysis. This allows us to define the computational tasks underlying verified ML in a model-agnostic way, and show that they are in principle computable.
\end{abstract}


\section{Introduction}

Machine Learning (ML) concerns the process of building both predictive and generative models through the use of optimisation procedures. The remarkable success of ML methods in various domains raises the question of how much trust one can put into the responses that a ML model provides. As ML models are also applied in critical domains, some form of verification might be adequate. 

However, due to the widespread use of non-discrete mathematics in ML, traditional verification techniques are hard to apply to its artefacts. Furthermore, many ML applications lack specifications in form of, say,  an input/output relationship, on which many `classical' verification approaches are based. A typical example of this would be an ML application that shall decide if a given picture depicts a cat. Lacking a specification, what kind of properties can be verified?

Using the toolset of computable analysis, in this paper we present an investigation into what kind of verification questions are answerable in principle about ML models -- irrespective of the particular ML framework applied. 

%% we are dealing with classification

For the purpose of this paper, we will focus upon the notion of a ML classifier. A classifier
takes as input some description of an object, say in 
form of a vector of real numbers. As an output a classifier either provides a
colour, which we take to be an integer from the set $\mathbf{k} = \{0,\ldots,k-1\}, k > 0$, or it does not give an answer, i.e., we consider classifiers to be semi-decision procedures. 
%
%% the properties we are interested in are of the kind ...
%
Simple examples of properties that such a classifier is $f$ might exhibit, include: 
\begin{itemize}
    \item Can $f$ output a specific colour for points stemming from a given region?
    \item Is $f$ constant on a given region?
    \item Are there two `close' points that $f$ maps to distinct colours?
\end{itemize}
We will show for these and other properties that they are indeed computable (under some `natural' assumptions). 

%% [why such properties would be interesting?] naturally answered by the adversarial examples
% in particular, we pay attention to adversarial examples, as they are prominently studies in the literature of DNNs
One specific verification task that has caught the attention of the ML community is to find \emph{adversarial examples} \cite{szegedy2013intriguing,Goodfellow-et-al-2016,Huang2016} or to prevent them from occurring. One says that an adversarial example occurs when a `small' change to the input results in a `large' change in output or a miss-classifications is made by the predictive model (i.e.~our third task above). For example, given a correctly classified image, small, even unnoticeable changes to the pixels in the image can vastly change the classification output. The existence of adversarial examples can be catastrophic in the context of fully-autonomous vehicles that use sensor-captured data as input for Deep Neural Network (DNN) models. In such situations, \cite{Huang2016} explains how lighting conditions and angles, as well as defects in sensor equipment themselves, may contain many realistic adversarial examples. We show that certain variations of this task are computable without the need to refer to specifics of an ML framework.

%% On the technical side, our investigation is based in classical notions of computable analysis 

Though we will present semi-decision procedures in this paper, our algorithms are not meant for implementation. For our foundational work on what kind of verification is possible in principle, it is beneficial to work in a framework which is agnostic to ML methods. However, when it comes to the task of actually developing `useful' verification algorithms, it will be essential to take into account the specifics of the method used.

While the language of computable analysis was a natural choice for us, we do not claim that this is the only way of how the challenge of `verified ML' can be addressed. Providing an abstract perspective will, we hope, help further researchers in the field to develop new ideas and communicate more efficiently.

The rest of the paper is organised as follows: in Section \ref{sec:background} we explore the concepts required for this paper within computable analysis. Classification tasks are considered within Section \ref{sec:AdvEx}, where we look into adversarial examples. We explore different techniques and how well they manage adversarial examples/attacks. The training of classifiers is discussed in Section \ref{sec:robust_learners}, where we explore the robustness of learners and sparsity of data. Finally, in Section \ref{sec:whatspaces} we examine different demands we could hold over our spaces, such as Hausdorffness and compactness. 






% Deep Neural Networks (DNNs), a sub-field of ML, has seen a remarkable rise in popularity due to their substantial gains in performance, as well as needing less data pre-processing or feature engineering compared to other ML techniques \cite{Goodfellow-et-al-2016}. Indeed, many of the current state-of-the-art models with some of the most accurate models for object recognition, text-to-speech and language translation, and generation of novel works of art all owe their roots back to DNNs \cite{Lecun2015}.

% DNNs, despite improvements in architectural designs and predictive performance, are shown to be susceptible to what is commonly referred to as adversarial examples \cite{szegedy2013intriguing}. These examples occur when small changes to the input result in a very large change in output or miss-classifications made by the predictive model. For example, given a correctly classified image, small, even unnoticable changes to the pixels in the image can vastly change the classification.

% The presence of adversarial examples has caught the attention of the ML community, where many methods for both the attack and defence of DNNs have been proposed. Many of these methods demonstrate the extent to which DNNs are susceptible to adversarial examples. The existence of adversarial examples may be then catastrophic in scenarios for fully-autonomous vehicles where sensor-captured data to the DNN models may be unpredictable due to changes in environment able factors such as various lighting conditions and angles, as well as defects in sensor equipment themselves \cite{Huang2016}.

% Furthermore, the existence of adversarial examples contradicts the notion of how well highly-trained DNNs are generalising when predicting accurately on the test set. Indeed, even with many proposed training methods to improve the robustness of DNNs to adversarial examples, many have shown to subsequently to be ineffective \cite{Carlini2017}. This disparity between testing well and generalising in the presence of small perturbations necessitates the use of verification techniques other than the test set. For DNNs to be used in safety-critical systems, formal guarantees on the properties of the DNN model must be made.

% \jmn{Should we explain some of the existing techniques for formal
%   verification of ML? For example, we may talk about the use of SMT
%   for verification of the existence of adversarials, but how, in
%   practice, the usage is limited due to time constraints and model
%   complexity.}

% \meta{I agree that bringing this up makes sense. In the introduction, we should probably stay more lightweight however. By the way, the way the computability proofs work kind of "suggests" that SMT-solvers can do the discretized version.}

% \jmn{We don't even have to just stick to Adversarial examples. There
%   are many more reasons to justify the use of verification. Take for
%   example, the use of ML for control of automated robotics. We could
%   use output range analysis of some other techniques. This should fall
%   under the principles of what questions we may want to ask.}

%   \meta{having more examples of verification is great, in particular if we can link them to general operations we've already considered, or we can considered, too}

% \paragraph*{Our ambitions}
% We hope to identify the core computational tasks in
% verified ML. While we will show
% that these tasks are computable, this is just evidence that
% we have indeed formalised things properly. Our algorithms
% are not meant for implementation. For foundational work as we present
% here, a framework which is agnostic to the
% ML methods is beneficial. Once it comes to the task of actually developing useful
% verification algorithms, taking into account the specifics of the
% methods used will be essential to yield usable algorithms.

% While the language of computable analysis is a natural choice for us,
% we are also not claiming that this is the way it has to be
% done. Providing an abstract perspective will, we hope, help some
% researchers engaged in verified ML to
% develop new ideas and communicate more efficiently.

\section{Computing with real numbers and other non-discrete data types}
\label{sec:background}

The mathematics behind the most popular notions of ML is  based on real numbers; whether it is on a fundamental level such as the use of Sigmoid activation functions in DNNs, the vector space structure for Support Vector Machine (SVM), or more abstract ideas such as the data manifold hypothesis \cite{narayanan2010sample}. While the actual implementations will use floating point numbers rather than exact real arithmetic, this still suggests that computable analysis is a reasonable framework for studying algorithmic issues in an abstract setting.

Computable analysis as a field is not just concerned with computability of operations on the real numbers, but far more generally with non-necessarily discrete domains. One question we strive to answer here is what properties of the domain are actually needed (or at least useful) to obtain the fundamental results by keeping track of what we are using where. Working with effectively locally-compact computable metric spaces turns out to be a promising candidate, as we will argue in Section \ref{sec:whatspaces}.

Standard sources for computable analysis include the textbook \cite{weihrauchd} by Weihrauch, around which the modern field coalesced; as well as the tutorial article \cite{brattkaintro}. The approach we follow is the one outlined in \cite{pauly-synthetic}, to which we refer to reader for details.

The basic structure in computable analysis is a \emph{represented space}, which carries a notion of computability via a coding of elements by infinite binary sequences. Core aspects relevant for us include the function space construction $\mathcal{C}(\mathbf{X},\mathbf{Y})$, where a function from $\mathbf{X}$ to $\mathbf{Y}$ is coded by just enough information to evaluate it. We refer to several hyperspaces of subsets, such as the open sets $\mathcal{O}(\mathbf{X})$ with just enough information to recognise membership (given an $U \in \mathcal{O}(\mathbf{X})$ and $x \in X$, the property $x\in U$ is semidecidable), compact subsets $\mathcal{K}(\mathbf{X})$ with just enough information to recognise containment in open sets (given $A \in \mathcal{K}(\mathbf{X})$ and $U \in \mathcal{O}(\mathbf{X})$, the property $A \subseteq U$ is semidedicable), and overt subsets $\mathcal{V}(\mathbf{X})$ with just enough information to recognise intersecting open sets (given $A \in \mathcal{V}(\mathbf{X})$ and $U \in \mathcal{O}(\mathbf{X})$, the property $A \cap U \neq \emptyset$ is semidedicable). Knowing $A \in (\mathcal{V} \wedge \mathcal{K})(\mathbf{X})$ means knowing $A \in \mathcal{V}(\mathbf{X})$ and $A \in \mathcal{K}(\mathbf{X})$ at the same time.

The spaces $\mathbb{R}^n$ can be considered as represented spaces by encoding real vectors as limits of sequences of rational vectors converging with speed $2^{-n}$. This in particular ensures that the maps $(x,r) \mapsto \overline{B}(x,r) : \mathbb{R}^n \times \mathbb{R} \to \mathcal{K}(\mathbf{\mathbb{R}^n})$ and $(x,r) \mapsto \overline{B}(x,r) : \mathbb{R}^n \times \mathbb{R} \to \mathcal{V}(\mathbf{\mathbb{R}^n})$ and computable, where $\overline{B}(x,r) = \{y \in \mathbb{R}^n \mid d(x,y) \leq r\}$ are the closed balls (with $d$ being any `reasonable' distance, e.g.~the Euclidean).

An important facet of computable analysis is that equality is typically not decidable; in particular it is not for the spaces $\mathbb{R}^n$. For these, inequality is semidecidable though (which makes them computably Hausdorff). If equality is also semidecidable, a space is called computably discrete. 

\subsection*{Some mathematical detail}
The following summarizes the formal definitions and key properties of the most important notions for our paper. It is taken near verbatim from \cite{paulydebrecht3-csl}. As mentioned above, a more comprehensive treatment is found in \cite{pauly-synthetic}.

\begin{definition}
A represented space is a pair $\mathbf{X} = (X, \delta_\mathbf{X})$ where $X$ is a set and $\delta_\mathbf{X} : \subseteq \Baire \to X$ is a partial surjection (the notation $:\subseteq$ denotes partial functions). A function between represented spaces is a function between the underlying sets.
\end{definition}

\begin{definition}
For $f : \subseteq \mathbf{X} \to \mathbf{Y}$ and $F : \subseteq \Baire \to \Baire$, we call $F$ a realizer of $f$ (notation $F \vdash f$), iff $\delta_Y(F(p)) = f(\delta_X(p))$ for all $p \in \dom(f\delta_X)$. A map between represented spaces is called computable (continuous), iff it has a computable (continuous) realizer.
\end{definition}

Two represented spaces of particular importance are the integers $\mathbb{N}$ and Sierpi\'nski space $\mathbb{S}$. The represented space $\mathbb{N}$ has as underlying set $\mathbb{N}$ and the representation $\delta_\mathbb{N} : \Baire \to \mathbb{N}$ defined by $\delta_\mathbb{N}(p) = p(0)$. The Sierpi\'nski space $\mathbb{S}$ has the underlying set $\{\top,\bot\}$ and the representation $\delta_\mathbb{S}$ with $\delta_\mathbb{S}(0^\omega) = \bot$ and $\delta_\mathbb{S}(p) = \top$ for $p \neq 0^\omega$.

Represented spaces have binary products, defined in the obvious way: The underlying set of $\mathbf{X} \times \mathbf{Y}$ is $X \times Y$, with the representation $\delta_{\mathbf{X} \times \mathbf{Y}}(\langle p, q\rangle) = (\delta_\mathbf{X}(p),\delta_\mathbf{Y}(q))$. Here $\langle \ , \ \rangle : \Baire \times \Baire \to \Baire$ is the pairing function defined via $\langle p, q\rangle(2n) = p(n)$ and $\langle p, q\rangle(2n+1) = q(n)$.

A central reason why the category of represented spaces is such a convenient setting lies in the fact that it is cartesian closed. We have available a function space construction $\mathcal{C}(\cdot, \cdot)$, where the represented space $\mathcal{C}(\mathbf{X},\mathbf{Y})$ has as underlying set the continuous functions from $\mathbf{X}$ to $\mathbf{Y}$, represented in such a way that the evaluation map $(f, x) : \mathcal{C}(\mathbf{X},\mathbf{Y}) \times \mathbf{X} \to \mathbf{Y}$ becomes computable. This can be achieved, e.g., by letting $nq$ represent $f$, if the $n$-th Turing machine equipped with oracle $q$ computes a realizer of $f$. This also makes currying, uncurrying and composition all computable maps.

Having available to us the space $\mathbb{S}$ and the function space construction, we can introduce the spaces $\mathcal{O}(\mathbf{X})$ and $\mathcal{A}(\mathbf{X})$ of open and closed subsets respectively of a given represented space $\mathbf{X}$. For this, we identify an open subset $U$ of $\mathbf{X}$ with its (continuous) characteristic function $\chi_U : \mathbf{X} \to \mathbb{S}$, and a closed subset with the characteristic function of the complement. As countable join (or) and binary meet (and) on $\mathbb{S}$ are computable, we can conclude that open sets are uniformly closed under countable unions, binary intersections and preimages under continuous functions by merely using elementary arguments about function spaces. The space $\mathcal{A}(\mathbf{X})$ corresponds to the upper Fell topology \cite{fell} on the hyperspace of closed sets.

Note that neither negation $\mathalpha{\neg} : \mathbb{S} \to \mathbb{S}$ (i.e.~mapping $\top$ to $\bot$ and $\bot$ to $\top$) nor countable meet (and) $\bigwedge : \mathcal{C}(\mathbb{N},\mathbb{S}) \to \mathbb{S}$ (i.e.~mapping the constant sequence $(\top)_{n \in \mathbb{N}}$ to $\top$ and every other sequence to $\bot$) are continuous or computable operations.

We need two further hyperspaces, which both will be introduced as subspaces of $\mathcal{O}(\mathcal{O}(\mathbf{X}))$. The space $\mathcal{K}(\mathbf{X})$ of saturated compact sets identifies $A \subseteq \mathbf{X}$ with $\{U \in \mathcal{O}(\mathbf{X}) \mid A \subseteq U\} \in \mathcal{O}(\mathcal{O}(\mathbf{X}))$. Recall that a set is saturated, iff it is equal to the intersection of all open sets containing it (this makes the identification work). The saturation of $A$ is denoted by $\sat{A} := \bigcap \{U \in \mathcal{O}(\mathbf{X}) \mid A \subseteq U\}$. Compactness of $A$ corresponds to $\{U \in \mathcal{O}(\mathbf{X}) \mid A \subseteq U\}$ being open itself. The dual notion to compactness is \emph{overtness}\footnote{This notion is much less known than compactness, as it is classically trivial. It is crucial in a uniform perspective, though. The term \emph{overt} was coined by Taylor \cite{taylor}, based on the observation that these sets share several closure properties with the open sets.}. We obtain the space $\mathcal{V}(\mathbf{X})$ of overt sets by identifying a closed set $A$ with $\{U \in \mathcal{O}(\mathbf{X}) \mid A \cap U \neq \emptyset\} \in \mathcal{O}(\mathcal{O}(\mathbf{X}))$. The space $\mathcal{V}(\mathbf{X})$ corresponds to the lower Fell (equivalently, the lower Vietoris) topology.

Aligned with the definition of the compact and overt subsets of a space, we can also define when a space itself is compact (respectively overt):

\begin{definition}
A represented space $\mathbf{X}$ is (computably) compact, iff $\textrm{isFull} : \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ mapping $X$ to $\top$ and any other open set to $\bot$ is continuous (computable). Dually, it is (computably) overt, iff $\textrm{isNonEmpty} : \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ mapping $\emptyset$ to $\bot$ and any non-empty open set to $\top$ is continuous (computable).
\end{definition}

The relevance of $\mathcal{K}(\mathbf{X})$ and $\mathcal{V}(\mathbf{X})$ is found in particular in the following characterisations, which show that compactness just makes universal quantification preserve open predicates, and dually, overtness makes existential quantification preserve open predicates.

\begin{proposition}[{\cite[Proposition 40]{pauly-synthetic}}]
\label{prop:exists}
The map $\exists : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{V}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ defined by $\exists(R, A) = \{y \in Y \mid \exists x \in A \ (x, y) \in R\}$ is computable. Moreover, whenever $\exists : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{S}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ is computable for some hyperspace $\mathcal{S}(\mathbf{X})$ and some space $\mathbf{Y}$ containing a computable element $y_0$, then $\overline{\phantom{A}} : \mathcal{S}(\mathbf{X}) \to \mathcal{V}(\mathbf{X})$ is computable, where $\overline{\phantom{A}}$ denotes topological closure.
\end{proposition}

\begin{proposition}[{\cite[Proposition 42]{pauly-synthetic}}]
\label{prop:forall}
The map $\forall : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{K}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ defined by $\forall(R, A) = \{y \in Y \mid \forall x \in A \ (x, y) \in R\}$ is computable. Moreover, whenever $\forall : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{S}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ is computable for some hyperspace $\mathcal{S}(\mathbf{X})$ and some space $\mathbf{Y}$ containing a computable element $y_0$, then $\sat{\id} : \mathcal{S}(\mathbf{X}) \to \mathcal{K}(\mathbf{X})$ is computable.
\end{proposition}


\section{Verifying classifiers and adversarial examples} \label{sec:AdvEx}
We consider classification tasks only. This means that a trained model will take as input some description of an object, and either outputs a class (which we take to be an integer from $\mathbf{k} = \{0,\ldots,k-1\}$, $k > 0$), or it does not give an answer. Here, not giving an answer can happen by the algorithm failing to terminate, rather than by an explicit refusal to select a class. This is important to handle connected domains such as the reals, in light of the continuity of all computable functions. Formally, we are dealing with the represented space $\mathbf{k}_{\bot}$ here, which contains the elements $\{0,\ldots,k-1,\bot\}$, where $0^\omega$ is the only name for $\bot$, and any $0^m1^\ell0^\omega$ is a name for $\ell < k$. 

\begin{definition}
\label{def:classifier}
A \emph{classifier} is a (computable) procedure that takes some $x \in \mathbf{X}$ as input, and either outputs a colour $j \in \mathbf{k}$, or diverges (which is seen as outputting $\bot$). The collection of classifiers is the space $\mathcal{C}(\mathbf{X},\mathbf{k}_\bot)$.
\end{definition}

At first, we do not need to put restrictions on the domain, and can work just with some represented space $\mathbf{X}$. We will discuss reasons for restrictions later on, in particular in Section \ref{sec:whatspaces}. 

\begin{example}
Let us consider the classifier we would obtain from Support Vector Machine \cite{hearst1998support}. Here, the relevant space $\mathbf{X}$ will be $\mathbb{R}^n$ for some $n \in \mathbb{N}$. The classifier is described by a hyperplane $P$ splitting $\mathbb{R}^n$ into two connected components $C_0$ and $C_1$. We have two colours (corresponding to the two connected components), so the classifier is a map $p : \mathbb{R}^n \to \mathbf{2}_\bot$. If $x \in C_i$, then $p(x) = i$. If $x \in P$, then $p(x) = \bot$.

One reason why the \emph{no-answer answer} $\bot$ has to be treated special is that we will never be able to be certain that a numerical input is exactly on the separating hyperplane (essentially as equality on reals is not decidable). If we are willing to compute with sufficiently high precision, then for any input not on the hyperplane we can eventually figure out on which side it is. 


This idealisation is not strictly necessary to make Definition \ref{def:classifier} applicable. One could apply floating point computations and keep track of the errors. In case the errors become too large to classify an input, the response is $\bot.$
This leads to an approximate classifier $\tilde{p} : \mathbb{R}^n \to \mathbf{2}_\bot$, where $\tilde{p}(x) = i$ signifies that with the available accuracy we can confirm that $x \in C_i$, and $\tilde{p}(x) = \bot$ stands for not being able to reach a conclusive answer at the given accuracy level. 
%While here we can in principle be certain that $\tilde{p}$ should output $\bot$, this answer is still not of the same nature than the definite classes $\{0,1\}$.
\end{example}

\begin{example}
Neural networks classifier compute a class score for every colour, which when these class scores are normalised, share similar properties as a probability distribution. This translates into our framework by fixing a threshold $p \geq 0.5$, and then assigning a particular colour to an input iff its class score exceeds the threshold $p$. If no colour has a sufficiently high score, the output is $\bot$. As long as the function computing the class scores is computable, so is the classifier we obtain in this fashion. If our class scores can use arbitrary real numbers, we cannot assign a colour for the inputs leading to the exact threshold.

We can, should we choose to do, however add a special colour for those inputs where all scores are strictly below the threshold. It is questionable how desirable assigning colour for the undecided cases is. For example, consider the binary classification task of predicting if an image depicts a cat or a dog. We start with a picture of a cat, and apply several iterations of destructive random image manipulations (such as rotations, sheers, and changing of individual pixel values). Should $\bot$ be a definite class, there would be a computable exact point at which the image ceases to depict a cat \cite{pmlr-v80-athalye18b}.
\end{example}

\paragraph*{Basic verification questions}
What are the verification-questions we could ask about a given classifier $f$ (that is, without considering the learning process that led us to it)? Obviously, we may want to confirm individual requirements of the form $f(x) = n$, but not much sophistication is needed here. It becomes more interesting once we deal with assertions regarding the behaviour of the classifier on an entire set or `region'. While this will generally involve a quantifier over infinite sets of inputs, we can do the following:

\begin{proposition}
\label{prop:basic}
The following maps are computable:
\begin{enumerate}
\item $\mathrm{existsValue} : \mathbf{k} \times \mathcal{V}(\mathbf{X}) \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbb{S}$, which answers $\operatorname{true}$ on input $(n,A,f)$ iff $\exists x \in A \ f(x) = n$.
\item $\mathrm{forallValue} : \mathbf{k} \times \mathcal{K}(\mathbf{X}) \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbb{S}$, which answers $\operatorname{true}$ on input $(n,A,f)$ iff $\forall x \in A \ f(x) = n$.
\item $\mathrm{fixedValue} : \mathbf{k} \times (\mathcal{V}\wedge\mathcal{K})(\mathbf{X}) \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbf{2}_\bot$, which on input $(n,A,f)$ answers $1$ iff $\forall x \in A \ f(x) = n$, and answer $0$ iff $\exists x \in A \ f(x) \in \mathbf{k} \setminus \{n\}$, and $\bot$ otherwise.
\item $\mathrm{constantValue} : (\mathcal{V}\wedge\mathcal{K})(\mathbf{X}) \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbf{2}_\bot$, which on input $(A,f)$ answers $1$ iff there is some $n \in \mathbf{k}$ such that $\mathrm{fixedValue}(n,A,f)$ answers $1$, and which answers $0$ iff $\mathrm{fixedValue}(n,A,f)$ answers $0$ for all $n \in \mathbf{k}$.
\end{enumerate}
\end{proposition}

\begin{proof}%[Proof of Proposition \ref{prop:basic}]
\begin{enumerate}
    \item Given a colour $k \in \mathbf{k}$ and a classifier $f \in \mathcal{C}(\mathbf{X},\mathbf{k}_{\bot})$, we can compute $f^{-1}(k) \in \mathcal{O}(\mathbf{X})$ due to the discreteness of $\mathbf{k}$. By definition of overtness, given $A \in \mathcal{V}(\mathbf{X})$, we can then semidecide whether $A \cap f^{-1}(k) \neq \emptyset$.
    \item As above, we can compute $f^{-1}(k) \in \mathcal{O}(\mathbf{X})$. By the definition of compactness, given $A \in \mathcal{K}(\mathbf{X})$ we can semidecide whether $A \subseteq f^{-1}(k)$.
    \item By running the algorithms from 1. and 2. in parallel.
    \item Since $\mathbf{k}$ is both compact and overt, we can quantify over it both existentially and universally. Or, more down to earth, we can simply run the algorithm from 3. for all finitely many cases in parallel, and answer once we have received enough information from those runs.
\end{enumerate}
\end{proof}

One useful application of the map $\mathrm{constantValue}$ is using it on some \emph{small} regions that we are interested in. In ML terms it addresses the question if there are adversarial examples for a classifier in the vicinity of $x$. To characterise small regions, we would have available a metric, and then wish to use closed balls $\overline{B}(x,r)$ as inputs to $\mathrm{constantValue}$.

To be able to do this, we need to obtain closed balls $\overline{B}(x,r)$ as elements of $(\mathcal{V}\wedge\mathcal{K})(\mathbf{X})$. Being able to obtain $\overline{B}(x,r) \in \mathcal{K}(\mathbf{X})$ for sufficiently small $r$, amounts to effective local compactness of $\mathbf{X}$. We generally get $\mathrm{cl} B(x,r)$, the closure of the open ball, as elements of $\mathcal{V}(\mathbf{X})$. For all but countably many radii $r$ we have that $\overline{B}(x,r) = \mathrm{cl} B(x,r)$, and we can effectively compute suitable radii within any interval
\cite{pauly-locallycompact}.

\begin{theorem}
\label{theo:adversarialcomputable}
Let $\mathbf{X}$ be an effectively locally compact computable metric space with metric $d$. There is a computable multi-valued function producing on input $x \in \mathbf{X}$ some real $E^x > 0$ as well as an operation $\mathrm{locallyConstant}_x : (0,E^x) \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbf{2}_\bot$, where $\mathrm{locallyConstant}_x(r,f) = 1$ iff $\forall y \in \overline{B}(x,r) \ f(x) = f(y) \neq \bot$, and $\mathrm{locallyConstant}_x(r,f) = 0$ iff $\exists y_0,y_1 \in B(x,r) \ \bot \neq f(y_0) \neq f(y_1) \neq \bot$.

If $(\mathbf{X},d)$ is such that every closed ball is compact, we can ignore the $E^x$ and have $\mathrm{locallyConstant} : \mathbf{X} \times \mathbb{R}^+ \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbf{2}_\bot$ instead, with $\mathrm{locallyConstant}(x,r,f) = \mathrm{locallyConstant}_x(r,f)$.
\end{theorem}

\begin{proof}%[Proof of Theorem \ref{theo:adversarialcomputable}]
By \cite[Proposition 12]{pauly-locallycompact}, given a point $x$ in an effectively locally compact computable metric space we can compute a radius $E^x$ and the corresponding closed ball $\overline{B}(x,E^x)$ as a compact set. Subsequently, we can obtain every closed ball as a compact set given a radius less than $E^x$. If every closed ball is compact, we can even obtain them computably as elements of $\mathcal{K}(\mathbf{X})$ by \cite[Proposition 10]{pauly-locallycompact}.

In either case, we have $\overline{B}(x,r)$ available to us as a compact set, and can thus correctly answer $1$ (if this is the case) by running $\mathrm{forallValue}$ from Proposition \ref{prop:basic} for all finitely many possible colours and $\overline{B}(x,r)$.

Without any constraints on the computable metric space $\mathbf{X}$ we can obtain $\mathrm{cl} B(x,r) \in \mathcal{V}(\mathbf{X})$, and we note that if there are two points assigned to distinct colours in $\mathrm{cl} B(x,r)$, there are already such points in $B(x,r)$. Hence, running $\mathrm{existsValue}$ from Proposition \ref{prop:basic} on all finitely many possible colours and $\mathrm{cl} B(x,r)$ lets us correctly answer $0$, if this is the case were are in.
\end{proof}

In the parlance of ML, the map $\mathrm{locallyConstant}_x$ states whether or not their are \emph{adversarial examples} for the classifier $f$ in the vicinity of $x$. If there are no adversarial examples, but some unclassified points (i.e.~where $f$ returns the value $\bot$), then $\mathrm{locallyConstant}_x$ cannot give an answer (because in general, we can never rule out that a classifier will give an answer later on).

\subsection{Adversarial examples}

An adversarial example is the result of a small change or perturbation to the original input that results in a change of classification made by the DNN. I.e.~given the classifier $f$ and an input $x$, an adversarial example is $f(x) \neq f(x + r)$ for $|| r || \leq \epsilon$ and $\epsilon > 0$. It was assumed that many DNNs followed many existing kernel methods in computer vision, where even with small perturbations, the output would still be given the correct class with a high probability. However, \cite{szegedy2013intriguing} demonstrates that this assumption does not hold, even for state-of-the-art image classifiers. They demonstrate the use of a box-constrained optimisation (L-BFGS) method to locate small low-probability `pockets' around the original image that produce miss-classification of a target class. Moreover, this work shows the susceptibility to adversarial examples are not simply an artefact of the training methods (as a model trained on a different subset of the data would produce the same miss-classification), but rather propose there is a deeper and not fully understood meaning behind the effect.

While \cite{szegedy2013intriguing} proposed to use the adversarial examples as part of the training set, the optimisation process was computationally expensive. Later work by \cite{Goodfellow2015} explores a possible reason for adversarial examples existing as well as creating a faster method of locating them. Their Fast Gradient Sign Method (FGSM) automatically generates adversarial examples by adding the sign of the gradient w.r.t. the cost function to the original image. By doing so, it is purposefully pushing the pixel value of the image in the direction that increases the loss of the prediction, and thus possibly creating a miss-classification. This approach allows for adversarial examples to be included in the set of training data, and thus aims to improve the robustness against adversarial examples at test time. 

A method by \cite{Papernot2016a} computes the Jacobian matrix of the input image to find the most significant or salient pixels of the image in which to perturb. The output of their method, Jacobian-based Saliency Map Attack or JSMA, differs from \cite{Goodfellow2015}. Where as in FGSM the perturbations are imperceptible changes to the entire image, JSMA makes a large and noticeable change to a small region, while the rest of the image remains untouched. While showing their method can outperform many other adversarial attacks in various scenarios, its reliance on computing the Jacobian, a potentially slow computation, restricts its usability in many applications. The C\&W attack designed by \cite{Carlini2017} is very effective against many adversarial defences. Their work improves upon the objective function of \cite{szegedy2013intriguing}, to improve the optimisation of the method. In this work they proposed many alternatives for objective function candidates as well as distance metrics.

Despite many methods being proposed to easily fool DNNs, very few quantify and compute the robustness w.r.t. adversarial examples. The \emph{DeepFool} algorithm proposed by \cite{moosavi2016deepfool} aims to compute adversarial examples and compare the robustness of various classifiers to particular types of perturbations. DeepFool iteratively perturbs the original image in the direction of the closest hyperplane and thus creating a miss-classification with a minimal Euclidean distance to the original image as well in a potentially quicker time as compared with \cite{Goodfellow2015}.

All of these methods demonstrate the key principle: image-based DNNs can be easily fooled with precise pixel manipulation. When combined with the realistic scenario of self-driving vehicles, safety concerns arise. \cite{Wicker2018a} use a Gaussian mixture model to identify keypoints in images that describe the saliency map of DNN classifiers. Modifying these keypoints may then change the classification label made by said DNN. They explore their approach on `traffic light challenges' (publicly available dashboard images of traffic lights with red/green annotations). In this challenge they find modifying a single pixel is enough to change neural network classification.

\subsection{Choosing the distance parameter}
  
If we want to use the operation from Theorem \ref{theo:adversarialcomputable} to test for the existence of adversarial examples, we need to know how to choose the parameter $r$ indicating how small perturbations need to be to potentially count as adversarial examples. For example, assume that we want to use our classifier to classify measurement results with some measurement errors. If our measurements are only precise up to $\varepsilon$, then having an adversarial example for $r = \varepsilon$ tells us that we cannot trust the answers from our classifier. We could just as well have received a slightly different answer from our measurement procedure, on which the classifier would have answered differently.

We could also use domain knowledge to select the radius $r$ \cite{morgan2021adaptive}. For example, in an image classification task we could assert a priori that changing a few pixels only can never turn a picture of an elephant into a picture of a car. If we use Hamming distance as metric on the pictures, stating what we mean with \emph{a few pixels} gives us the value $r$ such that any adversarial example demonstrates a fault in the classifier. Another example by \cite{Ruan2018} finds the upper and lower bounds of the input space via an optimisation procedure, following that DNNs are Lipschitz continuous functions and all values between these bounds are reachable.

Here we present a different route: rather than starting with a preconceived notion of small disturbance and then looking for adversarial examples, we can instead compute how much we would need to disturb a given point. In our results, we use the spaces $\mathbb{R}_<$ and $\mathbb{R}_>$, where reals are represented as limits of increasing respectively decreasing rational sequences; as well as the variant $\overline{\mathbb{R}}_<$ where $+\infty$ is included.

\begin{proposition}
\label{prop:radiuslower}
Let $\mathbf{X}$ be an effectively locally compact computable metric space with metric $d$ such that all closed balls are compact. Then the map $$(x,f) \mapsto \sup \{ r \in \mathbb{R} \mid \exists i \in \mathbf{k} \ \forall y \in \overline{B}(x,r) \quad f(y) = i\} : \mathbf{X} \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \overline{\mathbb{R}}_<$$ is computable.
\end{proposition}

\begin{proof}%[Proof of Proposition \ref{prop:radiuslower}]
As $\mathbf{k}$ is overt and $\overline{B}(x,r)$ is uniformly compact by assumption (via \cite[Proposition 10]{pauly-locallycompact}), it follows that $$ \exists i \in \mathbf{k} \ \forall y \in \overline{B}(x,r) \quad f(y) = i$$ is semidecidable in the parameters $x$, $f$ and $r$, thus $$\{ r \in \mathbb{R} \mid \exists i \in \mathbf{k} \ \forall y \in \overline{B}(x,r) \quad f(y) = i\} \in \mathcal{O}(\mathbb{R})$$ is an open set computable from $x$ and $f$. The supremum of a set $U \in \mathcal{O}(\mathbb{R})$ is computable as an element of $\overline{R}_<$ as shown in \cite[Section 10]{pauly-synthetic}.
\end{proof}

\begin{proposition}
\label{prop:radiusupper}
Let $(\mathbf{X},d)$ be a computable metric space. Then the map $$(x,f) \mapsto \inf \{r \in \mathbb{R}^{\geq 0} \mid \exists y \in B(x,r) \ \bot \neq f(x) \neq f(y) \neq \bot\} : \mathbf{X} \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbb{R}_>$$ is computable.
\end{proposition}

\begin{proof}%[Proof of Proposition \ref{prop:radiusupper}]
It makes no difference whether we use $\exists y \in B(x,r)$ or $\exists y \in \mathrm{cl} B(x,r)$ here. In a computable metric space $\mathbf{X}$, we can compute $\mathrm{cl} B(x,r) \in \mathcal{V}(\mathbf{X})$ from $x$ and $r$. In addition, we observe that $\bot \neq f(x) \neq f(y) \neq \bot$ is semidecidable in $x,y$ and $f$. Alltogether, this gives us access to $$\{r \in \mathbb{R}^{\geq 0} \mid \exists y \in B(x,r) \ \bot \neq f(x) \neq f(y) \neq \bot\} \in \mathcal{O}(\mathbb{R}^{\geq 0})\},$$ and we can then compute the infimum in $\mathbb{R}_>$. Since we consider the infimum in the space $\mathbb{R}^{\geq 0}$, it is at least $0$. 
\end{proof}

\begin{corollary}
Let $\mathbf{X}$ be an effectively locally compact computable metric space with metric $d$ such that all closed balls are compact. The map $\mathrm{OptimalRadius} : \subseteq \mathbf{X} \times \mathcal{C}(\mathbf{X},\mathbf{k}_\bot) \to \mathbb{R}$ defined by $(x,f) \in \dom(\mathrm{OptimalRadius})$ iff $f(x) \neq \bot$, $\exists y \ \bot \neq f(y) \neq f(x)$ and $\forall r, \varepsilon > 0 \ \exists z \in B(x,r+\varepsilon) \setminus B(x,r) \ f(z) \neq \bot$; and by $$\mathrm{OptimalRadius}(x,r) = \sup \{ r \in \mathbb{R} \mid \exists i \in \mathbf{k} \ \forall y \in \overline{B}(x,r) \quad f(y) = i\} = \inf \{r \mid \exists y \in B(x,r) \ \bot \neq f(x) \neq f(y) \neq \bot\}$$ is computable.
\end{corollary}



\section{Learners and their robustness}
\label{sec:robust_learners}

Let us now consider the process of training the classifier. To keep matters simple, we will not adopt a dynamic view, but rather model this as a one-step process. We also only consider supervised learning. This leads us to the conception of a learner as a map from finite sequences of labelled points to classifiers:

\begin{definition}
A \emph{learner} is a (computable) procedure that takes as an input a tuple $((x_0,n_0),\ldots,(x_\ell,n_\ell)) \in (\mathbf{X} \times \mathbf{k})^*$ and outputs a classifier $f \in \mathcal{C}(\mathbf{X},\mathbf{k}_\bot)$. The collection of all learners is the space $\mathcal{C}((\mathbf{X} \times \mathbf{k})^*,\mathcal{C}(\mathbf{X},\mathbf{k}_\bot))$.
\end{definition}


\paragraph*{Faithful reproduction of the training labels}
We do not prescribe any particular relation between the training data and the behaviour of the resulting classifier. It could seem reasonable to ask that a learner $L$ faithfully reproduces the training data, i.e.~satisfies $L((x_i,n_i)_{i \leq \ell})(x_m) = n_m$, but such a criterion is impossible to satisfy, as our notion of training data does not rule out having multiple occurrences of the same sample point with different labels. It would also not match applications, as it often is desirable that a model can disregard parts of its training data as being plausibly faulty.

\begin{proposition}
\label{prop:doesdeviate}
Let $\mathbf{X}$ be computably overt and computably Hausdorff. The operation \[\mathrm{doesDeviate} : \mathcal{C}((\mathbf{X} \times \mathbf{k})^*,\mathcal{C}(\mathbf{X},\mathbf{k}_\bot)) \to \mathbb{S}\] returning $\operatorname{true}$ on input $L$ iff there is some input $(x_i,n_i)_{i \leq \ell} \in (\mathbf{X} \times \mathbf{k})^*$ with $x_i \neq x_j$ for $i \neq j$, and some $m \leq \ell$ such that $L((x_i,n_i)_{i \leq \ell})(x_m) \in \mathbf{k} \setminus \{n_m\}$ is computable.
%\begin{enumerate}
%\item
%\item Let $\mathbf{X}$ be computably compact. The operation \[\mathrm{doesNotDeviate} : \mathbb{N} \times \mathcal{C}((\mathbf{X} \times \mathbf{k})^*,\mathcal{C}(\mathbf{X},\mathbf{k}_\bot)) \to \mathbb{S}\] returning $\operatorname{true}$ on input $\ell,L$ iff for every input  $(x_i,n_i)_{i \leq \ell} \in (\mathbf{X} \times \mathbf{k})^{\ell+1}$ and every $m \leq \ell$ it holds that  $L((x_i,n_i)_{i \leq \ell})(x_m) = n_m$ is computable.
%\item Let $\mathbf{X}$ be computably overt and computably compact. The  operation \[\mathrm{deviates} : \mathbb{N} \times \mathcal{C}((\mathbf{X} \times \mathbf{k})^*,\mathcal{C}(\mathbf{X},\mathbf{k}_\bot)) \to \mathbf{2}_\bot\] returning $0$ on input $\ell,L$ iff for every input  $(x_i,n_i)_{i \leq \ell} \in (\mathbf{X} \times \mathbf{k})^{\ell+1}$ and every $m \leq \ell$ it holds that  $L((x_i,n_i)_{i \leq \ell})(x_m) = n_m$, and returning $1$ iff  there is some input $(x_i,n_i)_{i \leq \ell} \in (\mathbf{X} \times \mathbf{k})^{\ell+1}$ and some $m \leq \ell$ such that $L((x_i,n_i)_{i \leq \ell})(x_m) \in \mathbf{k} \setminus \{n_m\}$; is computable.
%\end{enumerate}
\end{proposition}
\begin{proof}%[Proof of Proposition \ref{prop:doesdeviate}]
If $\mathbf{X}$ is overt, then so is $(\mathbf{X} \times \mathbf{k})^*$. Since the condition for answering  $\operatorname{true}$ is defined by existentially quantifying a semidecidable condition over an overt set, it is semidecidable itself.
\end{proof}

\paragraph*{Robustness under additional training data}
Generally, our goal will not be so much to algorithmically verify properties of learners for arbitrary training data, but rather be interested in the behaviour of the learner on the given training data and hypothetical small additions to it. One question here would be to ask how robust a classifier is under small additions to the training data. A basic version of this would be: 

\begin{proposition}
\label{prop:robustpoint}
Let $\mathbf{X}$ be computably compact and computably overt. The map \[\operatorname{robustPoint} : \mathbf{X} \times (\mathbf{X} \times \mathbf{k})^* \times \mathcal{C}((\mathbf{X} \times \mathbf{k})^*,\mathcal{C}(\mathbf{X},\mathbf{k}_\bot)) \to \mathbf{2}_\bot\] answering $1$ on input $x$, $(x_i,n_i)_{i \leq \ell}$ and $L$ iff \[\forall x_{\ell+1} \in \mathbf{X} \ \forall n_{\ell+1} \in \mathbf{k} \quad L((x_i,n_i)_{i \leq \ell})(x) = L((x_i,n_i)_{i \leq \ell + 1})(x) \in \mathbf{k}\] and answering $0$ iff \[\exists x_{\ell+1} \in \mathbf{X} \ \exists n_{\ell+1} \in \mathbf{k} \quad \bot \neq L((x_i,n_i)_{i \leq \ell})(x) \neq L((x_i,n_i)_{i \leq \ell + 1})(x) \neq \bot\]  is computable.
\end{proposition}
\begin{proof}%[Proof of Proposition \ref{prop:robustpoint}]
Since $\mathbf{X}$ and $\mathbf{k}$ are both compact and overt, both universal and existential quantification preserves semidecidable predicates. Since $\mathbf{k}$ is discrete and Hausdorff, the stem of our predicate definitions for answering $0$ and $1$ are both semidecidable.
\end{proof}

The map $\operatorname{robustPoint}$ can confirm for us that a particular classification would remain, even if we add another point to the sample; or that a labeling can change upon extending the training data a little bit. The very same reasoning would also work if we would consider adding two, three or $j$ additional samples. We can also lift this to compact and overt regions, and then ask whether either all values will stay unchanged, or whether there is one that does. As usual, if the point (or any point in the region) is assigned $\bot$, $\operatorname{robustPoint}$ will output $\bot$, too.

\paragraph*{Sparsity of training data}
Allowing arbitrary additional training data as above might not be too suitable -- for example, if we add the relevant query point together with another label to the training data, it would not be particularly surprising if the new classifier follows the new data. If we bring in a metric structure, we can exclude new training data which is too close to the given point.

\begin{definition}
\label{def:sparse}
Fix a learner $L : (\mathbf{X} \times \mathbf{k})^* \to \mathcal{C}(\mathbf{X},\mathbf{k}_\bot)$, some $N \in \mathbb{N}$ and $\varepsilon > 0$. We say that $(x_i,n_i)_{i \leq \ell}$ is \emph{sparse} at $x \in \mathbf{X}$, if there are $(y_i,m_i)_{i \leq j}$ and $(y'_i,m'_i)_{i \leq j'}$ such that $\ell + N \geq j,j' \geq \ell$, $y_i = y'_i = x_i$ and $m_i = m'_i = n_i$ for $i \leq \ell$, and $d(y_i,x),d(y'_i,x) > \varepsilon$ for $i > \ell$ satisfying $\bot \neq L((y_i,m_i)_{i \leq j})(x) \neq L((y'_i,m'_i)_{i \leq j'})(x) \neq \bot$. 

We say that $(x_i,n_i)_{i \leq \ell}$ is \emph{dense} at $x \in \mathbf{X}$ if for all $(y_i,m_i)_{i \leq j}$ and $(y'_i,m'_i)_{i \leq j'}$ such that $\ell + N \geq j,j' \geq \ell$, $y_i = y'_i = x_i$ and $m_i = m'_i = n_i$ for $i \leq \ell$, and $d(y_i,x),d(y'_i,x) \geq \varepsilon$ for $i > \ell$  it holds that $L((y_i,m_i)_{i \leq j})(x) = L((y'_i,m'_i)_{i \leq j'})(x) \neq \bot$.
\end{definition}

To put it in words: Training data is dense at a point whose label it determines, even if we add up to $N$ additional points to the training data, which have to be at least $\varepsilon$ away from that point. Conversely, at a sparse point we can achieve different labels by such an augmentation of the training data. If we have chosen the parameters $N$ and $\varepsilon$ well, then we can conclude that based on the training data we can make reasonable assertions about the dense query points, but unless we have some additional external knowledge of the true distribution of labels, we cannot draw reliable conclusion about the sparse query points. We concede that it would make sense to include points under \emph{sparse} where the classifiers will always output $\bot$ even if we enhance the training data, but this would destroy any hope of nice algorithmic properties.

\begin{theorem}
\label{theo:sparsity}
Let $\mathbf{X}$ be a computably compact computable metric space. The operation \[\operatorname{SprsOrDns} : \mathcal{C}((\mathbf{X} \times \mathbf{k})^*,\mathcal{C}(\mathbf{X},\mathbf{k}_\bot)) \times \mathbb{N} \times \mathbb{R}_+ \times (\mathbf{X} \times \mathbf{k})^* \times \mathbf{X} \to \mathbf{2}_\bot\] answering $0$ on input $L,N,\varepsilon$, $(x_i,n_i)_{i \leq \ell}$ and $x$ iff $(x_i,n_i)_{i \leq \ell}$ is sparse at $x$, and answers $1$ if $(x_i,n_i)_{i \leq \ell}$ is dense at $x$ is computable.
\end{theorem}

\begin{proof}%[Proof of Theorem \ref{theo:sparsity}]
Essentially, the claim is that both being sparse and being dense is a semidecidable property. We show this by examining Definition \ref{def:sparse} and in particular its quantification structure. 

We note that $\bot \neq L((y_i,m_i)_{i \leq j})(x) \neq L((y'_i,m'_i)_{i \leq j'})(x) \neq \bot$ is a semidecidable property. We are then adding existential quantifications over $\operatorname{cl} \{y \in \mathbf{X} \mid d(x,y) > \varepsilon\}$, and this set is always available to us as an overt set when working in a computable metric space. Thus, being sparse is semidecidable.

For density, note that $L((y_i,m_i)_{i \leq j})(x) = L((y'_i,m'_i)_{i \leq j'})(x) \neq \bot$ is again semidecidable. This time we add universal quantification over $\{y \in \mathbf{X} \mid d(x,y) \geq \varepsilon\}$. This set is available to us as a compact set thanks to the demand that $\mathbf{X}$ be a computably compact computable metric space.

\end{proof}

\paragraph*{Sparsity in the ML literature}

In the context of this study, sparsity refers to the \emph{measurement} of the input representation, and is not the representation itself (i.e.~sparse matrix forms). Indeed, we may evaluate the \emph{sparsity} of sampling from any underlying manifold that may have further implications on how ML algorithms learn and create classifications in these particular regions of space. In sparse regions of a manifold, there will be less direct information for the ML algorithm to learn from. Regions such as these then may harbour many more adversarial examples \cite{Lindenbaum2018GeometryGeneration}. Such methods of measurement may be useful for the creation of adversarial examples (or at least the search for adversarial examples) by manipulating the region size of our closed ball $\overline{B}(x, r)$ relative to the local information of sparsity of sampling of the manifold space. 

One typical measurement for sparsity of sampling within data with kernel density estimation (KDE), commonly with a Gaussian kernel $\text{exp} \left( \frac{|| \mathbf{x} - \mathbf{x}' ||^2}{2\sigma^2} \right)\,$  where $\sigma$ controls the width of the kernel. For small values of $\sigma$ the similarity matrix forms an identity matrix, or a matrix of all-ones for large values. Thus choosing an optimal bandwidth parameter is problem specific, and may require tuning to appropriately quantify the sparsity of the manifold space. However, some research has been conducted to alleviate this issue. Two methods are described within \cite{Lindenbaum2018} for the setting of bandwidth parameters in KDE that are locally adaptive to the regions, thereby optimally characterising the sparsity measurement with respect to each sample of the manifold space.

\section{Musings on the class of spaces to consider}
\label{sec:musings}

\label{sec:whatspaces}
In this section we focus on what demands we should make on the space $\mathbf{X}$. While we have seen that many operations can be defined and shown to be computable in a rather general setting, we will make the case here that effectively locally-compact computable metric spaces form a particular convenient setting. They have enough structure to make (almost) everything we have considered make sense and computable, are sufficiently general to contain conceivable settings from applications.

\paragraph*{Impact of names}
Usually, points $x \in \mathbf{X}$ will have multiple names $p \in \delta_\mathbf{X}^{-1}(\{x\})$. For example, Brattka and Hertling have shown that no admissible representation of the real numbers makes all reals have only finitely many names \cite{hertling7}. As a consequence, an algorithm can fail to compute an extensional function by giving different answers when reading different names of the input.

A desideratum would be that we could modify a given algorithm such that the modified algorithm gives a certain answer on an input only if the original algorithm gives that very same answer on all other names of the same input. Being able to do this means being able to universally quantify over names, which in turn means that we need an effectively fiber-compact representation (which is defined as letting us compute the compact set of names of a point from that point). It was shown in \cite{pauly-kihara-arxiv} that a space has an effectively fiber-compact computably admissible representation iff it is computably metrizable.

We could be less cautious, and instead decide to alter our algorithms such that producing a colour when reading some name of a point should lead to all names being mapped to that colour. This runs into the issue that there is no computable way to handle tie-breaks between different names mapping to different colours. We can handle this by adjoining to $\mathbf{k}_\bot$ a further element $\top$, where any sequence ending in $1^\omega$ denotes $\top$. This allows a computation to change from any colour to $\top$. While $\bot$ denotes \emph{no information}, $\top$ would denote \emph{contradictory information}. We still need to be able to existentially quantify over names, meaning that we need effectively fiber-overt representations. Having an effectively fiber-overt representation is equivalent to being effectively countably-based.


\paragraph*{Complexity theory}
While we have consciously avoided complexity issues in the present paper, we cannot deny that efficiency of algorithms will be a crucial issue for practical relevance. There is a complexity theory for computable analysis (see in particular Kawamura and Cook \cite{kawamura}, or for a generalisation \cite{eike2}), however, here the run-times of algorithms are no longer described by functions $\tau : \mathbb{N} \to \mathbb{N}$, but rather by second-order objects (e.g.~functions $\Upsilon : \mathbb{N}^\mathbb{N} \times \mathbb{N} \to \mathbb{N}$). However, this price is not due immediately when considering algorithms on non-discrete domains. Schr\"oder \cite{schroder6} has characterized the class of spaces admitting a complexity theory such that runtimes are still functions $\tau : \mathbb{N} \to \mathbb{N}$, these are the coPolish spaces\footnote{The term \emph{coPolish} was coined only after the publication of \cite{schroder6}. Its earliest occurrence in the publication record seems to be \cite{paulydebrecht4}.}. A separable metric space is locally compact iff it is coPolish, and a countably-based coPolish space is already metrizable. As such, we see that with the backdrop of our other desiderata, being locally compact and admitting a nice complexity theory are intricately linked.

\paragraph*{Hausdorffness}
Requiring spaces to be Hausdorff is certainly a popular requirement, even in the absence of necessity. In theoretical computer science we do, however, encounter many non-Hausdorff topological spaces (e.g.~\cite{goubault}). Most of the spaces we have referred to in this work are non-Hausdorff, such as the spaces $\mathbf{k}_\bot$. We do desire the space $\mathbf{X}$ to be Hausdorff, though. It is worth pointing out that this requirement can  be formulated in terms of discriminating classifiers:

\begin{proposition}
\label{prop:separatingfunctions}
The following are equivalent for a  countably-based  represented space $\mathbf{X}$:
\begin{enumerate}
\item $\mathbf{X}$ is Hausdorff.
\item For any $x, y \in \mathbf{X}$ with $x \neq y$ there exists a computable classifier $f_{xy} : \mathbf{X} \to \mathbf{2}_\bot$ with $f(x) = 0$ and $f(y) = 1$.
\end{enumerate}
\end{proposition}
\begin{proof}
If $\mathbf{X}$ is countably-based and Hausdorff, then distinct points are separated by disjoint basic open sets. If $U$ and $V$ are disjoint basic open sets, then $f_{UV} : \mathbf{X} \to \mathbf{2}_\bot$ with $f^{-1}(0) = U$ and $f^{-1}(1) = V$ is well-defined and computable.

A separating classifier $f_{xy} : \mathbf{X} \to \mathbf{2}_\bot$ immediately yields disjoint open sets separating $x$ and $y$, and thus witnesses the Hausdorff condition.
\end{proof}


\paragraph*{Schr\"oder's effective metrization theorem}
An effectively locally-compact, effectively countably-based computably Hausdorff is computably metrizable by Schr\"order's effective metrization theorem \cite{grubba3}. If we are willing to work relative to an oracle, a locally-compact represented space is already countably based by \cite[Corollary 6.11]{escardo7}. Thus, if we are asking for local compactness anyway to be able to verify that a classifier is locally constant (i.e.~has no adversarial examples in a neighbourhood of a given point), we can just as well as for the existence of a metric and make available to us a clear definition of adversarial examples. Of course, a metrization theorem does not tell us which metric to use. 

\paragraph*{Compactness}
Some of our results (Proposition~\ref{prop:robustpoint}, Theorem~\ref{theo:sparsity}) even require compactness, not just local compactness. While we could demand that the spaces we handle are compact, this could also be solved by restricting the universal quantification to a suitable compact neighbourhood of all the points concerned. We have not included this in our formal statements to avoid near-unreadable complexity.

\section{Future work}


The results in this paper assert that under certain conditions on the space, certain maps are computable. Dropping these conditions, or considering maps providing more information instead, generally will lead to non-computability. We leave the provision of counterexamples, as well as potentially a classification of \emph{how}\footnote{This would be answered using the formalism of Weihrauch degrees\cite{pauly-handbook}.} non-computable these maps are to future work. The notion of a \emph{maximal partial algorithm} recently proposed by Neumann \cite{eike-neumann2} also seems a promising approach to prove optimality of our results.

There is a trade-off between the robustness of a classifier and its `accuracy'. It seems possible to develop a computable quantitative notion of robustness on top of our function locallyConstant, which could then be used as part of the training process in a learner. This could be a next step to adversarial robustness \cite{Carlini2017,Goodfellow-et-al-2016}.

Rather than just asking questions about particular given classifiers or learners, we could start with a preconception regarding what classifier we would want to obtain for given training data. Natural algorithmic questions then are whether there is a learner in the first place that is guaranteed to meet our criteria for the classifiers, and whether we can compute such a learner from the criteria.


%% Potential future work from the perspective of the ML community:
%%  - Using locallyConstant as part of the training process for a learner as different distances to $x$. There is a trade-off between the robustness of the classifier and 'accuracy'. This seems like a natural evolution to adversarial robustness -- how can the measurement of robustness improve the learnt classifier?
%%  - Is it possible to describe a 'minimal' set of training data that is needed to reproduce the classifier with maximal performance.


% (i) what would we add, if we had 3 more month?
% (ii) what does it mean to the field?
% (iii) what can other fields learn from us?
% - and then leave out the categories, where I lack ideas

% to (i): 
% one might want to come up with a longer list of properties on ML classifiers
% one might want to provide some efficient algorithms for specific ML techniques for our theorems

% to (ii):
% when looking over our results, we provide only positive answers - provided the spaces and chosen sets are “nice”, we can compute something. what about negative answers? are there properties of classifiers that are not computable?

% to (iii):
% the abstract view from computable analysis allowed us to take a good step back and see the principles


\bibliography{references}

\end{document}

perspective_article.bbl
\begin{thebibliography}{10}

\bibitem{pmlr-v80-athalye18b}
Anish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok.
\newblock Synthesizing robust adversarial examples.
\newblock In {\em Proceedings of the 35th International Conference on Machine
  Learning}, volume~80, pages 284--293, 2018.

\bibitem{pauly-handbook}
Vasco Brattka, Guido Gherardi, and Arno Pauly.
\newblock Weihrauch complexity in computable analysis.
\newblock arXiv 1707.03202, 2017.

\bibitem{hertling7}
Vasco Brattka and Peter Hertling.
\newblock Topological properties of real number representations.
\newblock {\em Theoretical Computer Science}, 284:241--257, 2002.

\bibitem{brattkaintro}
Vasco Brattka, Peter Hertling, and Klaus Weihrauch.
\newblock A tutorial on computable analysis.
\newblock In Barry Cooper, Benedikt L\"owe, and Andrea Sorbi, editors, {\em New
  Computational Paradigms: Changing Conceptions of What is Computable}, pages
  425--491. Springer, 2008.

\bibitem{Carlini2017}
Nicholas Carlini and David Wagner.
\newblock {Towards Evaluating the Robustness of Neural Networks}.
\newblock In {\em IEEE Symposium on Security and Privacy}, pages 39--57, 2017.

\bibitem{paulydebrecht3-csl}
Matthew de~Brecht and Arno Pauly.
\newblock {Noetherian Quasi-Polish spaces}.
\newblock In {\em 26th EACSL Annual Conference on Computer Science Logic (CSL
  2017)}, volume~82 of {\em LIPIcs}, pages 16:1--16:17, 2017.

\bibitem{paulydebrecht4}
Matthew de~Brecht, Arno Pauly, and Matthias Schr\"oder.
\newblock Overt choice.
\newblock {\em Computability}, 2020.
\newblock available at https://arxiv.org/abs/1902.05926.

\bibitem{escardo7}
Mart\'in Escard\'o, Jimmie Lawson, and Alex Simpson.
\newblock Comparing cartesian closed categories of (core) compactly generated
  spaces.
\newblock {\em Topology and its Applications}, 143(1):105--145, 2004.

\bibitem{fell}
J.~M.~G. Fell.
\newblock A {H}ausdorff topology for the closed subsets of a locally compact
  non-{H}ausdorff space.
\newblock {\em Proceedings of the American Mathematical Society}, 13:472--476,
  1962.

\bibitem{Goodfellow-et-al-2016}
Ian Goodfellow, Yoshua Bengio, and Aaron Courville.
\newblock {\em {Deep Learning}}.
\newblock MIT Press, 2016.

\bibitem{Goodfellow2015}
Ian Goodfellow, Jonathon Shlens, and Szegedy Christian.
\newblock {Explaining and Harnessing Adversarial Examples}.
\newblock {\em arXiv preprint arXiv:1412.6572}, 2014.

\bibitem{goubault}
Jean Goubault-Larrecq.
\newblock {\em Non-Hausdorff Topology and Domain Theory}.
\newblock New Mathematical Monographs. Cambridge University Press, 2013.

\bibitem{grubba3}
Tanja Grubba, Matthias Schr\"oder, and Klaus Weihrauch.
\newblock Computable metrization.
\newblock {\em Mathematical Logic Quarterly}, 53(4-5):381--395, 2007.

\bibitem{hearst1998support}
Marti~A. Hearst, Susan~T Dumais, Edgar Osuna, John Platt, and Bernhard
  Scholkopf.
\newblock Support vector machines.
\newblock {\em IEEE Intelligent Systems and their applications}, 13(4):18--28,
  1998.

\bibitem{Huang2016}
Xiaowei Huang, Marta Kwiatkowska, Sen Wang, and Min Wu.
\newblock {Safety Verification of Deep Neural Networks}.
\newblock In {\em International Conference on Computer Aided Verification},
  pages 3--29, 2017.

\bibitem{kawamura}
Akitoshi Kawamura and Stephen Cook.
\newblock Complexity theory for operators in analysis.
\newblock {\em ACM Transactions on Computation Theory}, 4(2), 2012.

\bibitem{pauly-kihara-arxiv}
Takayuki Kihara and Arno Pauly.
\newblock Point degree spectra of represented spaces.
\newblock {\em arXiv preprint arXiv:1405.6866}, 2014.

\bibitem{Lindenbaum2018GeometryGeneration}
Ofir Lindenbaum, Jay~S. Stanley, Guy Wolf, and Smita Krishnaswamy.
\newblock {Geometry based data generation}.
\newblock In {\em Advances in Neural Information Processing Systems}, pages
  1400--1411, 2018.

\bibitem{Lindenbaum2018}
Ofir Lindenbaum, Jay~S. Stanley, Guy Wolf, and Smita Krishnaswamy.
\newblock {Geometry based data generation}.
\newblock In {\em Advances in Neural Information Processing Systems}, pages
  1400----1411, 2018.

\bibitem{moosavi2016deepfool}
Seyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, and Pascal Frossard.
\newblock {Deepfool: A Simple and Accurate Method to Fool Deep Neural
  Networks}.
\newblock In {\em IEEE conference on computer vision and pattern recognition},
  pages 2574--2582, 2016.

\bibitem{morgan2021adaptive}
Jay Morgan, Adeline Paiement, Arno Pauly, and Monika Seisenberger.
\newblock Adaptive neighbourhoods for the discovery of adversarial examples.
\newblock {\em arXiv preprint arXiv:2101.09108}, 2021.

\bibitem{narayanan2010sample}
Hariharan Narayanan and Sanjoy Mitter.
\newblock Sample complexity of testing the manifold hypothesis.
\newblock In {\em Advances in neural information processing systems}, pages
  1786--1794, 2010.

\bibitem{eike-neumann2}
Eike Neumann.
\newblock Decision problems for linear recurrences involving arbitrary real
  numbers.
\newblock arXiv 2008.00583, 2020.
\newblock URL: \url{https://arxiv.org/abs/2008.00583}.

\bibitem{eike2}
Eike Neumann and Florian Steinberg.
\newblock Parametrised second-order complexity theory with applications to the
  study of interval computation.
\newblock {\em Theoretical Computer Science}, 806:281 -- 304, 2020.

\bibitem{Papernot2016a}
Nicolas Papernot, Patrick Mcdaniel, Somesh Jha, Matt Fredrikson, Z.~Berkay
  Celik, and Ananthram Swami.
\newblock {The Limitations of Deep Learning in Adversarial Settings}.
\newblock In {\em IEEE European Symposium on Security and Privacy}, pages
  372----387, 2016.

\bibitem{pauly-synthetic}
Arno Pauly.
\newblock On the topological aspects of the theory of represented spaces.
\newblock {\em Computability}, 5(2):159--180, 2016.
\newblock URL: \url{http://arxiv.org/abs/1204.3763}, \href
  {https://doi.org/10.3233/COM-150049} {\path{doi:10.3233/COM-150049}}.

\bibitem{pauly-locallycompact}
Arno Pauly.
\newblock Effective local compactness and the hyperspace of located sets.
\newblock {\em arXiv preprint arXiv:1903.05490}, 2019.

\bibitem{Ruan2018}
Wenjie Ruan, Xiaowei Huang, and Marta Kwiatkowska.
\newblock Reachability analysis of deep neural networks with provable
  guarantees.
\newblock In {\em Proceedings of the Twenty-Seventh International Joint
  Conference on Artificial Intelligence}, pages 2651--2659, 7 2018.

\bibitem{schroder6}
Matthias Schr\"oder.
\newblock Spaces allowing type-2 complexity theory revisited.
\newblock {\em Mathematical Logic Quarterly}, 50(4/5):443--459, 2004.

\bibitem{szegedy2013intriguing}
Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan,
  Ian Goodfellow, and Rob Fergus.
\newblock {Intriguing Properties of Neural Networks}.
\newblock {\em arXiv preprint arXiv:1312.6199}, 2013.

\bibitem{taylor}
Paul Taylor.
\newblock A lambda calculus for real analysis.
\newblock {\em Journal of Logic \& Analysis}, 2(5):1--115, 2010.

\bibitem{weihrauchd}
Klaus Weihrauch.
\newblock {\em Computable Analysis}.
\newblock Springer-Verlag, 2000.

\bibitem{Wicker2018a}
Matthew Wicker, Xiaowei Huang, and Marta Kwiatkowska.
\newblock Feature-guided black-box safety testing of deep neural networks.
\newblock In {\em International Conference on Tools and Algorithms for the
  Construction and Analysis of Systems}, pages 408--426. Springer, 2018.

\end{thebibliography}

references.bib
@inproceedings{narayanan2010sample,
  title={Sample complexity of testing the manifold hypothesis},
  author={Narayanan, Hariharan and Mitter, Sanjoy},
  booktitle={Advances in neural information processing systems},
  pages={1786--1794},
  year={2010}
}


@InProceedings{paulydebrecht3-csl,
  author ={de Brecht, Matthew and Arno Pauly},
  title ={{Noetherian Quasi-Polish spaces}},
  booktitle ={26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
  pages ={16:1--16:17},
  series ={LIPIcs},
  year ={2017},
  volume ={82},
  }


@Article{hertling7,
  Title                    = {Topological properties of real number representations},
  Author                   = {Vasco Brattka and Peter Hertling},
  Journal                  = {Theoretical Computer Science},
  Year                     = {2002},
  Pages                    = {241--257},
  Volume                   = {284}
}

@InProceedings{brattkaintro,
  Title                    = {A tutorial on computable analysis},
  Author                   = {Vasco Brattka and Peter Hertling and Klaus Weihrauch},
  Booktitle                = {New Computational Paradigms: Changing Conceptions of What is Computable},
  Year                     = {2008},
  Editor                   = {Barry Cooper and Benedikt L\"owe and Andrea Sorbi},
  Pages                    = {425--491},
  Publisher                = {Springer}
}

@Article{paulydebrecht4,
  Title                    = {Overt choice},
  Author                   = {Matthew de Brecht and Arno Pauly and Matthias Schr\"oder},
  Journal                  = {Computability},
  Year                     = {2020},
  Note                     = {available at https://arxiv.org/abs/1902.05926},
}

@Article{escardo7,
  Title                    = {Comparing Cartesian closed categories of (core) compactly generated spaces},
  Author                   = {Mart\'in Escard\'o and Jimmie Lawson and Alex Simpson},
  Journal                  = {Topology and its Applications},
  Year                     = {2004},
  Number                   = {1},
  Pages                    = {105--145},
  Volume                   = {143},
}

@Article{fell,
  Title                    = {A {H}ausdorff topology for the closed subsets of a locally
 compact non-{H}ausdorff space},
  Author                   = {Fell, J. M. G.},
  Journal                  = {Proceedings of the American Mathematical Society},
  Year                     = {1962},
  Pages                    = {472--476},
  Volume                   = {13}
}

@Book{goubault,
  Title                    = {Non-Hausdorff Topology and Domain Theory},
  Author                   = {Jean Goubault-Larrecq},
  Publisher                = {Cambridge University Press},
  Year                     = {2013},
  Series                   = {New Mathematical Monographs}
}

@Article{grubba3,
  Title                    = {Computable Metrization},
  Author                   = {Tanja Grubba and Matthias Schr\"oder and Klaus Weihrauch},
  Journal                  = {Mathematical Logic Quarterly},
  Year                     = {2007},
  Number                   = {4-5},
  Pages                    = {381--395},
  Volume                   = {53}
}

@Article{kawamura,
  Title                    = {Complexity theory for operators in analysis},
  Author                   = {Akitoshi Kawamura and Stephen Cook},
  Journal                  = {ACM Transactions on Computation Theory},
  Year                     = {2012},
  Number                   = {2},
  Volume                   = {4}
}

@Article{pauly-kihara-arxiv,
  Title                    = {Point degree spectra of represented spaces},

  Author                   = {Takayuki Kihara and Arno Pauly},
  Year                     = {2014},
  journal = {arXiv preprint arXiv:1405.6866},
}

@Article{eike2,
  Title                    = {Parametrised second-order complexity theory with applications to the study of interval computation},
  Author                   = {Eike Neumann and Florian Steinberg},
  Journal                  = {Theoretical Computer Science},
  Year                     = {2020},
  Pages                    = {281 - 304},
  Volume                   = {806},
}

@Article{pauly-locallycompact,
  Title                    = {Effective local compactness and the hyperspace of located sets},

  Author                   = {Arno Pauly},
  HowPublished             = {arXiv 1903.05490},
  Year                     = {2019},
  journal = {arXiv preprint arXiv:1903.05490},
}

@misc{eike-neumann2,
  author    = {Eike Neumann},
  title     = {Decision problems for linear recurrences involving arbitrary real
               numbers},
  howpublished = {arXiv 2008.00583},
  year      = {2020},
  url       = {https://arxiv.org/abs/2008.00583},
}

@MISC{pauly-handbook,
  author =       {Vasco Brattka and Guido Gherardi and Arno Pauly},
  title =        {Weihrauch Complexity in Computable Analysis},
  howpublished = {arXiv 1707.03202},
  year =         {2017},
}

@Article{pauly-synthetic,
  Title                    = {On the topological aspects of the theory of represented spaces},
  Author                   = {Arno Pauly},
  Journal                  = {Computability},
  Year                     = {2016},
  Number                   = {2},
  Pages                    = {159--180},
  Volume                   = {5},
  Doi                      = {10.3233/COM-150049},
  Url                      = {http://arxiv.org/abs/1204.3763}
}

@Article{schroder,
  Title                    = {Extended admissibility},
  Author                   = {Matthias Schr\"{o}der},
  Journal                  = {Theoretical Computer Science},
  Year                     = {2002},
  Number                   = {2},
  Pages                    = {519-538},
  Volume                   = {284},

  Doi                      = {10.1016/S0304-3975(01)00109-8}
}

@Article{schroder6,
  Title                    = {Spaces allowing Type-2 Complexity Theory revisited},
  Author                   = {Matthias Schr\"oder},
  Journal                  = {Mathematical Logic Quarterly},
  Year                     = {2004},
  Number                   = {4/5},
  Pages                    = {443--459},
  Volume                   = {50}
}

@Article{taylor,
  Title                    = {A lambda calculus for real analysis},
  Author                   = {Paul Taylor},
  Journal                  = {Journal of Logic \& Analysis},
  Year                     = {2010},
  Number                   = {5},
  Pages                    = {1--115},
  Volume                   = {2}
}

@Book{weihrauchd,
  Title                    = {Computable Analysis},
  Author                   = {Klaus Weihrauch},
  Publisher                = {Springer-Verlag},
  Year                     = {2000}
}


@techreport{Tsipras,
abstract = {We show that there may exist an inherent tension between the goal of adversarial robustness and that of standard generalization. Specifically, training robust models may not only be more resource-consuming, but
also lead to a reduction of standard accuracy. We demonstrate that this trade-off between the standard accuracy of a model and its robustness to adversarial perturbations provably exists in a fairly simple and natural
setting. These findings also corroborate a similar phenomenon observed empirically in more complex settings. Further, we argue that this phenomenon is a consequence of robust classifiers learning fundamentally different
feature representations than standard classifiers. These differences, in particular, seem to result in unexpected benefits: the representations learned by robust models tend to align better with salient data characteristics
and human perception.},
archivePrefix = {arXiv},
arxivId = {1805.12152v5},
author = {Tsipras, Dimitris and Santurkar, Shibani and Engstrom, Logan and Mit, Alexander Turner and Mit, Aleksander M ˛ Adry},
eprint = {1805.12152v5},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Tsipras et al. - Unknown - Robustness May Be at Odds with Accuracy.pdf:pdf},
title = {{Robustness May Be at Odds with Accuracy}}
}
@inproceedings{dutta2019sherlock,
author = {Dutta, Souradeep and Chen, Xin and Jha, Susmit and Sankaranarayanan, Sriram and Tiwari, Ashish},
booktitle = {Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control},
pages = {262--263},
title = {{Sherlock-A tool for verification of neural network feedback systems: demo abstract}},
year = {2019}
}
@inproceedings{moosavi2016deepfool,
author = {Moosavi-Dezfooli, Seyed-Mohsen and Fawzi, Alhussein and Frossard, Pascal},
booktitle = {IEEE conference on computer vision and pattern recognition},
pages = {2574--2582},
title = {{Deepfool: A Simple and Accurate Method to Fool Deep Neural Networks}},
year = {2016}
}
@article{szegedy2013intriguing,
abstract = {Deep neural networks are highly expressive models that have recently achieved state of the art performance on speech and visual recognition tasks. While their expressiveness is the reason they succeed, it also
causes them to learn uninter-pretable solutions that could have counter-intuitive properties. In this paper we report two such properties. First, we find that there is no distinction between individual high level units and
random linear combinations of high level units, according to various methods of unit analysis. It suggests that it is the space, rather than the individual units, that contains the semantic information in the high layers of
neural networks. Second, we find that deep neural networks learn input-output mappings that are fairly discontinuous to a significant extent. We can cause the network to misclas-sify an image by applying a certain hardly
perceptible perturbation, which is found by maximizing the network's prediction error. In addition, the specific nature of these perturbations is not a random artifact of learning: the same perturbation can cause a different
network, that was trained on a different subset of the dataset, to misclassify the same input.},
author = {Szegedy, Christian and Zaremba, Wojciech and Sutskever, Ilya and Bruna, Joan and Erhan, Dumitru and Goodfellow, Ian and Fergus, Rob},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Szegedy et al. - Unknown - Intriguing properties of neural networks.pdf:pdf},
journal = {arXiv preprint arXiv:1312.6199},
title = {{Intriguing Properties of Neural Networks}},
year = {2013}
}
@techreport{Gopinath,
abstract = {Deep neural networks have become widely used, obtaining remarkable results in domains such as computer vision, speech recognition, natural language processing, audio recognition, social network filtering, machine
translation , and bio-informatics, where they have produced results comparable to human experts. However, these networks can be easily "fooled" by adversarial perturbations: minimal changes to correctly-classified inputs,
that cause the network to misclassify them. This phenomenon represents a concern for both safety and security, but it is currently unclear how to measure a network's robustness against such perturbations. Existing techniques
are limited to checking robustness around a few individual input points, providing only very limited guarantees. We propose a novel approach for automatically identifying safe regions of the input space, within which the
network is robust against adversarial perturbations. The approach is data-guided, relying on clustering to identify well-defined geometric regions as candidate safe regions. We then utilize verification techniques to confirm
that these regions are safe or to provide counterexamples showing that they are not safe. We also introduce the notion of targeted robustness which, for a given target label and region, ensures that a NN does not map any
input in the region to the target label. We evaluated our technique on the MNIST dataset and on a neural network implementation of a controller for the next-generation Airborne Collision Avoidance System for unmanned aircraft
(ACAS Xu). For these networks, our approach identified multiple regions which were completely safe as well as some which were only safe for specific labels. It also discovered several adversarial perturbations of interest.},
archivePrefix = {arXiv},
arxivId = {1710.00486v2},
author = {Gopinath, Divya and Katz, Guy and {P˘ As˘ Areanu}, Corina S and Barrett, Clark},
eprint = {1710.00486v2},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Gopinath et al. - Unknown - DeepSafe A Data-driven Approach for Checking Adversarial Robustness in Neural Networks.pdf:pdf},
title = {{DeepSafe: A Data-driven Approach for Checking Adversarial Robustness in Neural Networks}}
}
@techreport{Yuan,
abstract = {With rapid progress and significant successes in a wide spectrum of applications, deep learning is being applied in many safety-critical environments. However, deep neural networks have been recently found
vulnerable to well-designed input samples, called adversarial examples. Adversarial examples are imperceptible to human but can easily fool deep neural networks in the testing/deploying stage. The vulnerability to adversarial
examples becomes one of the major risks for applying deep neural networks in safety-critical environments. Therefore, attacks and defenses on adversarial examples draw great attention. In this paper, we review recent findings
on adversarial examples for deep neural networks, summarize the methods for generating adversarial examples, and propose a taxonomy of these methods. Under the taxonomy, applications for adversarial examples are investigated.
We further elaborate on countermeasures for ad-versarial examples and explore the challenges and the potential solutions.},
archivePrefix = {arXiv},
arxivId = {1712.07107v3},
author = {Yuan, Xiaoyong and Zhu, Qile and Li, Xiaolin},
eprint = {1712.07107v3},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Yuan, Zhu, Li - Unknown - Adversarial Examples Attacks and Defenses for Deep Learning.pdf:pdf},
keywords = {Index Terms-deep neural network,adversarial examples,deep learning,security},
title = {{Adversarial Examples: Attacks and Defenses for Deep Learning}}
}
@techreport{Amrani,
abstract = {Formal Veriication (FF) and Machine Learning (MM) can seem incompatible due to their opposite mathematical foundations and their use in real-life problems: FF mostly relies on discrete mathematics and aims at
ensuring correctness; MM ooen relies on proba-bilistic models and consists of learning paaerns from training data. In this paper, we postulate that they are complementary in practice, and explore how MM helps FF in its
classical approaches: static analysis, model-checking, theorem-proving, and SSS solving. We draw a landscape of the current practice and catalog some of the most prominent uses of MM inside FF tools, thus ooering a new
perspective on FF techniques that can help researchers and practitioners to beeer locate the possible synergies. We discuss lessons learned from our work, point to possible improvements and ooer visions for the future of the
domain in the light of the science of sooware and systems modeling.},
archivePrefix = {arXiv},
arxivId = {1806.03600v2},
author = {Amrani, Moussa and L{\'{u}}cio, Levi and Bibal, Adrien},
eprint = {1806.03600v2},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Amrani, L{\'{u}}cio, Bibal - Unknown - ML FV = ♥ A Survey on the Application of Machine Learning to Formal Verification.pdf:pdf},
isbn = {1806.03600v2},
keywords = {CCS CONCEPTS •General and reference → Validation,KEYWORDS Formal Veriication, Machine Learning,Veriication,•Computing methodologies → Machine learning},
title = {{ML + FV = ♥? A Survey on the Application of Machine Learning to Formal Verification}}
}
@article{Siau2018,
author = {Siau, Keng},
number = {March},
title = {{Building Trust in Artificial Intelligence, Machine Learning, and Robotics Supply Chain Management View project}},
year = {2018}
}
@article{Bruninghaus2003a,
author = {Br{\"{u}}ninghaus, Stephanie and Ashley, Kevin D},
doi = {10.1145/1047788.1047838},
isbn = {1581137478},
journal = {9h International Conference on Artificial intelligence and Law},
pages = {233--242},
title = {{Predicting Outcomes of Case-Based Legal Arguments}},
year = {2003}
}
@article{Choi2016,
abstract = {Accuracy and interpretability are two dominant features of successful predictive models. Typically, a choice must be made in favor of complex black box models such as recurrent neural networks (RNN) for accuracy
versus less accurate but more interpretable traditional models such as logistic regression. This tradeoff poses challenges in medicine where both accuracy and interpretability are important. We addressed this challenge by
developing the REverse Time AttentIoN model (RETAIN) for application to Electronic Health Records (EHR) data. RETAIN achieves high accuracy while remaining clinically interpretable and is based on a two-level neural attention
model that detects influential past visits and significant clinical variables within those visits (e.g. key diagnoses). RETAIN mimics physician practice by attending the EHR data in a reverse time order so that recent
clinical visits are likely to receive higher attention. RETAIN was tested on a large health system EHR dataset with 14 million visits completed by 263K patients over an 8 year period and demonstrated predictive accuracy and
computational scalability comparable to state-of-the-art methods such as RNN, and ease of interpretability comparable to traditional models.},
archivePrefix = {arXiv},
arxivId = {1608.05745},
author = {Choi, Edward and Bahadori, Mohammad Taha and Kulas, Joshua A. and Schuetz, Andy and Stewart, Walter F. and Sun, Jimeng},
eprint = {1608.05745},
isbn = {9781510838819},
issn = {10495258},
number = {Nips},
title = {{RETAIN: An Interpretable Predictive Model for Healthcare using Reverse Time Attention Mechanism}},
url = {http://arxiv.org/abs/1608.05745},
year = {2016}
}
@article{Malik2009a,
abstract = {THERE ARE MANY practical situations where we need to satisfy several potentially confl icting constraints. Simple examples of this abound in daily life, for example, determining a schedule for a series of games
that resolves the availability of players and venues, or fi nding a seating assignment at dinner consistent with various rules the host would like to impose. This also applies to applications in computing, for example,
ensuring that a hardware/software system functions correctly with its overall behavior constrained by the behavior of its components and their composition, or fi nding a plan for a robot to reach a goal that is consistent
with the moves it can make at any step. While the applications may seem varied, at the core they all have variables whose values we need to determine (for example, the person sitting at a given seat at dinner) and constraints
that these variables must satisfy (for example, the host's seating rules).},
author = {Malik, Sharad and Zhang, Lintao},
doi = {10.1145/1536616.1536637},
issn = {00010782},
journal = {Communications of the ACM},
number = {8},
pages = {76},
title = {{Boolean satisfiability from theoretical hardness to practical success}},
volume = {52},
year = {2009}
}
@misc{,
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Unknown - Unknown - DL-Learner Machine Learning in OWL, RDF and Description Logics.pdf:pdf},
title = {{DL-Learner Machine Learning in OWL, RDF and Description Logics}},
url = {https://dl-learner.org/}
}
@article{Hu2016a,
abstract = {Combining deep neural networks with structured logic rules is desirable to harness flexibility and reduce uninterpretability of the neural models. We propose a general framework capable of enhancing various types
of neural networks (e.g., CNNs and RNNs) with declarative first-order logic rules. Specifically, we develop an iterative distillation method that transfers the structured information of logic rules into the weights of neural
networks. We deploy the framework on a CNN for sentiment analysis, and an RNN for named entity recognition. With a few highly intuitive rules, we obtain substantial improvements and achieve state-of-the-art or comparable
results to previous best-performing systems.},
archivePrefix = {arXiv},
arxivId = {1603.06318},
author = {Hu, Zhiting and Ma, Xuezhe and Liu, Zhengzhong and Hovy, Eduard and Xing, Eric},
doi = {10.18653/v1/P16-1228},
eprint = {1603.06318},
isbn = {9781510827585},
issn = {1541-1672},
journal = {arXiv preprint arXiv:1603.06318},
pages = {1--20},
pmid = {18925972},
title = {{Harnessing Deep Neural Networks with Logic Rules}},
year = {2016}
}
@techreport{Damato2017,
author = {D'amato, Claudia},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/D'amato - 2017 - Ontology Mining by exploiting Machine Learning for Semantic Data Management.pdf:pdf},
title = {{Ontology Mining by exploiting Machine Learning for Semantic Data Management}},
year = {2017}
}
@article{Gehr2018,
abstract = {—We present AI 2 , the first sound and scalable ana-lyzer for deep neural networks. Based on overapproximation, AI 2 can automatically prove safety properties (e.g., robustness) of realistic neural networks (e.g.,
convolutional neural networks). The key insight behind AI 2 is to phrase reasoning about safety and robustness of neural networks in terms of classic abstract interpretation, enabling us to leverage decades of advances in
that area. Concretely, we introduce abstract transformers that capture the behavior of fully connected and convolutional neural network layers with rectified linear unit activations (ReLU), as well as max pooling layers. This
allows us to handle real-world neural networks, which are often built out of those types of layers. We present a complete implementation of AI 2 together with an extensive evaluation on 20 neural networks. Our results
demonstrate that: (i) AI 2 is precise enough to prove useful specifications (e.g., robustness), (ii) AI 2 can be used to certify the effectiveness of state-of-the-art defenses for neural networks, (iii) AI 2 is significantly
faster than existing analyzers based on symbolic analysis, which often take hours to verify simple fully connected networks, and (iv) AI 2 can handle deep convolutional networks, which are beyond the reach of existing
methods.},
author = {Gehr, Timon and Mirman, Matthew and Drachsler-Cohen, Dana and Tsankov, Petar and Chaudhuri, Swarat and Vechev, Martin},
doi = {10.1109/SP.2018.00058},
isbn = {9781538643525},
issn = {10816011},
journal = {Proceedings - IEEE Symposium on Security and Privacy},
keywords = {Abstract Interpretation,Neural Networks,Reliable Machine Learning,Robustness},
pages = {3--18},
title = {{AI2: Safety and Robustness Certification of Neural Networks with Abstract Interpretation}},
volume = {2018-May},
year = {2018}
}
@techreport{Qu,
abstract = {This paper studies semi-supervised object classification in relational data, which is a fundamental problem in relational data modeling. The problem has been extensively studied in the literature of both
statistical relational learning (e.g. rela-tional Markov networks) and graph neural networks (e.g. graph convolutional networks). Statistical relational learning methods can effectively model the dependency of object labels
through conditional random fields for collective classification , whereas graph neural networks learn effective object representations for classification through end-to-end training. In this paper, we propose the Graph Markov
Neural Network (GMNN) that combines the advantages of both worlds. A GMNN models the joint distribution of object labels with a conditional random field, which can be effectively trained with the variational EM algorithm. In
the E-step, one graph neural network learns effective object representations for approximating the posterior distributions of object labels. In the M-step, another graph neural network is used to model the local label
dependency. Experiments on object classification, link classification, and unsupervised node representation learning show that GMNN achieves state-of-the-art results.},
author = {Qu, Meng and Bengio, Yoshua and Tang, Jian},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Qu, Bengio, Tang - Unknown - GMNN Graph Markov Neural Networks.pdf:pdf},
title = {{GMNN: Graph Markov Neural Networks}}
}
@article{Hullermeier2005a,
abstract = {Over the past years, methods for the automated induction of models and the extraction of interesting patterns from empirical data have attracted considerable attention in the fuzzy set community. This paper
briefly reviews some typical applications and highlights potential contributions that fuzzy set theory can make to machine learning, data mining, and related fields. The paper concludes with a critical consideration of recent
developments and some suggestions for future research directions. {\textcopyright} 2005 Elsevier B.V. All rights reserved.},
author = {H{\"{u}}llermeier, Eyke},
doi = {10.1016/j.fss.2005.05.036},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/H{\"{u}}llermeier - 2005 - Fuzzy methods in machine learning and data mining Status and prospects.pdf:pdf},
issn = {01650114},
journal = {Fuzzy Sets and Systems},
keywords = {Data mining,Knowledge discovery,Machine learning},
month = {dec},
number = {3},
pages = {387--406},
publisher = {Elsevier},
title = {{Fuzzy methods in machine learning and data mining: Status and prospects}},
volume = {156},
year = {2005}
}
@inproceedings{Huang2016,
abstract = {Deep neural networks have achieved impressive experimental results in image classification, but can surprisingly be unstable with respect to adversarial perturbations, that is, minimal changes to the input image
that cause the network to misclassify it. With potential applications including perception modules and end-to-end controllers for self-driving cars, this raises concerns about their safety. We develop a novel automated
verification framework for feed-forward multi-layer neural networks based on Satisfiability Modulo Theory (SMT). We focus on safety of image classification decisions with respect to image manipulations, such as scratches or
changes to camera angle or lighting conditions that would result in the same class being assigned by a human, and define safety for an individual decision in terms of invariance of the classification within a small
neighbourhood of the original image. We enable exhaustive search of the region by employing discretisation, and propagate the analysis layer by layer. Our method works directly with the network code and, in contrast to
existing methods, can guarantee that adversarial examples, if they exist, are found for the given region and family of manipulations. If found, adversarial examples can be shown to human testers and/or used to fine-tune the
network. We implement the techniques using Z3 and evaluate them on state-of-the-art networks, including regularised and deep learning networks. We also compare against existing techniques to search for adversarial examples
and estimate network robustness.},
author = {Huang, Xiaowei and Kwiatkowska, Marta and Wang, Sen and Wu, Min},
booktitle = {International Conference on Computer Aided Verification},
pages = {3--29},
title = {{Safety Verification of Deep Neural Networks}},
year = {2017}
}
@article{Sen2018a,
author = {Sen, Shayak and Datta, Anupam and Carbonell, Jaime and Fredrikson, Matt and Rajamani, Sriram K and Wing, Jeannette M},
title = {{Influence-directed Explanations for Machine Learning}},
year = {2018}
}
@article{Samek2017,
abstract = {With the availability of large databases and recent improvements in deep learning methodology, the performance of AI systems is reaching or even exceeding the human level on an increasing number of complex tasks.
Impressive examples of this development can be found in domains such as image classification, sentiment analysis, speech understanding or strategic game playing. However, because of their nested non-linear structure, these
highly successful machine learning and artificial intelligence models are usually applied in a black box manner, i.e., no information is provided about what exactly makes them arrive at their predictions. Since this lack of
transparency can be a major drawback, e.g., in medical applications, the development of methods for visualizing, explaining and interpreting deep learning models has recently attracted increasing attention. This paper
summarizes recent developments in this field and makes a plea for more interpretability in artificial intelligence. Furthermore, it presents two approaches to explaining predictions of deep learning models, one method which
computes the sensitivity of the prediction with respect to changes in the input and one approach which meaningfully decomposes the decision in terms of the input variables. These methods are evaluated on three classification
tasks.},
archivePrefix = {arXiv},
arxivId = {1708.08296},
author = {Samek, Wojciech and Wiegand, Thomas and M{\"{u}}ller, Klaus-Robert},
doi = {10.1109/DSAA.2015.7344858},
eprint = {1708.08296},
isbn = {9781467382731},
keywords = {artificial intelligence,black-box models,deep neural networks,interpretability,layer-wise relevance,propagation,sensitivity analysis},
number = {1},
pages = {1--10},
title = {{Explainable Artificial Intelligence: Understanding, Visualizing and Interpreting Deep Learning Models}},
year = {2017}
}
@inproceedings{Edris2018,
author = {Edris, Saba S. and Zarka, Mohamed and Ouarda, Wael and Alimi, Adel M.},
booktitle = {Proceedings of: 2017 Sudan Conference on Computer Science and Information Technology, SCCSIT 2017},
doi = {10.1109/SCCSIT.2017.8293055},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Edris et al. - 2018 - A fuzzy ontology driven context classification system using large-scale image recognition based on deep CNN.pdf:pdf},
isbn = {9781538606674},
keywords = {Deep image understanding,Fuzzy Reasoning,Image classification,Semantic context,fuzzy Ontology},
month = {feb},
pages = {1--9},
publisher = {Institute of Electrical and Electronics Engineers Inc.},
title = {{A fuzzy ontology driven context classification system using large-scale image recognition based on deep CNN}},
volume = {2017-Novem},
year = {2018}
}
@article{Park2018a,
abstract = {Deep models that are both effective and explainable are desirable in many settings; prior explainable models have been unimodal, offering either image-based visualization of attention weights or text-based
generation of post-hoc justifications. We propose a multimodal approach to explanation, and argue that the two modalities provide complementary explanatory strengths. We collect two new datasets to define and evaluate this
task, and propose a novel model which can provide joint textual rationale generation and attention visualization. Our datasets define visual and textual justifications of a classification decision for activity recognition
tasks (ACT-X) and for visual question answering tasks (VQA-X). We quantitatively show that training with the textual explanations not only yields better textual justification models, but also better localizes the evidence
that supports the decision. We also qualitatively show cases where visual explanation is more insightful than textual explanation, and vice versa, supporting our thesis that multimodal explanation models offer significant
benefits over unimodal approaches.},
archivePrefix = {arXiv},
arxivId = {1802.08129},
author = {Park, Dong Huk and Hendricks, Lisa Anne and Akata, Zeynep and Rohrbach, Anna and Schiele, Bernt and Darrell, Trevor and Rohrbach, Marcus},
doi = {10.1016/0004-3702(92)90087-E},
eprint = {1802.08129},
isbn = {0004-3702},
issn = {00043702},
number = {Nips},
pages = {1--4},
title = {{Multimodal Explanations: Justifying Decisions and Pointing to the Evidence}},
year = {2018}
}
@article{Mirman2018,
abstract = {We introduce a scalable method for training robust neural networks based on abstract interpretation. We present several abstract transformers which balance efficiency with precision and show these can be used to
train large neural networks that are certifiably robust to adversarial perturbations.},
author = {Mirman, Matthew and Gehr, Timon and Vechev, Martin},
issn = {1938-7228},
journal = {International Conference on Machine Learning},
number = {Section 6},
pages = {3575--3583},
title = {{Differentiable Abstract Interpretation for Provably Robust Neural Networks}},
volume = {Internatio},
year = {2018}
}
@article{Kim2017,
abstract = {The interpretation of deep learning models is a challenge due to their size, complexity, and often opaque internal state. In addition, many systems, such as image classifiers, operate on low-level features rather
than high-level concepts. To address these challenges, we introduce Concept Activation Vectors (CAVs), which provide an interpretation of a neural net's internal state in terms of human-friendly concepts. The key idea is to
view the high-dimensional internal state of a neural net as an aid, not an obstacle. We show how to use CAVs as part of a technique, Testing with CAVs (TCAV), that uses directional derivatives to quantify the degree to which
a user-defined concept is important to a classification result--for example, how sensitive a prediction of "zebra" is to the presence of stripes. Using the domain of image classification as a testing ground, we describe how
CAVs may be used to explore hypotheses and generate insights for a standard image classification network as well as a medical application.},
archivePrefix = {arXiv},
arxivId = {1711.11279},
author = {Kim, Been and Wattenberg, Martin and Gilmer, Justin and Cai, Carrie and Wexler, James and Viegas, Fernanda and Sayres, Rory},
eprint = {1711.11279},
issn = {1938-7228},
title = {{Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)}},
year = {2017}
}
@article{Garnelo2018,
abstract = {Deep neural networks excel at function approximation, yet they are typically trained from scratch for each new function. On the other hand, Bayesian methods, such as Gaussian Processes (GPs), exploit prior
knowledge to quickly infer the shape of a new function at test time. Yet GPs are computationally expensive, and it can be hard to design appropriate priors. In this paper we propose a family of neural models, Conditional
Neural Processes (CNPs), that combine the benefits of both. CNPs are inspired by the flexibility of stochastic processes such as GPs, but are structured as neural networks and trained via gradient descent. CNPs make accurate
predictions after observing only a handful of training data points, yet scale to complex functions and large datasets. We demonstrate the performance and versatility of the approach on a range of canonical machine learning
tasks, including regression, classification and image completion.},
archivePrefix = {arXiv},
arxivId = {1807.01613},
author = {Garnelo, Marta and Rosenbaum, Dan and Maddison, Chris J. and Ramalho, Tiago and Saxton, David and Shanahan, Murray and Teh, Yee Whye and Rezende, Danilo J. and Eslami, S. M. Ali},
eprint = {1807.01613},
issn = {1938-7228},
title = {{Conditional Neural Processes}},
year = {2018}
}
@article{Bonanno2017,
abstract = {Deep Learning has proven to be an effective method for making highly accurate predictions from complex data sources. Convolutional neural networks continue to dominate image classification problems and recursive
neural networks have proven their utility in caption generation and language translations. While these approaches are powerful, they do not offer explanation for how the output is generated. Without understanding how deep
learning arrives at a solution there is no guarantee that these networks will transition from controlled laboratory environments to fieldable systems. This paper presents an approach for incorporating such rule based
methodology into neural networks by embedding fuzzy inference systems into deep learning networks.},
author = {Bonanno, David and Nock, Kristen and Smith, Leslie and Elmore, Paul and Petry, Fred},
doi = {10.1117/12.2268001},
journal = {Next-Generation Analyst V},
number = {May},
pages = {102070D},
title = {{An approach to explainable deep learning using fuzzy inference}},
volume = {10207},
year = {2017}
}
@article{Nguyen2006,
abstract = {approximate reasoning},
author = {Nguyen, Hung Son},
doi = {10.1007/11847465_16},
isbn = {978-3-540-39382-5},
issn = {03029743},
journal = {Transactions on Rough Sets V},
keywords = {association rules,boolean reasoning,data mining,decision,decision rule construction,discretization,feature selec-,large data tables,rough sets,tion and extraction,tree induction},
pages = {334--506},
title = {{Approximate Boolean Reasoning: Foundations and Applications in Data Mining}},
year = {2006}
}
@article{Deng2017,
abstract = {{\textcopyright} 2016 IEEE. Deep learning (DL) is an emerging and powerful paradigm that allows large-scale task-driven feature learning from big data. However, typical DL is a fully deterministic model that sheds
no light on data uncertainty reductions. In this paper, we show how to introduce the concepts of fuzzy learning into DL to overcome the shortcomings of fixed representation. The bulk of the proposed fuzzy system is a
hierarchical deep neural network that derives information from both fuzzy and neural representations. Then, the knowledge learnt from these two respective views are fused altogether forming the final data representation to be
classified. The effectiveness of the model is verified on three practical tasks of image categorization, high-frequency financial data prediction and brain MRI segmentation that all contain high level of uncertainties in the
raw data. The fuzzy dDL paradigm greatly outperforms other nonfuzzy and shallow learning approaches on these tasks.},
author = {Deng, Yue and Ren, Zhiquan and Kong, Youyong and Bao, Feng and Dai, Qionghai},
doi = {10.1109/TFUZZ.2016.2574915},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Deng et al. - 2017 - A Hierarchical Fused Fuzzy Deep Neural Network for Data Classification.pdf:pdf},
issn = {10636706},
journal = {IEEE Transactions on Fuzzy Systems},
keywords = {Feature learning,fuzzy neural networks,pattern classification},
month = {aug},
number = {4},
pages = {1006--1012},
publisher = {Institute of Electrical and Electronics Engineers Inc.},
title = {{A Hierarchical Fused Fuzzy Deep Neural Network for Data Classification}},
volume = {25},
year = {2017}
}
@article{Kulmanov2018,
abstract = {A large number of protein sequences are becoming available through the application of novel high-throughput sequencing technologies. Experimental functional characterization of these proteins is time-consuming and
expensive, and is often only done rigorously for few selected model organisms. Computational function prediction approaches have been suggested to fill this gap. The functions of proteins are classified using the Gene
Ontology (GO), which contains over 40,000 classes. Additionally, proteins have multiple functions, making function prediction a large-scale, multi-class, multi-label problem. We have developed a novel method to predict
protein function from sequence. We use deep learning to learn features from protein sequences as well as a cross-species protein-protein interaction network. Our approach specifically outputs information in the structure of
the GO and utilizes the dependencies between GO classes as background information to construct a deep learning model. We evaluate our method using the standards established by the Computational Assessment of Function
Annotation (CAFA) and demonstrate a significant improvement over baseline methods such as BLAST, with significant improvement for predicting cellular locations.},
archivePrefix = {arXiv},
arxivId = {1705.05919},
author = {Kulmanov, Maxat and Khan, Mohammed Asif and Hoehndorf, Robert},
doi = {10.1093/bioinformatics/btx624},
eprint = {1705.05919},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Kulmanov, Khan, Hoehndorf - 2018 - DeepGO Predicting protein functions from sequence and interactions using a deep ontology-aware
cla(2).pdf:pdf},
isbn = {13674811 (Electronic)},
issn = {14602059},
journal = {Bioinformatics},
month = {feb},
number = {4},
pages = {660--668},
pmid = {29028931},
publisher = {Oxford University Press},
title = {{DeepGO: Predicting protein functions from sequence and interactions using a deep ontology-aware classifier}},
volume = {34},
year = {2018}
}
@article{Hohenecker2017,
abstract = {In this work, we present a novel approach to ontology reasoning that is based on deep learning rather than logic-based formal reasoning. To this end, we introduce a new model for statistical relational learning
that is built upon deep recursive neural networks, and give experimental evidence that it can easily compete with, or even outperform, existing logic-based reasoners on the task of ontology reasoning. More precisely, we
compared our implemented system with one of the best logic-based ontology reasoners at present, RDFox, on a number of large standard benchmark datasets, and found that our system attained high reasoning quality, while being
up to two orders of magnitude faster.},
archivePrefix = {arXiv},
arxivId = {1705.10342},
author = {Hohenecker, Patrick and Lukasiewicz, Thomas},
eprint = {1705.10342},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Hohenecker, Lukasiewicz - 2017 - Deep Learning for Ontology Reasoning.pdf:pdf},
month = {may},
title = {{Deep Learning for Ontology Reasoning}},
year = {2017}
}
@article{Chen2018,
abstract = {Predicting traffic flow is one of the fundamental needs to comfortable travel, but this task is challenging in vehicular cyber–physical systems because of ever-increasing uncertain traffic big data. Although
deep-learning (DL) methods with outstanding performance recently have become popular, most existing DL models for traffic flow prediction are fully deterministic and shed no light on data uncertainty. In this study, a novel
fuzzy deep-learning approach called FDCN is proposed for predicting citywide traffic flow. This approach is built on the fuzzy theory and the deep residual network model. Our key idea is to introduce the fuzzy representation
into the DL model to lessen the impact of data uncertainty. A model of fuzzy deep convolutional network is established to improve traffic flow prediction while investigating the spatial and temporal correlation of traffic
flow. We further propose pre-training and fine-tuning strategies that efficiently learn parameters of the FDCN. To the best of our knowledge, this is the first time that a fuzzy DL approach has been applied to represent
traffic features for traffic flow prediction. Experimental results demonstrate that the proposed approach to traffic flow prediction has superior performance compared with state-of-the-art approaches.},
author = {Chen, Weihong and An, Jiyao and Li, Renfa and Fu, Li and Xie, Guoqi and Bhuiyan, Md Zakirul Alam and Li, Keqin},
doi = {10.1016/j.future.2018.06.021},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Chen et al. - 2018 - A novel fuzzy deep-learning approach to traffic flow prediction with uncertain spatial–temporal data features.pdf:pdf},
issn = {0167739X},
journal = {Future Generation Computer Systems},
keywords = {Deep learning,Fuzzy representation,Residual networks,Traffic flow prediction},
month = {dec},
pages = {78--88},
publisher = {Elsevier B.V.},
title = {{A novel fuzzy deep-learning approach to traffic flow prediction with uncertain spatial–temporal data features}},
volume = {89},
year = {2018}
}
@techreport{Ulsari1992,
abstract = {P roblems requiring inferenci ng with Boolean logic have been implemented in percept rons or feedforward networks, and some attempts have been made to imp lement fuzzy logic based inferencing in similar networks.
In this pap er, we pr esent producti ve networks , which are art ificial neur al networks, meant for fuzzy logic based inferencing. The nod es in t hese networks collect an offset product of the inpu ts, further offset by a
bias. A meaning can be assigned t o each node in such a network , since t he offsets must be eit her-1 , 0, or l. Earlier , it was shown t hat fuzzy logic inferencing could be perform ed in productive networks by manu ally
setting t he offsets. Thi s procedure, however , encountered crit icism, since t here is a feeling t hat neural networks should involve tr aining. We describe an algorit hm for training pro duct ive networks from a set of
training inst ances. Unlike feedforward neur al networks with sigmoida l neurons , these netwo rks can be t ra ined wit h a sma ll numb er of t ra ining instances. T he t hree main logical operations t hat form t he basis of
inferenc-ing-NOT, OR , and AND-can be implemented easily in pro du ctive networks. T he network s derive their nam e from t he way t he offset product of inpu ts forms t he act ivation of a node.},
author = {Ulsari, Abhay B},
booktitle = {Complex Systems},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Ulsari - 1992 - Training A rtificial Neural Networks for Fuzzy Logic.pdf:pdf},
pages = {443--457},
title = {{Training A rtificial Neural Networks for Fuzzy Logic}},
volume = {6},
year = {1992}
}
@book{Morgan2018a,
abstract = {This paper presents a novel method to address legal rights for children through a chatbot framework by integrating machine learning, a dialogue graph, and information extraction. The method addresses a significant
problem: we cannot presume that children have common knowledge about their rights or express themselves as an adult might. In our framework, a chatbot user begins a conversation, where based on the circumstance described, a
neural network predicts both speech acts, relating to a dialogue graph, and legal types. Information is extracted in order to create a case for a legal advisor. In collaboration with the Children's Legal Centre Wales, who
advocate for the improvement of legal rights in Wales, a corpus has been constructed and a prototype chatbot developed. The framework has been evaluated with classification measures and a user study.},
author = {Morgan, J. and Paiement, A. and Seisenberger, M. and Williams, J. and Wyner, A.},
booktitle = {Frontiers in Artificial Intelligence and Applications},
doi = {10.3233/978-1-61499-935-5-205},
isbn = {9781614999348},
issn = {09226389},
keywords = {Chatbot,Children's Legal Rights,Machine Learning,Natural Language Processing,Recurrent Neural Networks},
title = {{A chatbot framework for the children's legal centre}},
volume = {313},
year = {2018}
}
@article{Grossman2005,
archivePrefix = {arXiv},
arxivId = {arXiv:1011.1669v3},
author = {Grossman},
doi = {10.1533/9781845699789.5.663},
eprint = {arXiv:1011.1669v3},
isbn = {0301-4215},
issn = {03014215},
keywords = {city level trends,interactions,many studies have assessed,minority threat,nonlinear relationships,of minority populations and,police,police per capita,relationships between the size,the,threat theory by
investigating},
pmid = {21675331},
title = {{TECHNOLOGY-ASSISTED REVIEW IN E-DISCOVERY CAN BE MoRE EFFECTIVE AND MoRE EFFICIENT THAN EXHAUSTIVE MANUAL REVIEW}},
volume = {1},
year = {2005}
}
@book{Siekmann2008a,
author = {Siekmann, J and Wahlster, W},
doi = {10.1007/978-3-642-29066-4{_}11},
isbn = {9783540882688},
issn = {0104-6500},
pages = {247},
pmid = {1000198484},
title = {{Lecture Notes in Artificial Intelligence}},
year = {2008}
}


@inproceedings{Wicker2018a,
  title={Feature-guided black-box safety testing of deep neural networks},
  author={Wicker, Matthew and Huang, Xiaowei and Kwiatkowska, Marta},
  booktitle={International Conference on Tools and Algorithms for the Construction and Analysis of Systems},
  pages={408--426},
  year={2018},
  organization={Springer}
}


@inproceedings{Bastani2016,
abstract = {Despite having high accuracy, neural nets have been shown to be susceptible to adversarial examples, where a small perturbation to an input can cause it to become mislabeled. We propose metrics for measuring the
robustness of a neural net and devise a novel algorithm for approximating these metrics based on an encoding of robustness as a linear program. We show how our metrics can be used to evaluate the robustness of deep neural
nets with experiments on the MNIST and CIFAR-10 datasets. Our algorithm generates more informative estimates of robustness metrics compared to estimates based on existing algorithms. Furthermore, we show how existing
approaches to improving robustness "overfit" to adversarial examples generated using a specific algorithm. Finally, we show that our techniques can be used to additionally improve neural net robustness both according to the
metrics that we propose, but also according to previously proposed metrics.},
archivePrefix = {arXiv},
arxivId = {1605.07262},
author = {Bastani, Osbert and Ioannou, Yani and Lampropoulos, Leonidas and Vytiniotis, Dimitrios and Nori, Aditya and Criminisi, Antonio},
booktitle = {Advances in neural information processing systems},
eprint = {1605.07262},
issn = {10495258},
pages = {2613--2621},
title = {{Measuring Neural Net Robustness with Constraints}},
year = {2016}
}
@article{Baron2010a,
author = {Baron, Jason R.},
journal = {￼Richmond Journal of Law {\&} Technology},
number = {3},
pages = {1},
title = {{Law in the age of exabytes: Some further thoughts on information inflation and current issues in e-discovery search}},
volume = {17},
year = {2010}
}
@article{Fromherz2017,
abstract = {Agda is a dependently typed functional programming language and a proof assistant in which developing programs and proving their correctness is one activity. We show how this process can be enhanced by integrating
external automated theorem provers, provide a prototypical integration of the equational theorem prover Waldmeister, and give examples of how this proof automation works in practice.},
author = {Fromherz, Aymeric and Ouadjaout, Abdelraouf and Min, Antoine},
doi = {10.1007/978-3-642-38088-4},
isbn = {978-3-642-38087-7},
issn = {03029743},
journal = {NASA Formal Methods},
pages = {185--202},
title = {{Static Value Analysis of Python Programs by Abstract Interpretation}},
volume = {10811},
year = {2018}
}
@article{Leike2017a,
abstract = {We present a suite of reinforcement learning environments illustrating various safety properties of intelligent agents. These problems include safe interruptibility, avoiding side effects, absent supervisor,
reward gaming, safe exploration, as well as robustness to self-modification, distributional shift, and adversaries. To measure compliance with the intended safe behavior, we equip each environment with a performance function
that is hidden from the agent. This allows us to categorize AI safety problems into robustness and specification problems, depending on whether the performance function corresponds to the observed reward function. We evaluate
A2C and Rainbow, two recent deep reinforcement learning agents, on our environments and show that they are not able to solve them satisfactorily.},
archivePrefix = {arXiv},
arxivId = {1711.09883},
author = {Leike, Jan and Martic, Miljan and Krakovna, Victoria and Ortega, Pedro A. and Everitt, Tom and Lefrancq, Andrew and Orseau, Laurent and Legg, Shane},
eprint = {1711.09883},
title = {{AI Safety Gridworlds}},
year = {2017}
}
@article{Evans2018,
abstract = {Artificial Neural Networks are powerful function approximators capable of modelling solutions to a wide variety of problems, both supervised and unsupervised. As their size and expressivity increases, so too does
the variance of the model, yielding a nearly ubiquitous overfitting problem. Although mitigated by a variety of model regularisation methods, the common cure is to seek large amounts of training data---which is not
necessarily easily obtained---that sufficiently approximates the data distribution of the domain we wish to test on. In contrast, logic programming methods such as Inductive Logic Programming offer an extremely data-efficient
process by which models can be trained to reason on symbolic domains. However, these methods are unable to deal with the variety of domains neural networks can be applied to: they are not robust to noise in or mislabelling of
inputs, and perhaps more importantly, cannot be applied to non-symbolic domains where the data is ambiguous, such as operating on raw pixels. In this paper, we propose a Differentiable Inductive Logic framework, which can not
only solve tasks which traditional ILP systems are suited for, but shows a robustness to noise and error in the training data which ILP cannot cope with. Furthermore, as it is trained by backpropagation against a likelihood
objective, it can be hybridised by connecting it with neural networks over ambiguous data in order to be applied to domains which ILP cannot address, while providing data efficiency and generalisation beyond what neural
networks on their own can achieve.},
archivePrefix = {arXiv},
arxivId = {1711.04574},
author = {Evans, Richard and Grefenstette, Edward},
doi = {10.1613/jair.5477},
eprint = {1711.04574},
issn = {10769757},
journal = {Journal of Artificial Intelligence Research},
pages = {65--170},
title = {{Learning explanatory rules from noisy data}},
volume = {61},
year = {2018}
}
@inproceedings{Konev2016,
abstract = {{\textcopyright} Copyright 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. We investigate the problem of learning description logic (DL) ontologies in Angluin
et al.'s framework of exact learning via queries posed to an oracle. We consider membership queries of the form "is a tuple a of individuals a certain answer to a data retrieval query q in a given ABox and the unknown target
ontology?" and completeness queries of the form "does a hypothesis ontology entail the unknown target ontology?". Given a DL L and a data retrieval query language Q, we study polynomial learnability of ontologies in L using
data retrieval queries in Q and provide an almost complete classification for DLs that are fragments of EL with role inclusions and of DL-Lite and for data retrieval queries that range from atomic queries and EL/ELI-instance
queries to conjunctive queries. Some results are proved by non-Trivial reductions to learning from subsumption examples.},
author = {Konev, Boris and Ozaki, Ana and Wolter, Frank},
booktitle = {30th AAAI Conference on Artificial Intelligence, AAAI 2016},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Konev, Ozaki, Wolter - 2016 - A model for learning description logic ontologies based on exact learning.pdf:pdf},
isbn = {9781577357605},
pages = {1008--1015},
publisher = {AAAI press},
title = {{A model for learning description logic ontologies based on exact learning}},
year = {2016}
}
@article{Kulmanov2019,
abstract = {Protein function prediction is one of the major tasks of bioinformatics that can help in wide range of biological problems such as understanding disease mechanisms or finding drug targets. Many methods are
available for predicting protein functions from sequence based features, protein–protein interaction networks, protein structure or literature. However, other than sequence, most of the features are difficult to obtain or not
available for many proteins thereby limiting their scope. Furthermore, the performance of sequence-based function prediction methods is often lower than methods that incorporate multiple features and predicting protein
functions may require a lot of time. We developed a novel method for predicting protein functions from sequence alone which combines deep convolutional neural network (CNN) model with sequence similarity based predictions.
Our CNN model scans the sequence for motifs which are predictive for protein functions and combines this with functions of similar proteins. We evaluate the performance of DeepGOPlus on the CAFA3 dataset and significantly
improve the performance of predictions of biological processes and cellular components with F max of 0.47 and 0.70, respectively, using only the amino acid sequence of proteins as input. DeepGOPlus can annotate around 40
protein sequences per second, thereby making fast and accurate function predictions available for a wide range of proteins.},
author = {Kulmanov, Maxat and Hoehndorf, Robert},
doi = {10.1093/bioinformatics/btz595},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Kulmanov, Hoehndorf - 2019 - DeepGOPlus Improved protein function prediction from sequence.pdf:pdf},
issn = {1367-4803},
journal = {Bioinformatics},
month = {jul},
publisher = {Oxford University Press (OUP)},
title = {{DeepGOPlus: Improved protein function prediction from sequence}},
year = {2019}
}
@inproceedings{Godfrey2017,
abstract = {We present a deep learning architecture for learning fuzzy logic expressions. Our model uses an innovative, parameterized, differentiable activation function that can learn a number of logical operations by
gradient descent. This activation function allows a neural network to determine the relationships between its input variables and provides insight into the logical significance of learned network parameters. We provide a
theoretical basis for this parameterization and demonstrate its effectiveness and utility by successfully applying our model to five classification problems from the UCI Machine Learning Repository.},
author = {Godfrey, Luke B. and Gashler, Michael S.},
booktitle = {2017 IEEE International Conference on Systems, Man, and Cybernetics, SMC 2017},
doi = {10.1109/SMC.2017.8122696},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Godfrey, Gashler - 2017 - A parameterized activation function for learning fuzzy logic operations in deep neural networks.pdf:pdf},
isbn = {9781538616451},
month = {nov},
pages = {740--745},
publisher = {Institute of Electrical and Electronics Engineers Inc.},
title = {{A parameterized activation function for learning fuzzy logic operations in deep neural networks}},
volume = {2017-Janua},
year = {2017}
}
@article{Zhang2019,
abstract = {The adversarial training procedure proposed by Madry et al. (2018) is one of the most effective methods to defend against adversarial examples in deep neural networks (DNNs). In our paper, we shed some lights on
the practicality and the hardness of adversarial training by showing that the effectiveness (robustness on test set) of adversarial training has a strong correlation with the distance between a test point and the manifold of
training data embedded by the network. Test examples that are relatively far away from this manifold are more likely to be vulnerable to adversarial attacks. Consequentially, an adversarial training based defense is
susceptible to a new class of attacks, the "blind-spot attack", where the input images reside in "blind-spots" (low density regions) of the empirical distribution of training data but is still on the ground-truth data
manifold. For MNIST, we found that these blind-spots can be easily found by simply scaling and shifting image pixel values. Most importantly, for large datasets with high dimensional and complex data manifold (CIFAR,
ImageNet, etc), the existence of blind-spots in adversarial training makes defending on any valid test examples difficult due to the curse of dimensionality and the scarcity of training data. Additionally, we find that
blind-spots also exist on provable defenses including (Wong {\&} Kolter, 2018) and (Sinha et al., 2018) because these trainable robustness certificates can only be practically optimized on a limited set of training data.},
author = {Zhang, Huan and Chen, Hongge and Song, Zhao and Boning, Duane and Dhillon, Inderjit S. and Hsieh, Cho-Jui},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Zhang et al. - Unknown - THE LIMITATIONS OF ADVERSARIAL TRAINING AND THE BLIND-SPOT ATTACK.pdf:pdf},
journal = {arXiv preprint arXiv:1901.04684},
title = {{The limitations of adversarial training and the blind-spot attack}},
year = {2019}
}
@article{Vardia,
author = {Vardi, Moshe Y},
title = {{Machine Learning and Logic: Fast and Slow Thinking}},
}
@article{Konev2018,
abstract = {We study the problem of learning description logic (DL) ontologies in Angluin et al.'s framework of exact learning via queries. We admit membership queries ("is a given subsumption entailed by the target
ontology?") and equivalence queries ("is a given ontology equivalent to the target ontology?"). We present three main results: (1) ontologies formulated in (two relevant versions of) the description logic DL-Lite can be
learned with polynomially many queries of polynomial size; (2) this is not the case for ontologies formulated in the description logic EL, even when only acyclic ontologies are admitted; and (3) ontologies formulated in a
fragment of EL related to the web ontology language OWL 2 RL can be learned in polynomial time. We also show that neither membership nor equivalence queries alone are sufficient in cases (1) and (3).},
author = {Konev, Boris and Lutz, Carsten and Ozaki, Ana and Wolter, Frank},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Konev et al. - 2018 - Exact learning of lightweight description logic ontologies.pdf:pdf},
issn = {15337928},
journal = {Journal of Machine Learning Research},
keywords = {Complexity,Description Logic,Exact Learning},
month = {apr},
pages = {1--63},
publisher = {Microtome Publishing},
title = {{Exact learning of lightweight description logic ontologies}},
volume = {18},
year = {2018}
}
@article{Xiang2018c,
abstract = {This survey presents an overview of verification techniques for autonomous systems, with a focus on safety-critical autonomous cyber-physical systems (CPS) and subcomponents thereof. Autonomy in CPS is enabling by
recent advances in artificial intelligence (AI) and machine learning (ML) through approaches such as deep neural networks (DNNs), embedded in so-called learning enabled components (LECs) that accomplish tasks from
classification to control. Recently, the formal methods and formal verification community has developed methods to characterize behaviors in these LECs with eventual goals of formally verifying specifications for LECs, and
this article presents a survey of many of these recent approaches. 1 Motivation and Prevalence Artificial intelligence (AI) is in a renaissance, and AI methods, such as machine learning (ML), are now at a level of accuracy
and performance to be competitive or better than humans for many tasks. Deep neural networks (DNNs) in particular are increasingly effective at recognition and classification tasks. For instance, much of the sensing,
estimation, and fusion of such data that enables applications such as autonomous driving and other autonomous cyber-physical systems (CPS) increasingly relies on DNNs and similar ML techniques. However , this progress comes
at significant risk when these methods are deployed in operational safety-critical systems, especially those without direct human supervision. Building on applications of AI/ML in autonomous CPS, this paper surveys the
current state-of-the-art for safely integrating AI/ML components-which we term learning enabled components (LECs) into safety-critical CPS. This survey consists of several major parts, focused around how machine learning is
enabling autonomy in CPS, including: illustrative applications, intelligent control, safety architectures with AI/ML components, understanding AI/ML components and 1},
archivePrefix = {arXiv},
arxivId = {1810.01989v1},
author = {Xiang, Weiming and Musau, Patrick and Wild, Ayana A and Manzanas, Diego and Nathaniel, Lopez and Yang, Hamilton Xiaodong and Rosenfeld, Joel and Johnson, Taylor T},
doi = {arXiv:1810.01989v1},
eprint = {1810.01989v1},
pages = {1--51},
title = {{Verification for Machine Learning, Autonomy, and Neural Networks Survey}},
year = {2018}
}
@article{,
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Unknown - Unknown - IJCAI3.pdf:pdf},
title = {{IJCAI3}}
}
@article{Trask2018,
abstract = {Neural networks can learn to represent and manipulate numerical information, but they seldom generalize well outside of the range of numerical values encountered during training. To encourage more systematic
numerical extrapolation, we propose an architecture that represents numerical quantities as linear activations which are manipulated using primitive arithmetic operators, controlled by learned gates. We call this module a
neural arithmetic logic unit (NALU), by analogy to the arithmetic logic unit in traditional processors. Experiments show that NALU-enhanced neural networks can learn to track time, perform arithmetic over images of numbers,
translate numerical language into real-valued scalars, execute computer code, and count objects in images. In contrast to conventional architectures, we obtain substantially better generalization both inside and outside of
the range of numerical values encountered during training, often extrapolating orders of magnitude beyond trained numerical ranges.},
archivePrefix = {arXiv},
arxivId = {1808.00508},
author = {Trask, Andrew and Hill, Felix and Reed, Scott and Rae, Jack and Dyer, Chris and Blunsom, Phil},
eprint = {1808.00508},
title = {{Neural Arithmetic Logic Units}},
year = {2018}
}
@article{Fulton2018a,
abstract = {Formal verification provides a high degree of confidence in safe system operation, but only if reality matches the verified model. Although a good model will be accurate most of the time, even the best models are
incomplete. This is especially true in Cyber-Physical Systems because high-fidelity physi-cal models of systems are expensive to develop and often in-tractable to verify. Conversely, reinforcement learning-based controllers
are lauded for their flexibility in unmodeled envi-ronments, but do not provide guarantees of safe operation. This paper presents an approach for provably safe learning that provides the best of both worlds: the exploration
and op-timization capabilities of learning along with the safety guar-antees of formal verification. Our main insight is that formal verification combined with verified runtime monitoring can ensure the safety of a learning
agent. Verification results are preserved whenever learning agents limit exploration within the confounds of verified control choices as long as observed reality comports with the model used for off-line verification. When a
model violation is detected, the agent abandons ef-ficiency and instead attempts to learn a control strategy that guides the agent to a modeled portion of the state space. We prove that our approach toward incorporating
knowl-edge about safe control into learning systems preserves safety guarantees, and demonstrate that we retain the empirical per-formance benefits provided by reinforcement learning. We also explore various points in the
design space for these jus-tified speculative controllers in a simple model of adaptive cruise control model for autonomous cars.},
author = {Fulton, Nathan and Platzer, Andr{\'{e}}},
journal = {32nd AAAI International Conference on Artificial Intelligence},
title = {{Safe Reinforcement Learning via Formal Methods}},
year = {2018}
}
@techreport{Brachman,
author = {Brachman, Ronald J and Research, Yahoo ! and Domingos, Pedro and Lowd, Daniel},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Brachman et al. - Unknown - Markov Logic An Interface Layer for Artificial Intelligence Synthesis Lectures on Artificial Intelligence
an.pdf:pdf},
title = {{Markov Logic: An Interface Layer for Artificial Intelligence Synthesis Lectures on Artificial Intelligence and Machine Learning Markov Logic: An Interface Layer for Artificial Intelligence Introduction to
Semi-Supervised Learning}}
}
@inproceedings{Kulmanov2019a,
abstract = {An embedding is a function that maps entities from one algebraic structure into another while preserving certain characteristics. Embeddings are being used successfully for mapping relational data or text into
vector spaces where they can be used for machine learning, similarity search, or similar tasks. We address the problem of finding vector space embeddings for theories in the Description Logic ??⁺⁺ that are also models of the
TBox. To find such embeddings, we define an optimization problem that characterizes the model-theoretic semantics of the operators in ??⁺⁺ within ℝⁿ, thereby solving the problem of finding an interpretation function for an ??⁺⁺
theory given a particular domain $\Delta$. Our approach is mainly relevant to large ??⁺⁺ theories and knowledge bases such as the ontologies and knowledge graphs used in the life sciences. We demonstrate that our method can be
used for improved prediction of protein--protein interactions when compared to semantic similarity measures or knowledge graph embeddings.},
author = {Kulmanov, Maxat and Liu-Wei, Wang and Yan, Yuan and Hoehndorf, Robert},
doi = {10.24963/ijcai.2019/845},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Kulmanov et al. - 2019 - EL Embeddings Geometric Construction of Models for the Description Logic EL.pdf:pdf},
month = {jul},
pages = {6103--6109},
publisher = {International Joint Conferences on Artificial Intelligence},
title = {{EL Embeddings: Geometric Construction of Models for the Description Logic EL++}},
year = {2019}
}
@article{Prasad2005a,
abstract = {Dramatic improvements in SAT solver technology over the last decade and the growing need for more efficient and scalable verification solutions have fueled research in verification methods based on SAT solvers.
This paper presents a survey of the latest developments in SAT-based formal verification, including incomplete methods such as bounded model checking and complete methods for model checking. We focus on how the surveyed
techniques formulate the verification problem as a SAT problem and how they exploit crucial aspects of a SAT solver, such as application-specific heuristics and conflict-driven learning. Finally, we summarize the noteworthy
achievements in this area so far and note the major challenges in making this technology more pervasive in industrial design verification flows.},
author = {Prasad, Mukul R. and Biere, Armin and Gupta, Aarti},
doi = {10.1007/s10009-004-0183-4},
isbn = {1000900401},
issn = {1433-2779},
journal = {International Journal on Software Tools for Technology Transfer (STTT)},
keywords = {model checking,qbf,sat,verification},
number = {2},
pages = {156--173},
title = {{A survey of recent advances in SAT-based formal verification}},
volume = {7},
year = {2005}
}
@misc{,
title = {{Tensorflow - Neural Structured Learning}},
}
@article{Klopfenstein2018,
abstract = {The biological interpretation of gene lists with interesting shared properties, such as up- or down-regulation in a particular experiment, is typically accomplished using gene ontology enrichment analysis tools.
Given a list of genes, a gene ontology (GO) enrichment analysis may return hundreds of statistically significant GO results in a “flat” list, which can be challenging to summarize. It can also be difficult to keep pace with
rapidly expanding biological knowledge, which often results in daily changes to any of the over 47,000 gene ontologies that describe biological knowledge. GOATOOLS, a Python-based library, makes it more efficient to stay
current with the latest ontologies and annotations, perform gene ontology enrichment analyses to determine over- and under-represented terms, and organize results for greater clarity and easier interpretation using a novel
GOATOOLS GO grouping method. We performed functional analyses on both stochastic simulation data and real data from a published RNA-seq study to compare the enrichment results from GOATOOLS to two other popular tools: DAVID
and GOstats. GOATOOLS is freely available through GitHub:
                  https://github.com/tanghaibao/goatools

                .},
author = {Klopfenstein, D. V. and Zhang, Liangsheng and Pedersen, Brent S. and Ram{\'{i}}rez, Fidel and Vesztrocy, Alex Warwick and Naldi, Aur{\'{e}}lien and Mungall, Christopher J. and Yunes, Jeffrey M. and Botvinnik, Olga
and Weigel, Mark and Dampier, Will and Dessimoz, Christophe and Flick, Patrick and Tang, Haibao},
doi = {10.1038/s41598-018-28948-z},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Klopfenstein et al. - 2018 - GOATOOLS A Python library for Gene Ontology analyses.pdf:pdf},
issn = {20452322},
journal = {Scientific Reports},
month = {dec},
number = {1},
publisher = {Nature Publishing Group},
title = {{GOATOOLS: A Python library for Gene Ontology analyses}},
volume = {8},
year = {2018}
}

@inproceedings{Ruan2018,
  title     = {Reachability Analysis of Deep Neural Networks with Provable Guarantees},
  author    = {Wenjie Ruan and Xiaowei Huang and Marta Kwiatkowska},
  booktitle = {Proceedings of the Twenty-Seventh International Joint Conference on
               Artificial Intelligence},
  pages     = {2651--2659},
  year      = {2018},
  month     = {7},
}


@inproceedings{Lisi2009,
abstract = {In this paper we consider an extension of Logic Programming that tackles the SemanticWeb challenge of acquiring rules combined with ontologies. To face this bottleneck problem we propose a framework that resorts
to the expressive and deductive power of DL+log and adopts the methodological apparatus of Inductive Logic Programming. {\textcopyright} Copyright 2008 front matter by the editors.},
author = {Lisi, Francesca A. and Esposito, Floriana},
booktitle = {CEUR Workshop Proceedings},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Lisi, Esposito - 2009 - Combining logic programming with description logics and machine learning for the semantic web.pdf:pdf},
issn = {16130073},
title = {{Combining logic programming with description logics and machine learning for the semantic web}},
volume = {434},
year = {2009}
}
@article{,
title = {{Fuzzy Logic vs Bayesian Probability}},
}
@article{Morcos2018,
abstract = {Despite their ability to memorize large datasets, deep neural networks often achieve good generalization performance. However, the differences between the learned solutions of networks which generalize and those
which do not remain unclear. Additionally, the tuning properties of single directions (defined as the activation of a single unit or some linear combination of units in response to some input) have been highlighted, but their
importance has not been evaluated. Here, we connect these lines of inquiry to demonstrate that a network's reliance on single directions is a good predictor of its generalization performance, across networks trained on
datasets with different fractions of corrupted labels, across ensembles of networks trained on datasets with unmodified labels, across different hyperparameters, and over the course of training. While dropout only regularizes
this quantity up to a point, batch normalization implicitly discourages single direction reliance, in part by decreasing the class selectivity of individual units. Finally, we find that class selectivity is a poor predictor
of task importance, suggesting not only that networks which generalize well minimize their dependence on individual units by reducing their selectivity, but also that individually selective units may not be necessary for
strong network performance.},
archivePrefix = {arXiv},
arxivId = {1803.06959},
author = {Morcos, Ari S. and Barrett, David G. T. and Rabinowitz, Neil C. and Botvinick, Matthew},
eprint = {1803.06959},
pages = {1--15},
title = {{On the importance of single directions for generalization}},
year = {2018}
}
@article{Barrett2018,
abstract = {Whether neural networks can learn abstract reasoning or whether they merely rely on superficial statistics is a topic of recent debate. Here, we propose a dataset and challenge designed to probe abstract
reasoning, inspired by a well-known human IQ test. To succeed at this challenge, models must cope with various generalisation `regimes' in which the training and test data differ in clearly-defined ways. We show that popular
models such as ResNets perform poorly, even when the training and test sets differ only minimally, and we present a novel architecture, with a structure designed to encourage reasoning, that does significantly better. When we
vary the way in which the test questions and training data differ, we find that our model is notably proficient at certain forms of generalisation, but notably weak at others. We further show that the model's ability to
generalise improves markedly if it is trained to predict symbolic explanations for its answers. Altogether, we introduce and explore ways to both measure and induce stronger abstract reasoning in neural networks. Our
freely-available dataset should motivate further progress in this direction.},
archivePrefix = {arXiv},
arxivId = {1807.04225},
author = {Barrett, David G. T. and Hill, Felix and Santoro, Adam and Morcos, Ari S. and Lillicrap, Timothy},
eprint = {1807.04225},
issn = {1938-7228},
title = {{Measuring abstract reasoning in neural networks}},
year = {2018}
}
@techreport{VishwaVidyapeetham2012,
author = {{Vishwa Vidyapeetham}, Amrita},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Vishwa Vidyapeetham - 2012 - Fuzzy Based Machine Learning A Promising Approach.pdf:pdf},
title = {{Fuzzy Based Machine Learning: A Promising Approach}},
year = {2012}
}
@inproceedings{Frank1996,
author = {Frank, Lebourgeois and Hubert, Emptoz},
booktitle = {Proceedings of 13th International Conference on Pattern Recognition},
pages = {256----260},
title = {{Pretopological approach for supervised learning}},
volume = {4},
year = {1996}
}
@article{Blockeel1998,
abstract = {A first-order framework for top-down induction of logical decision trees is introduced. The expressivity of these trees is shown to be larger than that of the flat logic programs which are typically induced by
classical ILP systems, and equal to that of first-order decision lists. These results are related to predicate invention and mixed variable quantification. Finally, an implementation of this framework, the TILDE system, is
presented and empirically evaluated. {\textcopyright} 1998 Elsevier Science B.V. All rights reserved.},
author = {Blockeel, Hendrik and {De Raedt}, Luc},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Blockeel, De Raedt - 1998 - Top-down induction of first-order logical decision trees.pdf:pdf},
issn = {00043702},
journal = {Artificial Intelligence},
month = {may},
number = {1-2},
pages = {285--297},
title = {{Top-down induction of first-order logical decision trees}},
volume = {101},
year = {1998}
}
@incollection{Supek2017,
abstract = {{\textcopyright} The Author(s) 2017. Contemporary techniques in biology produce readouts for large numbers of genes simultaneously, the typical example being differential gene expression measurements. Moreover,
those genes are often richly annotated using GO terms that describe gene function and that can be used to summarize the results of the genome-scale experiments. However, making sense of such GO enrichment analyses may be
challenging. For instance, overrepresented GO functions in a set of differentially expressed genes are typically output as a flat list, a format not adequate to capture the complexities of the hierarchical structure of the GO
annotation labels. In this chapter, we survey various methods to visualize large, difficult-to-interpret lists of GO terms. We catalog their availability-Web-based or standalone, the main principles they employ in summarizing
large lists of GO terms, and the visualization styles they support. These brief commentaries on each software are intended as a helpful inventory, rather than comprehensive descriptions of the underlying algorithms. Instead,
we show examples of their use and suggest that the choice of an appropriate visualization tool may be crucial to the utility of GO in biological discovery.},
author = {Supek, Fran and {\v{S}}kunca, Nives},
booktitle = {Methods in Molecular Biology},
doi = {10.1007/978-1-4939-3743-1_15},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Supek, {\v{S}}kunca - 2017 - Visualizing GO annotations.pdf:pdf},
issn = {10643745},
keywords = {Enrichment,Gene ontology,Interpretation,Redundancy,Tools,Visualization},
pages = {207--220},
publisher = {Humana Press Inc.},
title = {{Visualizing GO annotations}},
volume = {1446},
year = {2017}
}
@inproceedings{Katz2017,
abstract = {Deep neural networks have emerged as a widely used and effective means for tackling complex, real-world problems. However, a major obstacle in applying them to safety-critical systems is the great difficulty in
providing formal guarantees about their behavior. We present a novel, scalable, and efficient technique for verifying properties of deep neural networks (or providing counter-examples). The technique is based on the simplex
method, extended to handle the non-convex Rectified Linear Unit (ReLU) activation function, which is a crucial ingredient in many modern neural networks. The verification procedure tackles neural networks as a whole, without
making any simplifying assumptions. We evaluated our technique on a prototype deep neural network implementation of the next-generation airborne collision avoidance system for unmanned aircraft (ACAS Xu). Results show that
our technique can successfully prove properties of networks that are an order of magnitude larger than the largest networks verified using existing methods.},
archivePrefix = {arXiv},
arxivId = {1702.01135},
author = {Katz, Guy and Barrett, Clark and Dill, David L. and Julian, Kyle and Kochenderfer, Mykel J.},
booktitle = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
doi = {10.1007/978-3-319-63387-9_5},
eprint = {1702.01135},
isbn = {9783319633862},
issn = {16113349},
pages = {97--117},
pmid = {4520227},
title = {{Reluplex: An efficient smt solver for verifying deep neural networks}},
volume = {10426 LNCS},
year = {2017}
}
@inproceedings{Petrucci2016,
abstract = {{\textcopyright} Springer International Publishing AG 2016. Recent developments in the area of deep learning have been proved extremely beneficial for several natural language processing tasks, such as sentiment
analysis, question answering, and machine translation. In this paper we exploit such advances by tailoring the ontology learning problem as a transductive reasoning task that learns to convert knowledge from natural language
to a logic-based specification. More precisely, using a sample of definitory sentences generated starting by a synthetic grammar, we trained Recurrent Neural Network (RNN) based architectures to extract OWL formulae from
text. In addition to the low feature engineering costs, our system shows good generalisation capabilities over the lexicon and the syntactic structure. The encouraging results obtained in the paper provide a first evidence of
the potential of deep learning techniques towards long term ontology learning challenges such as improving domain independence, reducing engineering costs, and dealing with variable language forms.},
author = {Petrucci, Giulio and Ghidini, Chiara and Rospocher, Marco},
booktitle = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
doi = {10.1007/978-3-319-49004-5_31},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Petrucci, Ghidini, Rospocher - 2016 - Ontology learning in the deep.pdf:pdf},
isbn = {9783319490038},
issn = {16113349},
pages = {480--495},
publisher = {Springer Verlag},
title = {{Ontology learning in the deep}},
volume = {10024 LNAI},
year = {2016}
}
@article{Doshi-Velez2017,
abstract = {As machine learning systems become ubiquitous, there has been a surge of interest in interpretable machine learning: systems that provide explanation for their outputs. These explanations are often used to
qualitatively assess other criteria such as safety or non-discrimination. However, despite the interest in interpretability, there is very little consensus on what interpretable machine learning is and how it should be
measured. In this position paper, we first define interpretability and describe when interpretability is needed (and when it is not). Next, we suggest a taxonomy for rigorous evaluation and expose open questions towards a
more rigorous science of interpretable machine learning.},
archivePrefix = {arXiv},
arxivId = {1702.08608},
author = {Doshi-Velez, Finale and Kim, Been},
eprint = {1702.08608},
number = {Ml},
pages = {1--13},
title = {{Towards A Rigorous Science of Interpretable Machine Learning}},
year = {2017}
}
@article{LeCun1998,
author = {LeCun, Yann and Bottou, L{\'{e}}on and Bengio, Yoshua and Haffner, Patrick},
journal = {Proceedings of the IEEE},
number = {11},
pages = {2278--2324},
publisher = {Ieee},
title = {{Gradient-based learning applied to document recognition}},
volume = {86},
year = {1998}
}
@article{LeCun2010,
author = {LeCun, Yann and Cortes, Corinna},
howpublished = {http://yann.lecun.com/exdb/mnist/},
keywords = {MSc {\_}checked character{\_}recognition mnist network n},
title = {{MNIST handwritten digit database}},
year = {2010}
}
@article{Fisher1936,
author = {Fisher, Ronald A},
journal = {Annals of eugenics},
number = {2},
pages = {179--188},
publisher = {Wiley Online Library},
title = {{The use of multiple measurements in taxonomic problems}},
volume = {7},
year = {1936}
}
@inproceedings{Sinha2019,
author = {Sinha, Uttaran and Joshi, Saurabh and Balasubramanian, Vineeth N},
booktitle = {ICML Workshop on Uncertainty {\&} Robustness in Deep Learning},
title = {{Defending deep neural networks against structural pertubations}},
year = {2019}
}
@inproceedings{Srinivasan2019,
author = {Srinivasan, Vignesh and Marban, Arturo and M{\"{u}}ller, Klaus-Robert and Samek, Wojciech and Nakajima, Shinichi},
booktitle = {ICML Workshop on Uncertainty {\&} Robustness in Deep Learning},
title = {{Defense against adversarial attacks by langevin dynamics}},
year = {2019}
}
@article{Ahmad2019,
author = {Ahmad, Subutai and Scheinkman, Luiz},
journal = {ICML Workshop on Uncertainty {\&} Robustness in Deep Learning},
title = {{How can we be so dense? The robustness of highly sparse representations}},
year = {2019}
}
@article{Brahma2016,
abstract = {Deep hierarchical representations of the data have been found out to provide better informative features for several machine learning applications. In addition, multilayer neural networks surprisingly tend to
achieve better performance when they are subject to an unsupervised pretraining. The booming of deep learning motivates researchers to identify the factors that contribute to its success. One possible reason identified is the
flattening of manifold-shaped data in higher layers of neural networks. However, it is not clear how to measure the flattening of such manifold-shaped data and what amount of flattening a deep neural network can achieve. For
the first time, this paper provides quantitative evidence to validate the flattening hypothesis. To achieve this, we propose a few quantities for measuring manifold entanglement under certain assumptions and conduct
experiments with both synthetic and real-world data. Our experimental results validate the proposition and lead to new insights on deep learning.},
author = {Brahma, Pratik Prabhanjan and Wu, Dapeng and She, Yiyuan},
file = {::},
journal = {IEEE Transactions on Neural Networks and Learning Systems},
keywords = {Deep learning,disentanglement,manifold learning,unsupervised feature transformation},
pages = {1997--2008},
title = {{Why deep learning works: A manifold disentanglement perspective}},
volume = {27},
year = {2016}
}
@book{Schubert1968,
author = {Schubert, Horst},
publisher = {Macdonald {\&} Company},
title = {{Topology}},
year = {1968}
}
@inproceedings{Narayanan2010,
abstract = {The hypothesis that high dimensional data tends to lie in the vicinity of a low dimensional manifold is the basis of a collection of methodologies termed Manifold Learning. In this paper, we study statistical
aspects of the question of fitting a manifold with a nearly optimal least squared error. Given upper bounds on the dimension, volume, and curvature, we show that Empirical Risk Minimization can produce a nearly optimal
manifold using a number of random samples that is independent of the ambient dimension of the space in which data lie. We obtain an upper bound on the required number of samples that depends polynomially on the curvature,
exponentially on the intrinsic dimension, and linearly on the intrinsic volume. For constant error, we prove a matching minimax lower bound on the sample complexity that shows that this dependence on intrinsic dimension,
volume and curvature is unavoidable. Whether the known lower bound of O(k 2},
author = {Narayanan, Hariharan and Mitter, Sanjoy},
booktitle = {Advances in neural information processing systems},
file = {::},
pages = {1786----1794},
title = {{Sample Complexity of Testing the Manifold Hypothesis}},
year = {2010}
}
@article{Grosse2017,
abstract = {Machine Learning (ML) models are applied in a variety of tasks such as network intrusion detection or Malware classification. Yet, these models are vulnerable to a class of malicious inputs known as adversarial
examples. These are slightly perturbed inputs that are classified incorrectly by the ML model. The mitigation of these adversarial inputs remains an open problem. As a step towards understanding adversarial examples, we show
that they are not drawn from the same distribution than the original data, and can thus be detected using statistical tests. Using thus knowledge, we introduce a complimentary approach to identify specific inputs that are
adversarial. Specifically, we augment our ML model with an additional output, in which the model is trained to classify all adversarial inputs. We evaluate our approach on multiple adversarial example crafting methods
(including the fast gradient sign and saliency map methods) with several datasets. The statistical test flags sample sets containing adversarial inputs confidently at sample sizes between 10 and 100 data points. Furthermore,
our augmented model either detects adversarial examples as outliers with high accuracy ({\textgreater} 80{\%}) or increases the adversary's cost - the perturbation added - by more than 150{\%}. In this way, we show that
statistical properties of adversarial examples are essential to their detection.},
archivePrefix = {arXiv},
arxivId = {1702.06280},
author = {Grosse, Kathrin and Manoharan, Praveen and Papernot, Nicolas and Backes, Michael and McDaniel, Patrick},
eprint = {1702.06280},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Grosse et al. - 2017 - On the (Statistical) Detection of Adversarial Examples.pdf:pdf},
title = {{On the (Statistical) Detection of Adversarial Examples}},
year = {2017}
}
@book{Goodfellow-et-al-2016,
annote = {$\backslash$url{\{}http://www.deeplearningbook.org{\}}},
author = {Goodfellow, Ian and Bengio, Yoshua and Courville, Aaron},
publisher = {MIT Press},
title = {{Deep Learning}},
year = {2016}
}
@article{Gong2016,
abstract = {In this study, we developed a method named GoFDR for predicting Gene Ontology (GO)-based protein functions. The input for GoFDR is simply a query sequence-based multiple sequence alignment (MSA) produced by
PSI-BLAST. For each GO term annotated to the sequences in the MSA, GoFDR identifies a number of functionally discriminating residues (FDRs) specific to the GO term, and scores the query sequence using a position specific
scoring matrix (PSSM) constructed for the FDRs. The raw score is then converted into a probability score according to a score-to-probability table prepared from training sequences. GoFDR outperformed three sequence-based
methods for predicting GO functions in a benchmark of 18,520 sequences. In addition, GoFDR was ranked one of the top methods according to the preliminary evaluation report released by the 2nd Critical Assessment of Function
Annotation (CAFA2) project. Finally, we applied GoFDR to the complete human proteome sequences, and showed that the predictions made by GoFDR with high confidence significantly expanded current annotations of human proteome.
As such, GoFDR is of great value not only for annotating protein functions in newly sequenced genomes, but also for characterizing the function of proteins of interest.},
author = {Gong, Qingtian and Ning, Wei and Tian, Weidong},
doi = {10.1016/j.ymeth.2015.08.009},
issn = {10959130},
journal = {Methods},
keywords = {Function prediction,Functional discriminating residues (FDRs),GO annotation,PSI-BLAST alignment,Raw score adjustment},
title = {{GoFDR: A sequence alignment based method for predicting protein functions}},
year = {2016}
}
@article{Aze2012,
author = {Az{\'{e}}, J{\'{e}}r{\^{o}}me and Prot{\'{e}}ine-prot{\'{e}}ine, Amarrage},
title = {{Pr{\'{e}}diction d ' Interactions et Amarrage Prot{\'{e}}ine-Prot{\'{e}}ine par combinaison de classifieurs To cite this version : ´ diction d ' Interactions et Pr e ´ ine-Prot e ´ ine par Amarrage Prot e
combinaison de classifieurs}},
year = {2012}
}
@article{Altschul1997,
abstract = {The BLAST programs are widely used tools for searching protein and DNA databases for sequence similarities. For protein comparisons, a variety of definitional, algorithmic and statistical refinements described
here permits the execution time of the BLAST programs to be decreased substantially while enhancing their sensitivity to weak similarities. A new criterion for triggering the extension of word hits, combined with a new
heuristic for generating gapped alignments, yields a gapped BLAST program that runs at approximately three times the speed of the original. In addition, a method is introduced for automatically combining statistically
significant alignments produced by BLAST into a position-specific score matrix, and searching the database using this matrix. The resulting Position-Specific Iterated BLAST (PSI-BLAST) program runs at approximately the same
speed per iteration as gapped BLAST, but in many cases is much more sensitive to weak but biologically relevant sequence similarities. PSI-BLAST is used to uncover several new and interesting members of the BRCT
superfamily.},
author = {Altschul, Stephen F. and Madden, Thomas L. and Sch{\"{a}}ffer, Alejandro A. and Zhang, Jinghui and Zhang, Zheng and Miller, Webb and Lipman, David J.},
doi = {10.1093/nar/25.17.3389},
issn = {03051048},
journal = {Nucleic Acids Research},
number = {17},
pages = {3389--3402},
pmid = {9254694},
title = {{Gapped BLAST and PSI-BLAST: A new generation of protein database search programs}},
volume = {25},
year = {1997}
}
@article{Lin2015,
abstract = {{\textcopyright} Copyright 2015, Association for the Advancement of Artificial Intelligence (www.aaa1.org). All rights reserved. Knowledge graph completion aims to perform link prediction between entities. In this
paper, we consider the approach of knowledge graph embeddings. Recently, models such as TransE and TransH build entity and relation embeddings by regarding a relation as translation from head entity to tail entity. We note
that these models simply put both entities and relations within the same semantic space. In fact, an entity may have multiple aspects and various relations may focus on different aspects of entities, which makes a common
space insufficient for modeling. In this paper, we propose TransR to build entity and relation embeddings in separate entity space and relation spaces. Afterwards, we learn embeddings by first projecting entities from entity
space to corresponding relation space and then building translations between projected entities. In experiments, we evaluate our models on three tasks including link prediction, triple classification and relational fact
extraction. Experimental results show significant and consistent improvements compared to state-of-the-art baselines including TransE and TransH. The source code of this paper can be obtained from
https://github.com/mrlyk423/relation-extraction.},
author = {Lin, Yankai and Liu, Zhiyuan and Sun, Maosong and Liu, Yang and Zhu, Xuan},
isbn = {9781577357018},
journal = {Proceedings of the National Conference on Artificial Intelligence},
keywords = {NLP and Knowledge Representation Track},
pages = {2181--2187},
title = {{Learning entity and relation embeddings for knowledge graph completion}},
volume = {3},
year = {2015}
}
@article{Shi2017,
abstract = {With the large volume of new information created every day, determining the validity of information in a knowledge graph and filling in its missing parts are crucial tasks for many researchers and practitioners.
To address this challenge, a number of knowledge graph completion methods have been developed using low-dimensional graph embeddings. Although researchers continue to improve these models using an increasingly complex feature
space, we show that simple changes in the architecture of the underlying model can outperform state-of-the-art models without the need for complex feature engineering. In this work, we present a shared variable neural network
model called ProjE that fills-in missing information in a knowledge graph by learning joint embeddings of the knowledge graph's entities and edges, and through subtle, but important, changes to the standard loss function. In
doing so, ProjE has a parameter size that is smaller than 11 out of 15 existing methods while performing {\$}37\backslash{\%}{\$} better than the current-best method on standard datasets. We also show, via a new fact checking
task, that ProjE is capable of accurately determining the veracity of many declarative statements.},
annote = {Why am I reading this?

Abstract: I would like to find out if the authors propose a good method for creating relations between nodes on a graph. It is not immediately clear if this is in the scope of the their work just from the abstract alone.

What are the authors trying to do in writing this?

Abstract: They are trying to show that simple changes such as joint embeddings of the entities and edges, in addition to the standard loss function (I don't yet know what they consider standard) can not only improve the
accuracy of knowledge graph completetion (filling in missing parts of a graph) but also drastically reduce the parameter size of the model.

What are the authors saying that is relevant to what I want to find out?

Abstract: The joint embeddings of the edges could present some interest to my work in creating existential relations.

How convincing is what the authors are saying?

Abstract: From the abstract they give some metrics such as 37{\%} better performance, and smaller than 11 out of the 15 existing models. Though, I need to know what they are considering for performance. Morover, does smaller
necessarilary mean faster?

In conclusion, what use can I make of this?

Abstract: I am only interested in the feature relations, and since this is not the main focus of the article, I will just look for the relevant sections.},
author = {Shi, Baoxu and Weninger, Tim},
journal = {31st AAAI Conference on Artificial Intelligence, AAAI 2017},
keywords = {Knowledge Representation and Reasoning},
number = {i},
pages = {1236--1242},
title = {{ProjE: Embedding projection for knowledge graph completion}},
year = {2017}
}
@techreport{Herzig2018,
author = {Herzig, Andreas and Baader, Franz and Calvanese, Diego and Mcguinness, Deborah L and Nardi, Daniele and Patel-Schneider, Peter F and Horrocks, Ian and Lutz, Carsten and Sattler, Ulrike},
file = {::},
title = {{Ontologies et web s{\'{e}}mantique}},
year = {2018}
}
@techreport{Franconi,
author = {Franconi, Enrico},
file = {::},
title = {{Description Logics Propositional Description Logics}},
}
@article{Klement2004,
abstract = {We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan
triples are also mentioned. Finally, a brief historical survey on triangular norms is given. {\textcopyright} 2003 Elsevier B.V. All rights reserved.},
author = {Klement, Erich Peter and Mesiar, Radko and Pap, Endre},
doi = {10.1016/j.fss.2003.06.007},
issn = {01650114},
journal = {Fuzzy Sets and Systems},
keywords = {Triangular norms},
number = {1},
pages = {5--26},
title = {{Triangular norms. Position paper I: Basic analytical and algebraic properties}},
volume = {143},
year = {2004}
}
@article{Zadeh1965,
author = {Zadeh, A and Riid, A and Mendel, M and Klir, J and Folger, A and Allendoerfer, B and Oakley, O and Zadeh, Lotfi A},
title = {{Fundamentals 2}},
year = {1965}
}
@article{Bobillo2016,
abstract = {Classical, two-valued, ontologies have been successfully applied to represent the knowledge in many domains. However, it has been pointed out that they are not suitable in domains where vague or imprecise pieces
of information play an important role. To overcome this limitation, several extensions to classical ontologies based on fuzzy logic have been proposed. We believe, however, that the success of fuzzy ontologies strongly
depends on the availability of effective reasoners able to deal with fuzzy ontologies. In this paper we describe fuzzyDL, an expressive fuzzy ontology reasoner with some unique features. We discuss its possibilities for fuzzy
ontology representation, the supported reasoning services, the different interfaces to interact with it, some implementation details, a comparison with other fuzzy ontology reasoners, and an overview of the main applications
that have used it so far.},
author = {Bobillo, Fernando and Straccia, Umberto},
doi = {10.1016/j.knosys.2015.11.017},
file = {::},
issn = {09507051},
journal = {Knowledge-Based Systems},
keywords = {Fuzzy description logics,Fuzzy ontologies,Fuzzy ontology reasoner},
month = {mar},
pages = {12--34},
publisher = {Elsevier},
title = {{The fuzzy ontology reasoner fuzzyDL}},
volume = {95},
year = {2016}
}
@techreport{Bobillo,
abstract = {In this paper we present fuzzyDL, an expressive fuzzy Description Logic reasoner. We present its salient features, including some novel concept constructs and queries, and examples of use cases: matchmaking and
fuzzy control.},
author = {Bobillo, Fernando and Straccia, Umberto},
file = {::},
title = {{fuzzyDL: An Expressive Fuzzy Description Logic Reasoner}}
}
@techreport{Stoilos2007,
abstract = {It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge
representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well
known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). We illustrate why transitive role axioms are
difficult to handle in the presence of fuzzy interpretations and how to handle them properly. Then we extend these results by adding role hierarchies and finally number restrictions. The main contributions of the paper are
the decidability proof of the fuzzy DL languages fuzzy-SI and fuzzy-SHIN , as well as decision procedures for the knowledge base satisfiability problem of the fuzzy-SI and fuzzy-SHIN .},
author = {Stoilos, Giorgos and Stamou, Giorgos and Pan, Jeff Z and Tzouvaras, Vassilis and Horrocks, Ian},
booktitle = {Journal of Artificial Intelligence Research},
file = {::},
pages = {273--320},
title = {{Reasoning with Very Expressive Fuzzy Description Logics}},
volume = {30},
year = {2007}
}
@techreport{Sanchez,
abstract = {In this paper we introduce reasoning procedures for ALCQ + F , a fuzzy description logic with extended qualified quantification. The language allows for the definition of fuzzy quantifiers of the absolute and
relative kind by means of piecewise linear functions on N and Q ∩ [0, 1] respectively. In order to reason about instances, the semantics of quantified expressions is defined based on recently developed measures of the
cardinality of fuzzy sets. A procedure is described to calculate the fuzzy satisfiability of a fuzzy assertion, which is a very important reasoning task. The procedure considers several different cases and provides direct
solutions for the most frequent types of fuzzy assertions.},
author = {S{\'{a}}nchez, Daniel and Tettamanzi, Andrea G B},
file = {::},
title = {{Reasoning and Quantification in Fuzzy Description Logics}}
}
@techreport{Lukasiewicz,
abstract = {Vagueness and imprecision abound in multimedia information processing and retrieval. In this paper, we present an approach to fuzzy description logic programs under the answer set semantics for the Semantic Web,
which is an integration of description logics with nonmonotonic logic programs under the answer set semantics (with default negation in rule bodies) that also allows for representing and reasoning with vagueness and
imprecision. More concretely , we define a canonical semantics of positive and stratified fuzzy dl-programs in terms of a unique least model and iterative least models, respectively. We then define the answer set semantics of
general fuzzy dl-programs, and show in particular that all answer sets of a fuzzy dl-program are minimal models, and that the answer set semantics of positive and stratified fuzzy dl-programs coincides with their canonical
least model and iterative least model semantics, respectively. Furthermore, we also provide a characterization of the canonical semantics of positive and stratified fuzzy dl-programs in terms of a fixpoint and an iterative
fixpoint semantics, respectively.},
author = {Lukasiewicz, Thomas},
file = {::},
title = {{Fuzzy Description Logic Programs under the Answer Set Semantics for the Semantic Web}}
}
@techreport{Tsatsou,
abstract = {In this paper we present LiFR, a lightweight DL reasoner capable of performing in resource-constrained devices, that implements a fuzzy extension of Description Logic Programs. Preliminary evaluation against two
existing fuzzy DL reasoners and within a real-world use case has shown promising results.},
author = {Tsatsou, Dorothea and Dasiopoulou, Stamatia and Kompatsiaris, Ioannis and Mezaris, Vasileios},
file = {::},
title = {{LiFR: A Lightweight Fuzzy DL Reasoner}},
}
@techreport{Turksen1989,
abstract = {An interval-valued fuzzy set approach is proposed for approximate reasoning. There are a number of possible interpretations of the interval-valued fuzzy modus ponens. Four of these are discussed. Sufficiency
conditions are identified in order to reach a conclusion similar to the classical result. Numerical examples are given where appropriate. A prototype study is reviewed that shows an application of the interval-valued fuzzy
set approach to aggregate production planning. The results of this approach are shown to fairly well approximate the results of currently accepted methods of aggregate production planning. Furthermore, the approach provides a
much more "'user friendly'" interface.},
author = {Turksen, I B},
file = {::},
keywords = {approximate reasoning,crisp connectives,interval-valued fuzzy sets,linguistic connectives,modus ponens,production planning,sufficiency condi-tions},
title = {{Four Methods of Approximate Reasoning with Interval-Valued Fuzzy Sets}},
year = {1989}
}
@article{Gasmi2016,
abstract = {A DF-ALC (Decomposing fuzzy ALC) is proposed in this paper to satisfy the need for representing and reasoning with fuzzy ontologies in the context of semantic Web. A DF-ALC is also proposed to satisfy the need for
seeing the necessity of decomposing ontology into several sub-ontologies in order to optimize the fuzzy reasoning process. The main contribution of this work is to decompose the axioms of the ontology into sub-axioms
according to a degree of certainty which is assigned to the fuzzy concepts and roles. It is also to define the syntax and semantics and to propose a local reasoning algorithm and a way of using gateways to infer between local
TBox.},
author = {Gasmi, Mohamed and Bourahla, Mustapha},
doi = {10.1016/j.jides.2016.05.003},
file = {::},
issn = {23526645},
journal = {Journal of Innovation in Digital Ecosystems},
month = {jun},
number = {1},
pages = {30--36},
publisher = {Elsevier BV},
title = {{Reasoning over decomposing fuzzy description logic}},
volume = {3},
year = {2016}
}
@techreport{Guevara2014,
abstract = {Embedding non-vectorial data into a normed vec-torial space is very common in machine learning, aiming to perform tasks such classification, regression, clustering and so on. Fuzzy datasets or datasets whose
observations are fuzzy sets, are an example of non-vectorial data and, many of fuzzy pattern recognition algorithms analyze them in the space formed by the set of fuzzy sets. However, the analysis of fuzzy data in such space
has the limitation of not being a vectorial space. To overcome such limitation, in this work, we propose the embedding of fuzzy data into a proper Hilbert space of functions called the Reproducing Kernel Hilbert Space or
RKHS. This embedding is possible using a positive definite kernel function defined on fuzzy sets. As a result, we present a formulation of a real-valued kernels on fuzzy sets, particularly, we define the intersection kernel
and the cross product kernel on fuzzy sets giving some examples of them using T-norm operators. Also, we analyze the nonsingleton TSK fuzzy kernel and, finally, we gave several examples of kernels on fuzzy sets, that can be
easily constructed from the previous ones.},
author = {Guevara, Jorge and {Hirata Jr}, Roberto and Canu, Stephane and {Etienne du Rouvray France}, St},
file = {::},
keywords = {()},
title = {{Positive Definite Kernel Functions on Fuzzy Sets}},
year = {2014}
}
@techreport{Nickel,
abstract = {Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, state-of-the-art embedding methods typically do not account for latent hierarchical
structures which are characteristic for many complex symbolic datasets. In this work, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space-or more
precisely into an n-dimensional Poincar{\'{e}} ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We
present an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar{\'{e}} embeddings can outperform Euclidean embeddings significantly on data with latent hierarchies,
both in terms of representation capacity and in terms of generalization ability.},
author = {Nickel, Maximilian and Kiela, Douwe},
file = {::},
title = {{Poincar{\'{e}} Embeddings for Learning Hierarchical Representations}}
}
@article{Jayaram2009,
abstract = {Special implications were introduced by H{\'{a}}jek and Kohout [Fuzzy implications and generalized quantifiers, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4 (1996) 225-233] in their investigations on some
statistics on marginals. They have either suggested or only partially answered three important questions, especially related to special implications and residuals of t-norms. In this work we investigate these posers in-depth
and give complete answers. Toward this end, firstly we show that many of the properties considered as part of the definition of special implications are redundant. Then, a geometric interpretation of the specialty property is
given, using which many results and bounds for such implications are obtained. We have obtained a characterization of general binary operations whose residuals become special. Finally, some constructive procedures to obtain
special fuzzy implications are proposed and methods of obtaining special implications from existing ones are given, showing that there are infinitely many fuzzy implications that are special but cannot be obtained as
residuals of t-norms. {\textcopyright} 2008 Elsevier B.V. All rights reserved.},
author = {Jayaram, Balasubramaniam and Mesiar, Radko},
doi = {10.1016/j.fss.2008.11.004},
file = {::},
issn = {01650114},
journal = {Fuzzy Sets and Systems},
keywords = {(S,N)-implication,1-Lipschitzianity,Fuzzy connectives,Fuzzy implication,Residual of conjunction,Special implication},
month = {jul},
number = {14},
pages = {2063--2085},
title = {{On special fuzzy implications}},
volume = {160},
year = {2009}
}
@book{Dessimoz,
abstract = {Primer on Ontologies -- The Gene Ontology and the Meaning of Biological Function -- Primer on the Gene Ontology -- Best Practices in Manual Annotation with the Gene Ontology -- Computational Methods for Annotation
Transfers from Sequence -- Text Mining to Support Gene Ontology Curation and Vice Versa -- How Does the Scientific Community Contribute to Gene Ontology? -- Evaluating Computational Gene Ontology Annotations -- Evaluating
Functional Annotations of Enzymes Using the Gene Ontology -- Community-Based Evaluation of Computational Function Prediction -- Get GO!: Retrieving GO Data Using AmiGO, QuickGO, API, Files, and Tools -- Semantic Similarity in
the Gene Ontology -- Gene-Category Analysis -- Gene Ontology: Pitfalls, Biases, and Remedies -- Visualizing GO Annotations -- A Gene Ontology Tutorial in Python -- Annotation Extensions -- The Evidence and Conclusion Ontology
(ECO): Supporting GO Annotations -- Complementary Sources of Protein Functional Information: The Far Side of GO -- Integrating Bio-Ontologies and Controlled Clinical Terminologies: From Base Pairs to Bedside Phenotypes -- The
Vision and Challenges of the Gene Ontology.},
author = {Dessimoz, Christophe and Škunca, Nives},
file = {::},
isbn = {9781493937417},
pages = {305},
title = {{The gene ontology handbook}}
}
@inproceedings{Hoehndorf2016,
abstract = {A large number of biomedical resources have been developed to represent the functions of biological entities, and these resources are widely used for data integration and analysis. Expressing functions in
biomedical ontologies currently uses formal representation patterns that renders basic reasoning tasks to fall in complexity classes beyond polynomial time, thereby limiting the potential of using knowledge-based methods for
data integration, querying or quality control. Here, we propose an alternative representation pattern for expressing knowledge about biological functions, together with a biological and ontological justification, which can be
expressed using the description logic EL++ and implemented using the OWL 2 EL profile. To demonstrate the utility of our account of biological functions, we apply it to all proteins contained in the SwissProt database and
evaluate its utility with respect to answering complex queries as well with respect to the classification and query times.},
author = {Hoehndorf, Robert and Mencel, Liam and Gkoutos, Georgios V. and Schofield, Paul N.},
booktitle = {Frontiers in Artificial Intelligence and Applications},
doi = {10.3233/978-1-61499-660-6-299},
file = {::},
isbn = {9781614996590},
issn = {09226389},
keywords = {Big ontologies,Biological function,Protein,Tractable reasoning},
pages = {299--312},
publisher = {IOS Press},
title = {{Large-scale reasoning over functions in biomedical ontologies}},
volume = {283},
year = {2016}
}
@techreport{Goebel,
author = {Goebel, Randy and Wahlster, Wolfgang and Siekmann, Joerg},
file = {::},
title = {{Lecture Notes in Artificial Intelligence 7364 Subseries of Lecture Notes in Computer Science LNAI Series Editors LNAI Founding Series Editor}}
}
@techreport{Straccia2001,
abstract = {Description Logics (DLs) are suitable, well-known, logics for managing structured knowledge. They allow reasoning about individuals and well defined concepts, i.e. set of individuals with common properties. The
experience in using DLs in applications has shown that in many cases we would like to extend their capabilities. In particular, their use in the context of Multimedia Information Retrieval (MIR) leads to the convincement that
such DLs should allow the treatment of the inherent imprecision in multimedia object content representation and retrieval. In this paper we will present a fuzzy extension of ALC, combining Zadeh's fuzzy logic with a classical
DL. In particular, concepts becomes fuzzy and, thus, reasoning about imprecise concepts is supported. We will define its syntax, its semantics, describe its properties and present a constraint propagation calculus for
reasoning in it.},
author = {Straccia, Umberto},
booktitle = {Journal of Artificial Intelligence Research},
file = {::},
pages = {137--166},
title = {{Reasoning within Fuzzy Description Logics}},
volume = {14},
year = {2001}
}
@inproceedings{Serafini2017,
abstract = {Copyright 2017 ACM. This paper presents a revision of Real Logic and its implementation with Logic Tensor Networks and its application to Semantic Image Interpretation. Real Logic is a framework where learning
from numerical data and logical reasoning are integrated using first order logic syntax. The symbols of the signature of Real Logic are interpreted in the data-space, i.e, on the domain of real numbers. The integration of
learning and reasoning obtained in Real Logic allows us to formalize learning as approximate satisfiability in the presence of logical constraints, and to perform inference on symbolic and numerical data. After introducing a
refined version of the formalism, we describe its implementation into Logic Tensor Networks which uses deep learning within Google's TensorFlowTM. We evaluate LTN on the task of classifying objects and their parts in images,
where we combine state-of-the-art-object detectors with a part-of ontology. LTN outperforms the state-of-the-art on object classification, and improves the performances on part-of relation detection with respect to a
rule-based baseline.},
author = {Serafini, Luciano and Donadello, Ivan and {D'Avila Garcez}, Artur},
booktitle = {Proceedings of the ACM Symposium on Applied Computing},
doi = {10.1145/3019612.3019642},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Serafini, Donadello, D'Avila Garcez - 2017 - Learning and reasoning in logic tensor networks Theory and application to semantic image
in.pdf:pdf},
isbn = {9781450344869},
keywords = {Learning with logical constraints,Logic and neural networks},
month = {apr},
pages = {125--130},
publisher = {Association for Computing Machinery},
title = {{Learning and reasoning in logic tensor networks: Theory and application to semantic image interpretation}},
volume = {Part F1280},
year = {2017}
}
@article{FanFenlei,
author = {{Fan Fenlei} and {Wang Ge}},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Fan Fenlei, Wang Ge - Unknown - Fuzzy Logic Interpretation of Quadratic Networks.pdf:pdf},
title = {{Fuzzy Logic Interpretation of Quadratic Networks}}
}
@misc{,
title = {http://www.umbertostraccia.it/cs/}
}
@techreport{Serratosa2000,
author = {Serratosa, Francesc and Director, Casanelles and Alberto, :},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Serratosa, Director, Alberto - 2000 - UNIVERSITAT POLIT{\`{E}}CNICA DE CATALUNYA FUNCTION-DESCRIBED GRAPHS FOR STRUCTURAL PATTERN
RECOGNITION.pdf:pdf},
title = {{UNIVERSITAT POLIT{\`{E}}CNICA DE CATALUNYA FUNCTION-DESCRIBED GRAPHS FOR STRUCTURAL PATTERN RECOGNITION}},
year = {2000}
}
@book{Tan,
abstract = {Held jointly with Extending Database Technology (EDBT 2013).},
author = {Tan, Wang-Chiew and {Association for Computing Machinery} and {ACM Digital Library.} and {International Conference on Extending Database Technology (16th : 2013 : Genoa}, Italy)},
file = {:home/jaymorgan/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Tan et al. - Unknown - Database Theory ICDT 2013 16th International Conference on Database Theory Genoa, Italy, March 18-22, 2013
pr.pdf:pdf},
isbn = {9781450315982},
pages = {301},
title = {{Database Theory : ICDT 2013 : 16th International Conference on Database Theory : Genoa, Italy, March 18-22, 2013 : proceedings}}
}
@article{Wang2012,
author = {Wang, Jianzhong and Wang, Jianzhong},
doi = {10.1007/978-3-642-27497-8_8},
journal = {Geometric Structure of High-Dimensional Data and Dimensionality Reduction},
number = {December},
pages = {151--180},
title = {{Isomaps}},
volume = {290},
year = {2012}
}
@article{Bengio2009,
author = {Bengio, Yoshua and Vincent, Pascal and Delalleau, Olivier and Roux, Nicolas Le and Ouimet, Marie},
pages = {1--8},
title = {{Bengio{\_}Extension{\_}Nips{\_}2003}},
year = {2009}
}
@article{Erichson2018,
abstract = {Diffusion maps are an emerging data-driven technique for non-linear dimensionality reduction, which are especially useful for the analysis of coherent structures and nonlinear embeddings of dynamical systems.
However, the computational complexity of the diffusion maps algorithm scales with the number of observations. Thus, long time-series data presents a significant challenge for fast and efficient embedding. We propose
integrating the Nystr$\backslash$"om method with diffusion maps in order to ease the computational demand. We achieve a speedup of roughly two to four times when approximating the dominant diffusion map components.},
archivePrefix = {arXiv},
arxivId = {1802.08762},
author = {Erichson, N. Benjamin and Mathelin, Lionel and Brunton, Steven L. and Kutz, J. Nathan},
eprint = {1802.08762},
title = {{Diffusion Maps meet Nystr$\backslash$"om}},
year = {2018}
}
@article{MendozaQuispe2016,
abstract = {Manifold learning techniques have shown a great potential for computer vision problems; however, they do not extend easily to points different from the ones on which they were trained (out-of-sample). On the other
hand, extreme learning machine (ELM) is a powerful method that allows to perform nonlinear, multivariate regression. This paper discusses the effectiveness of ELM for the out-of-sample problem and compares it to the
state-of-the-art solution: the Nystr{\"{o}}m extension. Both methods are evaluated through the reconstruction of the manifold learnt using Laplacian eigenmaps, via experiments on a wide range of publicly available image
datasets. We show that when reducing the data dimension to its intrinsic dimension, the ELM offers a better approximation of the embedded coordinates, also with reduced computational costs during testing.},
author = {{Mendoza Quispe}, Arturo and Petitjean, Caroline and Heutte, Laurent},
doi = {10.1016/j.patrec.2016.01.024},
issn = {01678655},
journal = {Pattern Recognition Letters},
keywords = {Dimensionality reduction,Extreme learning machine,Laplacian eigenmaps,Manifold learning,Out-of-sample extension},
pages = {68--73},
publisher = {Elsevier B.V.},
title = {{Extreme learning machine for out-of-sample extension in Laplacian eigenmaps}},
volume = {74},
year = {2016}
}
@book{Godinho2014,
abstract = {Many developments in the design and analysis of environmental epidemiology have been made in air pollution studies. In the analysis of the short-term effects of particulate matter on daily mortality, Poisson
regression models with flexible smoothing methods have been developed for the analysis of time-series data. Another option for such studies is the use of case-crossover designs, and there have been extensive discussions on
the selection of control periods. In the Study on Respiratory Disease and Automobile Exhaust project conducted by the Japanese Ministry of the Environment, we adopted a new 2-stage case-control design that is efficient when
both exposure and disease are rare. Based on our experience in conducting air pollution epidemiologic studies, we review 2-stage case-control designs, case-crossover designs, generalized linear models, generalized additive
models, and generalized estimating equations, all of which are useful approaches in environmental epidemiology.},
author = {Godinho, Leonor and Nat{\'{a}}rio, Jos{\'{e}}},
doi = {10.1007/978-3-319-08666-8},
isbn = {978-3-319-08665-1},
title = {{An Introduction to Riemannian Geometry}},
year = {2014}
}
@article{Nagy1946,
author = {Nagy, B{\'{e}}la de Sz},
doi = {10.1007/BF02565964},
issn = {00102571},
journal = {Commentarii Mathematici Helvetici},
number = {1},
pages = {347--366},
title = {{Perturbations des transformations autoadjointes dans l'espace de Hilbert}},
volume = {19},
year = {1946}
}
@article{Schmidt,
abstract = {This report mainly illustrates a way to compute the Laplace-Beltrami-Operator on a Rie- mannian Manifold and gives information to why and where it is used in the Analysis of 3D Shapes. After a brief introduction,
an overview over the necessary properties of manifolds for calculating the Laplacian is given. Furthermore the two operators needed for defining the Laplace-Beltrami-Operator - the gradient and the divergence - are
introduced. Both of them are individually derived from the geometric meaning of the operator in Euclidean space, which then is translated into a Riemannian Manifold. With these operators defined, their formulas are combined
into a specification used to calculate the Laplace-Beltrami-Operator. Finally, some examples for its applications are presented. Most of this report uses [1] as a reference, for this reason it won't be specifically noted.},
author = {Schmidt, Frank and Schmidt, Frank and Schmidt, Frank},
number = {1},
pages = {1--6},
title = {{The Laplace-Beltrami-Operator on Riemannian Manifolds}}
}
@article{Porte2008a,
abstract = {This paper describes a mathematical technique [1] for dealing with dimensionality reduction. Given data in a high-dimensional space, we show how to find parameters that describe the lower-dimensional structures of
which it is comprised. Unlike other popular methods such as Principle Component Analysis and Multi-dimensional Scaling, diffusion maps are non-linear and focus on discovering the underlying manifold (lower-dimensional
constrained “surface” upon which the data is embedded). By integrating local similarities at different scales, a global description of the data-set is obtained. In comparisons, it is shown that the technique is robust to
noise perturbation and is computationally inexpensive. Illustrative examples and an open implementation are given.},
author = {Porte, J De La and Herbst, Bm},
doi = {10019.1/45333},
journal = {Proceedings of the 19th Symposium of the Pattern Recognition Association of South Africa (PRASA 2008)},
number = {November},
pages = {15--25},
title = {{An introduction to diffusion maps}},
year = {2008}
}
@article{Fasshauer2007,
abstract = {Abstract  Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method. In most papers the authors end up choosing this shape parameter by
trial and error or some other ad hoc means. The method of cross validation has long been used in the statistics literature, and the special case of leave-one-out cross validation forms the basis of the algorithm for choosing
an optimal value of the shape parameter proposed by Rippa in the setting of scattered data interpolation with RBFs. We discuss extensions of this approach that can be applied in the setting of iterated approximate moving
least squares approximation of function value data and for RBF pseudo-spectral methods for the solution of partial differential equations. The former method can be viewed as an efficient alternative to ridge regression or
smoothing spline approximation, while the latter forms an extension of the classical polynomial pseudo-spectral approach. Numerical experiments illustrating the use of our algorithms are included.},
author = {Fasshauer, Gregory E. and Zhang, Jack G.},
doi = {10.1007/s11075-007-9072-8},
issn = {10171398},
journal = {Numerical Algorithms},
keywords = {Approximate moving least squares,Cross validation,Pseudo-spectral methods,Radial basis functions,Shape parameter},
number = {1-4},
pages = {345--368},
title = {{On choosing "optimal" shape parameters for RBF approximation}},
volume = {45},
year = {2007}
}
@article{Rippa1999,
abstract = {The accuracy of many schemes for interpolating scattered data with radial basis functions depends on a shape parameter c of the radial basis function. In this paper we study the effect of c on the quality of fit
of the multiquadric, inverse multiquadric and Gaussian interpolants. We show, numerically, that the value of the optimal c (the value of c that minimizes the interpolation error) depends on the number and distribution of data
points, on the data vector, and on the precision of the computation. We present an algorithm for selecting a good value for c that implicitly takes all the above considerations into account. The algorithm selects c by
minimizing a cost function that imitates the error between the radial interpolant and the (unknown) function from which the data vector was sampled. The cost function is defined by taking some norm of the error vector E = (E
1, ... , EN)T where E k = Ek = fk - Sk xk) and S k is the interpolant to a reduced data set obtained by removing the point x k and the corresponding data value f k from the original data set. The cost function can be defined
for any radial basis function and any dimension. We present the results of many numerical experiments involving interpolation of two dimensional data sets by the multiquadric, inverse multiquadric and Gaussian interpolants
and we show that our algorithm consistently produces good values for the parameter c.},
archivePrefix = {arXiv},
arxivId = {arXiv:astro-ph/0005074v1},
author = {Rippa, Shmuel},
doi = {10.1023/A:1018975909870},
eprint = {0005074v1},
isbn = {1019-7168},
issn = {1019-7168},
journal = {Adv. Comp. Math.},
pages = {193--210},
pmid = {1284},
primaryClass = {arXiv:astro-ph},
title = {{An algorithm for selecting a good value for the parameter c in radial basis function interpolation}},
volume = {11},
year = {1999}
}
@inproceedings{Lindenbaum2018,
abstract = {Many generative models attempt to replicate the density of their input data. However, this approach is often undesirable, since data density is highly affected by sampling biases, noise, and artifacts. We propose
a method called SUGAR (Synthesis Using Geometrically Aligned Random-walks) that uses a diffusion process to learn a manifold geometry from the data. Then, it generates new points evenly along the manifold by pulling randomly
generated points into its intrinsic structure using a diffusion kernel. SUGAR equalizes the density along the manifold by selectively generating points in sparse areas of the manifold. We demonstrate how the approach corrects
sampling biases and artifacts, while also revealing intrinsic patterns (e.g. progression) and relations in the data. The method is applicable for correcting missing data, finding hypothetical data points, and learning
relationships between data features.},
author = {Lindenbaum, Ofir and Stanley, Jay S. and Wolf, Guy and Krishnaswamy, Smita},
booktitle = {Advances in Neural Information Processing Systems},
pages = {1400----1411},
title = {{Geometry based data generation}},
year = {2018}
}
@article{Ozakin2009,
abstract = {Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens
exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much lower-dimensional intrinsic
structure. We propose a small modification to ker- nel density estimation for estimating probability density functions on Riemannian submanifolds of Euclidean space. Using ideas from Riemannian geometry, we prove the
consistency of this modified estimator and show that the convergence rate is determined by the intrinsic dimension of the submanifold. We conclude with empirical results demonstrating the behavior predicted by our theory.},
author = {Ozakin, Arkadas and Gray, AG},
journal = {Advances in Neural Information Processing Systems},
pages = {1375----1382},
title = {{Submanifold density estimation}},
year = {2009}
}
@article{Chigirev2004,
abstract = {We introduce an information theoretic method for nonparametric, non- linear dimensionality reduction, based on the infinite cluster limit of rate distortion theory. By constraining the information available to
manifold coordinates, a natural probabilistic map emerges that assigns original data to corresponding points on a lower dimensional manifold. With only the information-distortion trade off as a parameter, our method de-
termines the shape of the manifold, its dimensionality, the probabilistic map and the prior that provide optimal description of the data.},
author = {Chigirev, Denis and Bialek, William},
isbn = {0262201526},
issn = {1049-5258},
journal = {Advances in neural information processing systems 16: proceedings of the 2003 conference},
pages = {161},
title = {{Optimal manifold representation of data: an information theoretic approach}},
year = {2004}
}
@article{cummins2013combining,
author = {Cummins, Lisa},
publisher = {University College Cork},
title = {{Combining and choosing case base maintenance algorithms}},
year = {2013}
}
@article{orriols2010documentation,
author = {Orriols-Puig, Albert and Macia, N{\'{u}}ria and Ho, Tin Kam},
journal = {Universitat Ramon Llull, La Salle},
pages = {1----40},
title = {{Documentation for the data complexity library in C++}},
volume = {196},
year = {2010}
}
@article{Karoubi2017,
abstract = {We introduce the notion of the " covering type " of a space, which is more subtle that the notion of Lusternik Schnirel-man category. It measures the complexity of a space which arises from coverings by
contractible subspaces whose non-empty inter-sections are also contractible. From the point of view of Algebraic Topology, the simplest spaces are the contractible ones. A crude measure of the complexity of a space X is the
size of a finite open covering of X by contractible subspaces; this idea goes back to the work of Lusternick and Schnirelman in 1934. A more subtle invariant is the size of good covers, i.e., covers by con-tractible subspaces
such that each of their non-empty intersections is also contractible. The idea of a good cover appears in a 1952 paper by Andr{\'{e}} Weil [17], but is preceded by Leray's notion of a convexoid cover [12, p. 139] which uses
closed covers with acyclic intersections. We define the covering type of X to be the minimum size of a good cover of a space homotopy equivalent to X. See Definition 1.2 below for a precise description using open covers; for
finite CW complexes, an equivalent version using closed covers by subcomplexes is given in Theorem 2.5. We will see that this is an interesting measure of the complexity of a space. For connected graphs, the covering type is
approximately √ 2h, where h is the number of circuits. Thus it is very different from the Lusternick–Schnirelman category (which is 2 for non-tree graphs), and Farber's topological complexity [4] (which is at most 3 for
graphs). The covering type of a surface is related to its chromatic number. The chromatic number of a surface is the smallest number n such that every map on the surface is n-colorable (see Definition 6.1); it was first
described in 1890 by Haewood [7]. Finding the chromatic number of a surface was long known as the map coloring problem. The chromatic number of the 2-sphere is 4; this is the " four color theorem, " settled in 1976. The other
cases of the map coloring problem were settled in 1968; see [15]. For the 2-sphere, torus and projective plane, the covering type equals the chromatic number: 4, 7 and 6, respectively. Special solutions to the map coloring
problem give an upper bound for the covering type of a surface: it is at most one more than the},
author = {Karoubi, Max and Weibel, Charles},
doi = {10.4171/lem/62-3/4-4},
journal = {L'Enseignement Math{\'{e}}matique},
number = {3},
pages = {457--474},
title = {{On the covering type of a space}},
volume = {62},
year = {2017}
}
@article{Lorena2019,
author = {Lorena, Ana C and Garcia, Lu{\{}$\backslash$'$\backslash$i{\}}s PF and Lehmann, Jens and Souto, Marcilio CP and Ho, Tin Kam},
journal = {ACM Computing Surveys},
number = {5},
pages = {1----34},
title = {{How complex is your classification problem? A survey on measuring classification complexity}},
volume = {52},
year = {2019}
}
@article{Li2018,
abstract = {Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But
how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented
subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement,
computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of
models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the
solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum
problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by
parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing
networks, in some cases by more than 100 times.},
archivePrefix = {arXiv},
arxivId = {1804.08838},
author = {Li, Chunyuan and Farkhoor, Heerad and Liu, Rosanne and Yosinski, Jason},
eprint = {1804.08838},
title = {{Measuring the Intrinsic Dimension of Objective Landscapes}},
year = {2018}
}
@article{Ho2002,
abstract = {We studied a number of measures that characterize the difficulty of a classification problem, focusing on the geometrical complexity of the class boundary. We compared a set of real-world problems to random
labelings of points and found that real problems contain structures in this measurement space that are significantly different from the random sets. Distributions of problems in this space show that there exist at least two
independent factors affecting a problem's difficulty. We suggest using this space to describe a classifier's domain of competence. This can guide static and dynamic selection of classifiers for specific problems as well as
subproblems formed by confinement, projection, and transformations of the feature vectors.},
author = {Ho, Tin Kam and Basu, Mitra},
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
keywords = {Classification,Clustering,Complexity,Linear separability,Mixture identifiability},
number = {3},
pages = {289----300},
title = {{Complexity measures of supervised classification problems}},
volume = {24},
year = {2002}
}
@inproceedings{DBLP:journals/corr/abs-1905-07299,
author = {Branchaud-Charron, Frederic and Achkar, Andrew and Jodoin, Pierre-Marc},
booktitle = {IEEE Conference on Computer Vision and Pattern Recognition},
pages = {3215----3224},
title = {{Spectral metric for dataset complexity assessment}},
year = {2019}
}
@article{Donoser2007,
abstract = {This paper presents a novel unsupervised color segmentation scheme named ROI-SEG, which is based on the main idea of combining a set of different sub-segmentation results. We propose an efficient algorithm to
compute sub-segmentations by an integral image approach for calculating Bhattacharyya distances and a modified version of the maximally stable extremal region (MSER) detector. The sub-segmentation algorithm gets a
region-of-interest (ROI) as input and detects connected regions having similar color appearance as the ROI. We further introduce a method to identify ROIs representing the predominant color and texture regions of an image.
Passing each of the identified ROIs to the sub-segmentation algorithm provides a set of different segmentations, which are then combined by analyzing a local quality criterion. The entire approach is fully unsupervised and
does not need a priori information about the image scene. The method is compared to state-of-the-art algorithms on the Berkeley image database, where it shows competitive results at reduced computational costs.},
author = {Donoser, Michael and Bischof, Horst},
doi = {10.1109/CVPR.2007.383231},
isbn = {1424411807},
issn = {10636919},
journal = {Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition},
title = {{ROI-SEG: Unsupervised color segmentation by combining differently focused sub results}},
year = {2007}
}
@article{Dwaraknath2016,
abstract = {guided bp 结合vgg 做物体弱监督定位},
author = {Dwaraknath, Anjan and Menghani and Deepak and Mongia},
title = {{Fast Unsupervised Object Localization}},
year = {2016}
}
@article{Kim2009,
abstract = {This paper proposes a fast and scalable alternating optimization technique to detect regions of interest (ROIs) in cluttered Web images without labels. The proposed approach discovers highly probable regions of
object instances by iteratively repeating the following two functions: (1) choose the exemplar set (i.e. a small number of highly ranked reference ROIs) across the dataset and (2) refine the ROIs of each image with respect to
the exemplar set. These two subproblems are formulated as ranking in two different similarity networks of ROI hypotheses by link analysis. The experiments with the PASCAL 06 dataset show that our unsupervised localization
performance is better than one of state-of-the-art techniques and comparable to supervised methods. Also, we test the scalability of our approach with five objects in Flickr dataset consisting of more than 200K images.},
author = {Kim, Gunhee and Torralba, Antonio},
journal = {Advances in neural information processing {\ldots}},
pages = {1--9},
title = {{Unsupervised detection of regions of interest using iterative link analysis}},
year = {2009}
}
@inproceedings{Derakhshani2019,
author = {Derakhshani, Mohammad Mahdi and Masoudnia, Saeed and Shaker, Amir Hossein and Mersa, Omid},
title = {{Assisted Excitation of Activations : A Learning Technique to Improve Object Detectors}},
year = {2019}
}
@article{Chenb,
abstract = {Many machine learning models can be fooled by adversarial examples. The adversarial examples are the subtly perturbed inputs crafted to fool the machine learning models. While the models correctly classify the
original inputs, the perturbed adversarial inputs are misclassified to other classes. The perturbations added to the original inputs are usually not easily perceived by human eyes. Therefore the malicious attacks with the
adversarial examples are serious security risks because the attacks can remain stealth. This paper compares various denoising autoencoders applied to inputs before the classifiers to filter the fast gradient sign method
(FGSM) adversarial perturbations in the inputs. Both the capacity of removing the attacks and performance change on the original un-attacked inputs are analyzed.},
author = {Chen, I-Ting and Sirkeci-Mergen, Birsen},
isbn = {1601324855},
keywords = {adversarial examples,autoencoders,machine learning models,security},
pages = {132--136},
title = {{A Comparative Study of Autoencoders against Adversarial Attacks}},
}
@article{Gu2014,
abstract = {Recent work has shown deep neural networks (DNNs) to be highly susceptible to well-designed, small perturbations at the input layer, or so-called adversarial examples. Taking images as an example, such distortions
are often imperceptible, but can result in 100{\%} mis-classification for a state of the art DNN. We study the structure of adversarial examples and explore network topology, pre-processing and training strategies to improve
the robustness of DNNs. We perform various experiments to assess the removability of adversarial examples by corrupting with additional noise and pre-processing with denoising autoencoders (DAEs). We find that DAEs can remove
substantial amounts of the adversarial noise. How- ever, when stacking the DAE with the original DNN, the resulting network can again be attacked by new adversarial examples with even smaller distortion. As a solution, we
propose Deep Contractive Network, a model with a new end-to-end training procedure that includes a smoothness penalty inspired by the contractive autoencoder (CAE). This increases the network robustness to adversarial
examples, without a significant performance penalty.},
author = {Gu, Shixiang and Rigazio, Luca},
journal = {arXiv preprint arXiv:1412.5068},
title = {{Towards deep neural network architectures robust to adversarial examples}},
year = {2014}
}
@article{karaboga2018adaptive,
author = {Karaboga, Dervis and Kaya, Ebubekir},
journal = {Artificial Intelligence Review},
pages = {1--31},
publisher = {Springer},
title = {{Adaptive network based fuzzy inference system (ANFIS) training approaches: a comprehensive survey}},
year = {2018}
}
@article{Jang1991,
abstract = {1 Abstract We propose a new approach to build a fuzzy infer-ence system of which the parameters can be updated to achieve a desired input-output mapping. The struc-ture of the proposed fuzzy inference system is
called generalized neural networks, and its learning procedure (rules to update parameters) is basically composed of a gradient descent algorithm and Kalman filter algo-rithm. Specifically, we first introduce the concept of
generalized neural networks (GNN's) and develop a gradient-descent-based supervised learning procedure to update the GNN's parameters. Secondly, we ob-serve that if the overall output of a GNN is a linear combination of some
of its parameters, then these pa-rameters can be identified by one-time application of Kalman filter algorithm to minimize the squared er-ror. According to the simulation results, it is concluded that the proposed new fuzzy
inference system can not only incorporate prior knowledge about the original system but also fine-tune the membership functions of the fuzzy rules as the training data set varies.},
author = {Jang, Jyh-Shing},
isbn = {978-0-262-51059-2},
journal = {Proceedings of the 9th National Conference on Artificial Intelligence},
pages = {762 -- 767},
title = {{Fuzzy Modeling Using Generalized Neural Networks and Kalman Filter Algorithm.}},
volume = {91},
year = {1991}
}
@article{Qin2019,
author = {Qin, Yi and Hunt, Ryan and Yue, Chuan},
doi = {10.1145/3309182.3309190},
isbn = {9781450361781},
keywords = {Adversarial Examples, Adversarial Training, Advers,acm reference format,adversarial examples,adversarial machine,adversarial training,deep learning,learning,neural networks},
pages = {5--13},
title = {{On Improving the Effectiveness of Adversarial Training}},
year = {2019}
}
@article{pulina2012challenging,
author = {Pulina, Luca and Tacchella, Armando},
journal = {Ai Communications},
number = {2},
pages = {117--135},
publisher = {IOS Press},
title = {{Challenging SMT solvers to verify neural networks}},
volume = {25},
year = {2012}
}
@article{Picon2009,
abstract = {Hyperspectral data allows the construction of more elaborate models to sample the properties of the nonferrous materials than the standard RGB color representation. In this paper, the nonferrous waste materials
are studied as they cannot be sorted by classical procedures due to their color, weight and shape similarities. The experimental results presented in this paper reveal that factors such as the various levels of oxidization of
the waste materials and the slight differences in their chemical composition preclude the use of the spectral features in a simplistic manner for robust material classification. To address these problems, the proposed FUSSER
(fuzzy spectral and spatial classifier) algorithm detailed in this paper merges the spectral and spatial features to obtain a combined feature vector that is able to better sample the properties of the nonferrous materials
than the single pixel spectral features when applied to the construction of multivariate Gaussian distributions. This approach allows the implementation of statistical region merging techniques in order to increase the
performance of the classification process. To achieve an efficient implementation, the dimensionality of the hyperspectral data is reduced by constructing bio-inspired spectral fuzzy sets that minimize the amount of redundant
information contained in adjacent hyperspectral bands. The experimental results indicate that the proposed algorithm increased the overall classification rate from 44{\%} using RGB data up to 98{\%} when the spectral-spatial
features are used for nonferrous material classification.},
author = {Picon, Artzai and Ghita, Ovidiu and Whelan, Paul F. and Iriondo, Pedro M.},
doi = {10.1109/TII.2009.2031238},
issn = {15513203},
journal = {IEEE Transactions on Industrial Informatics},
keywords = {Hyperspectral image processing,Image classification,Spectral fuzzy sets},
number = {4},
pages = {483--494},
title = {{Fuzzy spectral and spatial feature integration for classification of nonferrous materials in hyperspectral data}},
volume = {5},
year = {2009}
}
@article{Khalifa2016,
abstract = {Fuzzy logic is a powerful tool to model knowledge uncertainty, measurements imprecision, and vagueness. However, there is another type of vagueness that arises when data have multiple forms of expression that
fuzzy logic does not address quite well. This is the case for multiple instance learning problems (MIL). In MIL, an object is represented by a collection of instances, called a bag. A bag is labeled negative if all of its
instances are negative, and positive if at least one of its instances is positive. Positive bags encode ambiguity since the instances themselves are not labeled. In this paper, we introduce fuzzy inference systems and neural
networks designed to handle bags of instances as input and capable of learning from ambiguously labeled data. First, we introduce the Multiple Instance Sugeno style fuzzy inference (MI-Sugeno) that extends the standard Sugeno
style inference to handle reasoning with multiple instances. Second, we use MI-Sugeno to define and develop Multiple Instance Adaptive Neuro Fuzzy Inference System (MI-ANFIS). We expand the architecture of the standard ANFIS
to allow reasoning with bags and derive a learning algorithm using backpropagation to identify the premise and consequent parameters of the network. The proposed inference system is tested and validated using synthetic and
benchmark datasets suitable for MIL problems. We also apply the proposed MI-ANFIS to fuse the output of multiple discrimination algorithms for the purpose of landmine detection using Ground Penetrating Radar.},
archivePrefix = {arXiv},
arxivId = {1610.04973},
author = {Khalifa, Amine Ben and Frigui, Hichem},
eprint = {1610.04973},
pages = {1--28},
title = {{Multiple Instance Fuzzy Inference Neural Networks}},
year = {2016}
}
@article{Vieira2004,
abstract = {The techniques of artificial intelligence based in fuzzy logic and neural networks are frequently applied together. The reasons to combine these two paradigms come out of the difficulties and inherent limitations
of each isolated paradigm. Generically, when they are used in a combined way, they are called Neuro-Fuzzy Systems. This term, however, is often used to assign a specific type of system that integrates both techniques. This
type of system is characterised by a fuzzy system where fuzzy sets and fuzzy rules are adjusted using input output patterns. There are several different implementations of neuro-fuzzy systems, where each author defined its
own model. This article summarizes a general vision of the area describing the most known hybrid neuro-fuzzy techniques, its advantages and disadvantages.},
author = {Vieira, Jos{\'{e}} and Dias, Morgado and Mota, Alexandre},
journal = {WSEAS Transactions on Systems},
keywords = {Concurrent Systems,Cooperative Systems,Hybrid Systems,Neuro-Fuzzy Architectures,Non-Linear Modelling},
pages = {414--419},
title = {{Neuro-fuzzy systems: a survey}},
volume = {3},
year = {2004}
}
@book{vanderbei2015linear,
author = {Vanderbei, Robert J and Others},
publisher = {Springer},
title = {{Linear programming}},
year = {2015}
}
@article{owen2003quasi,
author = {Owen, Art B},
journal = {Monte Carlo Ray Tracing: Siggraph},
pages = {69--88},
title = {{Quasi-monte carlo sampling}},
volume = {1},
year = {2003}
}
@article{singhee2010quasi,
author = {Singhee, Amith and Rutenbar, Rob A},
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems},
number = {11},
pages = {1763--1776},
publisher = {IEEE},
title = {{Why quasi-monte carlo is better than monte carlo or latin hypercube sampling for statistical circuit analysis}},
volume = {29},
year = {2010}
}
@book{azar2018mathematical,
author = {Azar, Ahmad Taher and Radwan, Ahmed G and Vaidyanathan, Sundarapandian},
publisher = {Elsevier},
title = {{Mathematical Techniques of Fractional Order Systems}},
year = {2018}
}
@article{zadeh1965fuzzy,
author = {Zadeh, Lotfi A},
journal = {Information and control},
number = {3},
pages = {338--353},
publisher = {Elsevier},
title = {{Fuzzy sets}},
volume = {8},
year = {1965}
}
@article{lee1990fuzzy,
author = {Lee, Chuen-Chien},
journal = {IEEE Transactions on systems, man, and cybernetics},
number = {2},
pages = {419--435},
publisher = {IEEE},
title = {{Fuzzy logic in control systems: fuzzy logic controller. II}},
volume = {20},
year = {1990}
}
@book{klir1995fuzzy,
author = {Klir, George J and Yuan, Baozung},
publisher = {Prentice Hall PTR New Jersey},
title = {{Fuzzy sets and fuzzy logic: theory and applications}},
volume = {574},
year = {1995}
}
@inproceedings{bianchi2019capabilities,
author = {Bianchi, Federico and Hitzler, Pascal},
booktitle = {AAAI Spring Symposium: Combining Machine Learning with Knowledge Engineering},
title = {{On the Capabilities of Logic Tensor Networks for Deductive Reasoning.}},
year = {2019}
}
@article{serafini2016logic,
author = {Serafini, Luciano and d'Avila Garcez, Artur},
journal = {arXiv preprint arXiv:1606.04422},
title = {{Logic tensor networks: Deep learning and logical reasoning from data and knowledge}},
year = {2016}
}
@article{ghosh2018trusted,
author = {Ghosh, Shalini and Mercier, Amaury and Pichapati, Dheeraj and Jha, Susmit and Yegneswaran, Vinod and Lincoln, Patrick},
journal = {arXiv preprint arXiv:1805.07075},
title = {{Trusted Neural Networks for Safety-Constrained Autonomous Control}},
year = {2018}
}
@inproceedings{wang2016learning,
author = {Wang, William Yang and Cohen, William W},
booktitle = {IJCAI},
pages = {2132--2138},
title = {{Learning First-Order Logic Embeddings via Matrix Factorization.}},
year = {2016}
}
@article{sarker2017explaining,
author = {Sarker, Md Kamruzzaman and Xie, Ning and Doran, Derek and Raymer, Michael and Hitzler, Pascal},
journal = {arXiv preprint arXiv:1710.04324},
title = {{Explaining trained neural networks with semantic web technologies: First steps}},
year = {2017}
}
@article{DBLP:journals/corr/abs-1905-00441,
archivePrefix = {arXiv},
arxivId = {1905.00441},
author = {Li, Yandong and Li, Lijun and Wang, Liqiang and Zhang, Tong and Gong, Boqing},
eprint = {1905.00441},
journal = {CoRR},
title = {{{\{}NATTACK:{\}} Learning the Distributions of Adversarial Examples for an Improved Black-Box Attack on Deep Neural Networks}},
volume = {abs/1905.0},
year = {2019}
}
@article{DBLP:journals/corr/abs-1905-07121,
author = {Guo, Chuan and Gardner, Jacob R and You, Yurong and Wilson, Andrew Gordon and Weinberger, Kilian Q},
journal = {arXiv preprint arXiv:1905.07121},
title = {{Simple black-box adversarial attacks}},
year = {2019}
}
@article{DBLP:journals/corr/abs-1905-07387,
archivePrefix = {arXiv},
arxivId = {1905.07387},
author = {Ko, Ching-Yun and Lyu, Zhaoyang and Weng, Tsui-Wei and Daniel, Luca and Wong, Ngai and Lin, Dahua},
eprint = {1905.07387},
journal = {CoRR},
title = {{{\{}POPQORN:{\}} Quantifying Robustness of Recurrent Neural Networks}},
volume = {abs/1905.0},
year = {2019}
}
@article{kononenko2001machine,
author = {Kononenko, Igor},
journal = {Artificial Intelligence in medicine},
number = {1},
pages = {89--109},
publisher = {Elsevier},
title = {{Machine learning for medical diagnosis: history, state of the art and perspective}},
volume = {23},
year = {2001}
}
@article{kanevsky2016big,
author = {Kanevsky, Jonathan and Corban, Jason and Gaster, Richard and Kanevsky, Ari and Lin, Samuel and Gilardino, Mirko},
journal = {Plastic and reconstructive surgery},
number = {5},
pages = {890e----897e},
publisher = {Wolters Kluwer},
title = {{Big data and machine learning in plastic surgery: a new frontier in surgical innovation}},
volume = {137},
year = {2016}
}
@article{bojarski2016end,
author = {Bojarski, Mariusz and {Del Testa}, Davide and Dworakowski, Daniel and Firner, Bernhard and Flepp, Beat and Goyal, Prasoon and Jackel, Lawrence D and Monfort, Mathew and Muller, Urs and Zhang, Jiakai and Others},
journal = {arXiv preprint arXiv:1604.07316},
title = {{End to end learning for self-driving cars}},
year = {2016}
}
@article{huval2015empirical,
author = {Huval, Brody and Wang, Tao and Tandon, Sameep and Kiske, Jeff and Song, Will and Pazhayampallil, Joel and Andriluka, Mykhaylo and Rajpurkar, Pranav and Migimatsu, Toki and Cheng-Yue, Royce and Others},
journal = {arXiv preprint arXiv:1504.01716},
title = {{An empirical evaluation of deep learning on highway driving}},
year = {2015}
}
@inproceedings{reiley2010motion,
author = {Reiley, Carol E and Plaku, Erion and Hager, Gregory D},
booktitle = {2010 Annual International Conference of the IEEE Engineering in Medicine and Biology},
organization = {IEEE},
pages = {967--970},
title = {{Motion generation of robotic surgical tasks: Learning from expert demonstrations}},
year = {2010}
}
@article{greenspan2016guest,
author = {Greenspan, Hayit and {Van Ginneken}, Bram and Summers, Ronald M},
journal = {IEEE Transactions on Medical Imaging},
number = {5},
pages = {1153--1159},
publisher = {IEEE},
title = {{Guest editorial deep learning in medical imaging: Overview and future promise of an exciting new technique}},
volume = {35},
year = {2016}
}
@book{sutton2007fuzzing,
author = {Sutton, Michael and Greene, Adam and Amini, Pedram},
publisher = {Pearson Education},
title = {{Fuzzing: brute force vulnerability discovery}},
year = {2007}
}
@book{jia2018simplex,
author = {Jia, Yan-Bin},
title = {{The Simplex Method}},
year = {2018}
}
@article{Huangfu2018,
abstract = {This paper introduces the design and implementation of two par-allel dual simplex solvers for general large scale sparse linear pro-gramming problems. One approach, called PAMI, extends a relatively unknown
pivoting strategy called suboptimization and exploits paral-lelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible.
Computational results show that the performance of PAMI is superior to that of the leading open-source simplex solver, and that SIP complements PAMI in achieving speedup when PAMI results in slowdown. One of the authors has
implemented the techniques un-derlying PAMI within the FICO Xpress simplex solver and this paper presents computational results demonstrating their value. This perfor-mance increase is sufficiently valuable for the
achievement to be used as the basis of promotional material by FICO. In developing the first parallel revised simplex solver of general utility and commercial impor-tance, this work represents a significant achievement in
computational optimization.},
author = {Huangfu, Q. and Hall, J. A.J.},
doi = {10.1007/s12532-017-0130-5},
issn = {18672957},
journal = {Mathematical Programming Computation},
keywords = {Linear programming,Parallel computing,Revised simplex method},
number = {1},
pages = {119--142},
publisher = {Springer Berlin Heidelberg},
title = {{Parallelizing the dual revised simplex method}},
volume = {10},
year = {2018}
}
@book{maros2012computational,
author = {Maros, Istv{\'{a}}n},
publisher = {Springer Science {\&} Business Media},
title = {{Computational techniques of the simplex method}},
volume = {61},
year = {2012}
}
@article{nelder1965simplex,
author = {Nelder, John A and Mead, Roger},
journal = {The computer journal},
number = {4},
pages = {308--313},
publisher = {Oxford University Press},
title = {{A simplex method for function minimization}},
volume = {7},
year = {1965}
}
@article{rifkin2003regularized,
author = {Rifkin, Ryan and Yeo, Gene and Poggio, Tomaso and Others},
journal = {Nato Science Series Sub Series III Computer and Systems Sciences},
pages = {131--154},
publisher = {IOS PRESS},
title = {{Regularized least-squares classification}},
volume = {190},
year = {2003}
}
@inproceedings{cousot1977abstract,
author = {Cousot, Patrick and Cousot, Radhia},
booktitle = {Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages},
organization = {ACM},
pages = {238--252},
title = {{Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints}},
year = {1977}
}
@article{franzle2007efficient,
author = {Fr{\"{a}}nzle, Martin and Herde, Christian and Teige, Tino and Ratschan, Stefan and Schubert, Tobias},
journal = {JSAT},
number = {3-4},
pages = {209--236},
title = {{Efficient Solving of Large Non-linear Arithmetic Constraint Systems with Complex Boolean Structure.}},
volume = {1},
year = {2007}
}
@article{solow2007linear,
author = {Solow, Daniel},
journal = {Wiley Encyclopedia of Computer Science and Engineering},
publisher = {Wiley Online Library},
title = {{Linear and nonlinear programming}},
year = {2007}
}
@article{Bjorner2011,
abstract = {Constraint-satisfaction problems arise in diverse application areas, including software and hardware verification, type inference, static program analysis, test-case generation, scheduling, planning, and graph
problems, and share a common trait—a core component using logical formulas for describing},
author = {Bjorner, Nikolaj and Moura, Leonardo De},
doi = {10.1145/1995376},
isbn = {9781586039295},
issn = {00010782},
journal = {Comm. ACM},
pages = {69--77},
pmid = {12778541},
title = {{Satisfiability Modulo Theories: Introduction and Applications}},
year = {2011}
}
@inproceedings{10.1007/978-3-540-78800-3_24,
abstract = {Satisfiability Modulo Theories (SMT) problem is a decision problem for logical first order formulas with respect to combinations of background theories such as: arithmetic, bit-vectors, arrays, and uninterpreted
functions. Z3 is a new and efficient SMT Solver freely available from Microsoft Research. It is used in various software verification and analysis applications.},
address = {Berlin, Heidelberg},
author = {de Moura, Leonardo and Bj{\o}rner, Nikolaj},
booktitle = {Tools and Algorithms for the Construction and Analysis of Systems},
editor = {Ramakrishnan, C R and Rehof, Jakob},
isbn = {978-3-540-78800-3},
pages = {337--340},
publisher = {Springer Berlin Heidelberg},
title = {{Z3: An Efficient SMT Solver}},
year = {2008}
}
@article{Wing1990,
abstract = {Formal methods used in developing computer systems (i.e. mathematically based techniques for describing system properties) are defined, and their role is delineated. Formal specification languages, which provide
the formal method's mathematical basis, are examined. Certain pragmatic concerns about formal methods and their users, uses, and characteristics are discussed. Six well-known or commonly used formal methods are illustrated by
simple examples. They are Z, VDM, Larch, temporal logic, CSP, and transition axioms},
author = {Wing, Jeannette M.},
doi = {10.1109/2.58215},
issn = {00189162},
journal = {Computer},
number = {9},
pages = {8--22},
title = {{A Specifier's Introduction to Formal Methods}},
volume = {23},
year = {1990}
}
@article{hall1990seven,
author = {Hall, Anthony},
journal = {IEEE software},
number = {5},
pages = {11--19},
publisher = {IEEE},
title = {{Seven myths of formal methods}},
volume = {7},
year = {1990}
}
@article{Ostroff1992,
abstract = {Safety-critical computers increasingly affect nearly every aspect of our lives. Computers control the planes we fly on, monitor our health in hospitals and do our work in hazardous environments. Computers with
software deficiencies that fail to meet stringent timing constraints have resulted in catastrophic failures. This article surveys formal methods for specifying, designing, and verifying real-time systems so as to improve
their safety and reliability. {\textcopyright} 1992.},
author = {Ostroff, Jonathan S.},
doi = {10.1016/0164-1212(92)90045-L},
issn = {01641212},
journal = {The Journal of Systems and Software},
number = {1},
pages = {33--60},
title = {{Formal methods for the specification and design of real-time safety critical systems}},
volume = {18},
year = {1992}
}
@article{Clarke1996,
abstract = {Hardware and software systems will inevitably grow in scale and functionality. Because of this increase in complexity, the likelihood of subtle errors is much greater. Moreover, some of these errors may cause
catastrophic loss of money, time or even human life. A major goal of software engineering is to enable developers to construct systems that operate reliably despite this complexity. One way of achieving this goal is by using
formal methods, which are mathematically based languages, techniques and tools for specifying and verifying such systems. Use of formal methods does not, a priori, guarantee correctness. However, they can greatly increase our
understanding of a system by revealing inconsistencies, ambiguities and incompleteness that might otherwise go undetected. The first part of this report assesses the state of the art in specification and verification. For
verification, we highlight advances in model checking and theorem proving. In the three sections on specification, model checking and theorem proving, we explain what we mean by the general technique and briefly describe some
successful case studies and well-known tools. The second part of this report outlines future directions in fundamental concepts, new methods and tools, integration of methods, and education and technology transfer. We close
with summary remarks and pointers to resources for more information},
author = {Clarke, Edmund M. and Wing, Jeannette M.},
doi = {10.1145/242223.242257},
issn = {03600300},
journal = {ACM Computing Surveys},
number = {4},
pages = {626--643},
title = {{Formal methods: state of the art and future directions}},
volume = {28},
year = {1996}
}
@incollection{GIBBINS1990278,
abstract = {Publisher Summary
This chapter elaborates the different aspects of formal methods. A formal system is a well-defined system of notation together with well-defined rules for manipulating that notation, which are based on mathematical theory. A
formal method is a set of rigorous engineering practices which are generally based on formal systems, and which are applied to the development of engineering products such as software, or hardware. A formal language is a
language which has a precise set of semantics, as opposed to natural languages in which it is possible to construct ambiguous sentences. A formal language is needed to support a formal method. Formal methods are partly
because they stress that what a software engineer designs, and what a programmer implements is a formal, mathematical structure. They are formal partly because they describe these structures in formal terms. Formal
specification techniques secure the software development process in the well-understood mathematical basis of computer science, namely formal logic, and discrete mathematics.},
author = {Gibbins, Peter},
booktitle = {The Software Life Cycle},
doi = {https://doi.org/10.1016/B978-0-408-03741-9.50020-9},
editor = {Ince, Darrel and Andrews, Derek},
isbn = {978-0-408-03741-9},
pages = {278--290},
publisher = {Butterworth-Heinemann},
title = {{Chapter 13 - What are formal methods?}},
year = {1990}
}
@article{huang2017adversarial,
author = {Huang, Sandy and Papernot, Nicolas and Goodfellow, Ian and Duan, Yan and Abbeel, Pieter},
journal = {arXiv preprint arXiv:1702.02284},
title = {{Adversarial attacks on neural network policies}},
year = {2017}
}
@article{ganin2016domain,
author = {Ganin, Yaroslav and Ustinova, Evgeniya and Ajakan, Hana and Germain, Pascal and Larochelle, Hugo and Laviolette, Fran{\c{c}}ois and Marchand, Mario and Lempitsky, Victor},
journal = {The Journal of Machine Learning Research},
number = {1},
pages = {2030--2096},
publisher = {JMLR. org},
title = {{Domain-adversarial training of neural networks}},
volume = {17},
year = {2016}
}
@article{cybenko1989approximation,
author = {Cybenko, George},
journal = {Mathematics of control, signals and systems},
number = {4},
pages = {303--314},
publisher = {Springer},
title = {{Approximation by superpositions of a sigmoidal function}},
volume = {2},
year = {1989}
}
@article{Lin2017,
abstract = {We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well,
the class of functions of practical interest can frequently be approximated through "cheap learning" with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as
symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical
form prevalent in physics and machine-learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We
prove various "no-flattening theorems" showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss, for example, we show that {\$}n{\$} variables cannot be multiplied
using fewer than 2{\^{}}n neurons in a single hidden layer.},
author = {Lin, Henry W. and Tegmark, Max and Rolnick, David},
doi = {10.1007/s10955-017-1836-5},
issn = {00224715},
journal = {Journal of Statistical Physics},
keywords = {Artificial neural networks,Deep learning,Statistical physics},
number = {6},
pages = {1223--1247},
publisher = {Springer US},
title = {{Why Does Deep and Cheap Learning Work So Well?}},
volume = {168},
year = {2017}
}
@article{Karlik2015,
abstract = {See, stats, and : https : / / www . researchgate . net / publication / 228813985 Performance Functions Neural Article CITATIONS 61 READS 1 , 147 2 : Ahmet 1 SEE Bekir Selcuk 80 SEE All - text , letting . Available
: Bekir Retrieved : 19 Abstract The activation function used to transform the activation level of a unit (neuron) into an output signal . There are a number of common activation functions in use with artificial neural
networks (ANN) . The most common choice of activation functions for multi layered perceptron (MLP) is used as transfer functions in research and engineering . Among the reasons for this popularity are its boundedness in the
unit interval , the function ' s and its derivative ' s fast computability , and a number of amenable mathematical properties in the realm of approximation theory . However , considering the huge variety of problem domains
MLP is applied in , it is intriguing to suspect that specific problems call for single or a set of specific activation functions . The aim of this study is to analyze the performance of generalized MLP architectures which has
back - propagation algorithm using various different activation functions for the neurons of hidden and output layers . For experimental comparisons , Bi - polar sigmoid , Uni - polar sigmoid , Tanh , Conic Section , and
Radial Bases Function (RBF) were used .},
author = {Karlik, Bekir and Olgac, A Vehbi},
journal = {International Journal of Artificial Intelligence and Expert Systems},
keywords = {activation functions,multi layered perceptron,neural networks,performance analysis},
number = {4},
pages = {111--122},
title = {{Performance analysis of various activation functions in generalized MLP architectures of neural networks}},
volume = {1},
year = {2015}
}
@book{alpaydin2009introduction,
author = {Alpaydin, Ethem},
publisher = {MIT press},
title = {{Introduction to machine learning}},
year = {2009}
}
@inproceedings{Schulze2008,
author = {Schulze, Lothar and Behling, Sebastian and Buhrs, Stefan},
booktitle = {International MultiConference of Engineers and Computer Scientists},
isbn = {978-988-17012-1-3},
title = {{Automated Guided Vehicle Systems: a Driver for Increased Business Performance}},
volume = {2},
year = {2008}
}
@article{Lecun2015,
abstract = {Deep learning (DL) is a high dimensional data reduction technique for constructing high-dimensional predictors in input-output models. DL is a form of machine learning that uses hierarchical layers of latent
features. In this article, we review the state-of-the-art of deep learning from a modeling and algorithmic perspective. We provide a list of successful areas of applications in Artificial Intelligence (AI), Image Processing,
Robotics and Automation. Deep learning is predictive in its nature rather then inferential and can be viewed as a black-box methodology for high-dimensional function estimation.},
author = {Lecun, Yann and Bengio, Yoshua and Hinton, Geoffrey},
journal = {Nature},
number = {7553},
pages = {436----444},
title = {{Deep learning}},
volume = {521},
year = {2015}
}
@article{Liang2018,
abstract = {In this paper, we present an effective method to craft text adversarial samples, revealing one important yet underestimated fact that DNN-based text classifiers are also prone to adversarial sample attack.
Specifically, confronted with different adversarial scenarios, the text items that are important for classification are identified by computing the cost gradients of the input (white-box attack) or generating a series of
occluded test samples (black-box attack). Based on these items, we design three perturbation strategies, namely insertion, modification, and removal, to generate adversarial samples. The experiment results show that the
adversarial samples generated by our method can successfully fool both state-of-the-art character-level and word-level DNN-based text classifiers. The adversarial samples can be perturbed to any desirable classes without
compromising their utilities. At the same time, the introduced perturbation is difficult to be perceived.},
author = {Liang, Bin and Li, Hongcheng and Su, Miaoqiang and Bian, Pan and Li, Xirong and Shi, Wenchang},
isbn = {9780999241127},
issn = {10450823},
journal = {IJCAI International Joint Conference on Artificial Intelligence},
keywords = {Multidisciplinary Topics and Applications: Securit,Natural Language Processing: Text Classification},
pages = {4208--4215},
title = {{Deep text classification can be fooled}},
volume = {2018-July},
year = {2018}
}
@article{Haidar2019,
abstract = {Text generation with generative adversarial networks (GANs) can be divided into the text-based and code-based categories according to the type of signals used for discrimination. In this work, we introduce a novel
text-based approach called Soft-GAN to effectively exploit GAN setup for text generation. We demonstrate how autoencoders (AEs) can be used for providing a continuous representation of sentences, which we will refer to as
soft-text. This soft representation will be used in GAN discrimination to synthesize similar soft-texts. We also propose hybrid latent code and text-based GAN (LATEXT-GAN) approaches with one or more discriminators, in which
a combination of the latent code and the soft-text is used for GAN discriminations. We perform a number of subjective and objective experiments on two well-known datasets (SNLI and Image COCO) to validate our techniques. We
discuss the results using several evaluation metrics and show that the proposed techniques outperform the traditional GAN-based text-generation methods.},
archivePrefix = {arXiv},
arxivId = {1904.07293},
author = {Haidar, Md. Akmal and Rezagholizadeh, Mehdi and Do-Omri, Alan and Rashid, Ahmad},
eprint = {1904.07293},
title = {{Latent Code and Text-based Generative Adversarial Networks for Soft-text Generation}},
year = {2019}
}
@article{Ebrahimi2017,
abstract = {We propose an efficient method to generate white-box adversarial examples to trick a character-level neural classifier. We find that only a few manipulations are needed to greatly decrease the accuracy. Our method
relies on an atomic flip operation, which swaps one token for another, based on the gradients of the one-hot input vectors. Due to efficiency of our method, we can perform adversarial training which makes the model more
robust to attacks at test time. With the use of a few semantics-preserving constraints, we demonstrate that HotFlip can be adapted to attack a word-level classifier as well.},
archivePrefix = {arXiv},
arxivId = {1712.06751},
author = {Ebrahimi, Javid and Rao, Anyi and Lowd, Daniel and Dou, Dejing},
eprint = {1712.06751},
journal = {arXiv preprint arXiv:1712.06751},
title = {{HotFlip: White-Box Adversarial Examples for Text Classification}},
year = {2017}
}
@article{Madry2017,
abstract = {Recent work has demonstrated that neural networks are vulnerable to adversarial examples, i.e., inputs that are almost indistinguishable from natural data and yet classified incorrectly by the network. In fact,
some of the latest findings suggest that the existence of adversarial attacks may be an inherent weakness of deep learning models. To address this problem, we study the adversarial robustness of neural networks through the
lens of robust optimization. This approach provides us with a broad and unifying view on much of the prior work on this topic. Its principled nature also enables us to identify methods for both training and attacking neural
networks that are reliable and, in a certain sense, universal. In particular, they specify a concrete security guarantee that would protect against any adversary. These methods let us train networks with significantly
improved resistance to a wide range of adversarial attacks. They also suggest the notion of security against a first-order adversary as a natural and broad security guarantee. We believe that robustness against such
well-defined classes of adversaries is an important stepping stone towards fully resistant deep learning models.},
archivePrefix = {arXiv},
arxivId = {1706.06083},
author = {Madry, Aleksander and Makelov, Aleksandar and Schmidt, Ludwig and Tsipras, Dimitris and Vladu, Adrian},
doi = {10.1227/01.NEU.0000255452.20602.C9},
eprint = {1706.06083},
isbn = {9781405161251},
issn = {1607-551X},
pages = {1--27},
pmid = {17460516},
title = {{Towards Deep Learning Models Resistant to Adversarial Attacks}},
year = {2017}
}
@inproceedings{Papernot2016a,
abstract = {Deep learning takes advantage of large datasets and computationally efficient training algorithms to outperform other approaches at various machine learning tasks. However, imperfections in the training phase of
deep neural networks make them vulnerable to adversarial samples: inputs crafted by adversaries with the intent of causing deep neural networks to misclassify. In this work, we formalize the space of adversaries against deep
neural networks (DNNs) and introduce a novel class of algorithms to craft adversarial samples based on a precise understanding of the mapping between inputs and outputs of DNNs. In an application to computer vision, we show
that our algorithms can reliably produce samples correctly classified by human subjects but misclassified in specific targets by a DNN with a 97{\%} adversarial success rate while only modifying on average 4.02{\%} of the
input features per sample. We then evaluate the vulnerability of different sample classes to adversarial perturbations by defining a hardness measure. Finally, we describe preliminary work outlining defenses against
adversarial samples by defining a predictive measure of distance between a benign input and a target classification.},
author = {Papernot, Nicolas and Mcdaniel, Patrick and Jha, Somesh and Fredrikson, Matt and Celik, Z. Berkay and Swami, Ananthram},
booktitle = {IEEE European Symposium on Security and Privacy},
pages = {372----387},
title = {{The Limitations of Deep Learning in Adversarial Settings}},
year = {2016}
}
@inproceedings{Carlini2017,
abstract = {Neural networks provide state-of-the-art results for most machine learning tasks. Unfortunately, neural networks are vulnerable to adversarial examples: given an input {\$}x{\$} and any target classification
{\$}t{\$}, it is possible to find a new input {\$}x'{\$} that is similar to {\$}x{\$} but classified as {\$}t{\$}. This makes it difficult to apply neural networks in security-critical areas. Defensive distillation is a
recently proposed approach that can take an arbitrary neural network, and increase its robustness, reducing the success rate of current attacks' ability to find adversarial examples from {\$}95\backslash{\%}{\$} to
{\$}0.5\backslash{\%}{\$}. In this paper, we demonstrate that defensive distillation does not significantly increase the robustness of neural networks by introducing three new attack algorithms that are successful on both
distilled and undistilled neural networks with {\$}100\backslash{\%}{\$} probability. Our attacks are tailored to three distance metrics used previously in the literature, and when compared to previous adversarial example
generation algorithms, our attacks are often much more effective (and never worse). Furthermore, we propose using high-confidence adversarial examples in a simple transferability test we show can also be used to break
defensive distillation. We hope our attacks will be used as a benchmark in future defense attempts to create neural networks that resist adversarial examples.},
author = {Carlini, Nicholas and Wagner, David},
booktitle = {IEEE Symposium on Security and Privacy},
pages = {39--57},
title = {{Towards Evaluating the Robustness of Neural Networks}},
year = {2017}
}
@inproceedings{Wong2018,
abstract = {We propose a method to learn deep ReLU-based classifiers that are provably robust against norm-bounded adversarial perturbations on the training data. For previously unseen examples, the approach is guaranteed to
detect all adversarial examples, though it may flag some non-adversarial examples as well. The basic idea is to consider a convex outer approximation of the set of activations reachable through a norm-bounded perturbation,
and we develop a robust optimization procedure that minimizes the worst case loss over this outer region (via a linear program). Crucially, we show that the dual problem to this linear program can be represented itself as a
deep network similar to the backpropagation network, leading to very efficient optimization approaches that produce guaranteed bounds on the robust loss. The end result is that by executing a few more forward and backward
passes through a slightly modified version of the original network (though possibly with much larger batch sizes), we can learn a classifier that is provably robust to any norm-bounded adversarial attack. We illustrate the
approach on a number of tasks to train classifiers with robust adversarial guarantees (e.g. for MNIST, we produce a convolutional classifier that provably has less than 5.8{\%} test error for any adversarial attack with
bounded {\$}\backslashell{\_}\backslashinfty{\$} norm less than {\$}\backslashepsilon = 0.1{\$}), and code for all experiments in the paper is available at https://github.com/locuslab/convex{\_}adversarial.},
archivePrefix = {arXiv},
arxivId = {1711.00851},
author = {Wong, Eric and Kolter, J. Zico},
booktitle = {International Conference on Machine Learning},
eprint = {1711.00851},
isbn = {9781510867963},
issn = {1938-7228},
pages = {5283--5292},
title = {{Provable defenses against adversarial examples via the convex outer adversarial polytope}},
year = {2018}
}
@inproceedings{Zheng2016,
abstract = {In this paper we address the issue of output instability of deep neural networks: small perturbations in the visual input can significantly distort the feature embeddings and output of a neural network. Such
instability affects many deep architectures with state-of-the-art performance on a wide range of computer vision tasks. We present a general stability training method to stabilize deep networks against small input distortions
that result from various types of common image processing, such as compression, rescaling, and cropping. We validate our method by stabilizing the state-of-the-art Inception architecture against these types of distortions. In
addition, we demonstrate that our stabilized model gives robust state-of-the-art performance on large-scale near-duplicate detection, similar-image ranking, and classification on noisy datasets.},
archivePrefix = {arXiv},
arxivId = {1604.04326},
author = {Zheng, Stephan and Song, Yang and Leung, Thomas and Goodfellow, Ian},
booktitle = {Proceedings of the ieee conference on computer vision and pattern recognition},
doi = {10.1109/CVPR.2016.485},
eprint = {1604.04326},
isbn = {978-1-4673-8851-1},
issn = {10636919},
pages = {4480--4488},
pmid = {21358929},
title = {{Improving the Robustness of Deep Neural Networks via Stability Training}},
year = {2016}
}
@article{Gonzalez,
author = {Gonz{\'{a}}lez, Bachelor-thesis Von Camila},
title = {{Reconstruction of Boolean Functions from Deep Neural Networks}},
}
@article{Garcez2014,
abstract = {The goal of neural-symbolic computation is to integrate ro- bust connectionist learning and sound symbolic reasoning. With the recent advances in connectionist learning, in par- ticular deep neural networks, forms
of representation learn- ing have emerged. However, such representations have not become useful for reasoning. Results from neural-symbolic computation have shown to offer powerful alternatives for knowledge representation,
learning and reasoning in neural computation. This paper recalls the main contributions and discusses key challenges for neural-symbolic integration which have been identified at a recent Dagstuhl seminar.},
author = {Garcez, Avila and Besold, Tarek R and Raedt, Luc De and F{\"{o}}ldiak, Peter and Hitzler, Pascal and Icard, Thomas and K{\"{u}}hnberger, Kai-uwe and Lamb, Luis C and Miikkulainen, Risto and Silver, Daniel L},
doi = {10.13140/2.1.1779.4243},
isbn = {9781577357070},
journal = {Proceedings of the AAAI Spring Symposium on Knowledge Representation and Reasoning: Integrating Symbolic and Neural Approaches, Stanford},
pages = {18--21},
title = {{Neural-Symbolic Learning and Reasoning : Contributions and Challenges}},
year = {2014}
}
@article{Parodi2018,
abstract = {This study evaluates the performance of a set of machine learning techniques in predicting the prognosis of Hodgkin's lymphoma using clinical factors and gene expression data. Analysed samples from 130 Hodgkin's
lymphoma patients included a small set of clinical variables and more than 54,000 gene features. Machine learning classifiers included three black-box algorithms (k-nearest neighbour, Artificial Neural Network, and Support
Vector Machine) and two methods based on intelligible rules (Decision Tree and the innovative Logic Learning Machine method). Support Vector Machine clearly outperformed any of the other methods. Among the two rule-based
algorithms, Logic Learning Machine performed better and identified a set of simple intelligible rules based on a combination of clinical variables and gene expressions. Decision Tree identified a non-coding gene (XIST)
involved in the early phases of X chromosome inactivation that was overexpressed in females and in non-relapsed patients. XIST expression might be responsible for the better prognosis of female Hodgkin's lymphoma patients.
{\textcopyright} 2016, {\textcopyright} The Author(s) 2016.},
author = {Parodi, Stefano and Manneschi, Chiara and Verda, Damiano and Ferrari, Enrico and Muselli, Marco},
doi = {10.1177/1460458216655188},
issn = {17412811},
journal = {Health Informatics Journal},
keywords = {Decision Tree,Hodgkin's lymphoma,Logic Learning Machine,Support Vector Machine,artificial neural network,cancer prognosis},
number = {1},
pages = {54--65},
pmid = {27354395},
title = {{Logic Learning Machine and standard supervised methods for Hodgkin's lymphoma prognosis using gene expression data and clinical variables}},
volume = {24},
year = {2018}
}
@article{DeLuigi2016,
abstract = {We improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of
multidimensional vanilla options in the Black-Scholes framework which emphasizes the numerical problems of integrating non-smooth functions. In high dimensions, this new algorithm is used as a control variate after a
dimension reduction based on principal component analysis. Numerical tests are performed on the Genz package, on the pricing of basket, put on minimum and digital options in dimensions up to ten.},
author = {{De Luigi}, Christophe and Lelong, J{\'{e}}r{\^{o}}me and Maire, Sylvain},
doi = {10.1016/j.apnum.2015.11.001},
issn = {01689274},
journal = {Applied Numerical Mathematics},
keywords = {Adaptive numerical integration,Control variates,Genz package,Option pricing},
pages = {14--30},
publisher = {Elsevier B.V.},
title = {{Robust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options}},
url = {http://dx.doi.org/10.1016/j.apnum.2015.11.001},
volume = {100},
year = {2016}
}
@article{Varatharajan2017,
abstract = {Multiple Kernel Learning with Adaptive Neuro-Fuzzy Inference System (MKL with ANFIS) based deep learning method is proposed in this paper for heart disease diagnosis. The proposed MKL with ANFIS based deep
learning method follows two-fold approach. MKL method is used to divide parameters between heart disease patients and normal individuals. The result obtained from the MKL method is given to the ANFIS classifier to classify
the heart disease and healthy patients. Sensitivity, Specificity and Mean Square Error (MSE) are calculated to evaluate the proposed MKL with ANFIS method. The proposed MKL with ANFIS is also compared with various existing
deep learning methods such as Least Square with Support Vector Machine (LS with SVM), General Discriminant Analysis and Least Square Support Vector Machine (GDA with LS-SVM), Principal Component Analysis with Adaptive
Neuro-Fuzzy Inference System (PCA with ANFIS) and Latent Dirichlet Allocation with Adaptive Neuro-Fuzzy Inference System (LDA with ANFIS). The results from the proposed MKL with ANFIS method has produced high sensitivity
(98{\%}), high specificity (99{\%}) and less Mean Square Error (0.01) for the for the KEGG Metabolic Reaction Network dataset.},
author = {Varatharajan, R and Priyan, M},
doi = {10.1007/s11042-017-5515-y},
isbn = {1104201755},
issn = {13807501},
journal = {Multimedia Tools and Applications},
keywords = {Medical Diagnosis ; Mean Square Errors ; Adaptive},
number = {December},
pages = {1--21},
publisher = {Multimedia Tools and Applications},
title = {{Hybrid Recommendation System for Heart Disease Diagnosis based on Multiple Kernel Learning with Adaptive Neuro-Fuzzy Inference System}},
url = {http://search.proquest.com/docview/1979725042/?pq-origsite=primo},
year = {2017}
}
@inproceedings{Bietti2018,
abstract = {Despite their success, deep neural networks suffer from several drawbacks: they lack robustness to small changes of input data known as "adversarial examples" and training them with small amounts of annotated data
is challenging. In this work, we study the connection between regularization and robustness by viewing neural networks as elements of a reproducing kernel Hilbert space (RKHS) of functions and by regularizing them using the
RKHS norm. Even though this norm cannot be computed, we consider various approximations based on upper and lower bounds. These approximations lead to new strategies for regularization, but also to existing ones such as
spectral norm penalties or constraints, gradient penalties, or adversarial training. Besides, the kernel framework allows us to obtain margin-based bounds on adversarial generalization. We study the obtained algorithms for
learning on small datasets, learning adversarially robust models, and discuss implications for learning implicit generative models.},
archivePrefix = {arXiv},
arxivId = {1810.00363},
author = {Bietti, Alberto and Mialon, Gr{\'{e}}goire and Mairal, Julien},
booktitle = {arXiv:1810.00363},
doi = {arXiv:1810.00363v2},
eprint = {1810.00363},
pages = {1--16},
title = {{On Regularization and Robustness of Deep Neural Networks}},
url = {http://arxiv.org/abs/1810.00363},
year = {2018}
}
@article{Tobergte2013,
abstract = {applicability for this approach.},
archivePrefix = {arXiv},
arxivId = {arXiv:1011.1669v3},
author = {Tobergte, David R. and Curtis, Shirley},
doi = {10.1017/CBO9781107415324.004},
eprint = {arXiv:1011.1669v3},
isbn = {9788578110796},
issn = {1098-6596},
journal = {Journal of Chemical Information and Modeling},
keywords = {icle},
number = {9},
pages = {1689--1699},
pmid = {25246403},
title = {{Neural-Symbolic Cognitive Reasoning}},
volume = {53},
year = {2013}
}
@article{Alshahrani2017,
abstract = {Motivation: Biological data and knowledge bases increasingly rely on Semantic Web technologies and the use of knowledge graphs for data integration, retrieval and federated queries. In the past years, feature
learning methods that are applicable to graph-structured data are becoming available, but have not yet widely been applied and evaluated on structured biological knowledge. Results: We develop a novel method for feature
learning on biological knowledge graphs. Our method combines symbolic methods, in particular knowledge representation using symbolic logic and automated reasoning, with neural networks to generate embeddings of nodes that
encode for related information within knowledge graphs. Through the use of symbolic logic, these embeddings contain both explicit and implicit information. We apply these embeddings to the prediction of edges in the knowledge
graph representing problems of function prediction, finding candidate genes of diseases, protein-protein interactions, or drug target relations, and demonstrate performance that matches and sometimes outperforms traditional
approaches based on manually crafted features. Our method can be applied to any biological knowledge graph, and will thereby open up the increasing amount of Semantic Web based knowledge bases in biology to use in machine
learning and data analytics. Availability and Implementation: https://github.com/bio-ontology-research-group/walking-rdf-and-owl Contact: robert.hoehndorf@kaust.edu.sa},
archivePrefix = {arXiv},
arxivId = {1612.04256},
author = {Alshahrani, Mona and Khan, Mohammad Asif and Maddouri, Omar and Kinjo, Akira R. and Queralt-Rosinach, N{\'{u}}ria and Hoehndorf, Robert},
doi = {10.1093/bioinformatics/btx275},
eprint = {1612.04256},
isbn = {13674811 (Electronic)},
issn = {13674811},
journal = {Bioinformatics (Oxford, England)},
number = {17},
pages = {2723--2730},
pmid = {28449114},
title = {{Neuro-symbolic representation learning on biological knowledge graphs}},
volume = {33},
year = {2017}
}
@article{Montavon2018,
abstract = {This paper provides an entry point to the problem of interpreting a deep neural network model and explaining its predictions. It is based on a tutorial given at ICASSP 2017. As a tutorial paper, the set of methods
covered here is not exhaustive, but sufficiently representative to discuss a number of questions in interpretability, technical challenges, and possible applications. The second part of the tutorial focuses on the recently
proposed layer-wise relevance propagation (LRP) technique, for which we provide theory, recommendations, and tricks, to make most efficient use of it on real data.},
archivePrefix = {arXiv},
arxivId = {1706.07979},
author = {Montavon, Gr{\'{e}}goire and Samek, Wojciech and M{\"{u}}ller, Klaus Robert},
doi = {10.1016/j.dsp.2017.10.011},
eprint = {1706.07979},
issn = {10512004},
journal = {Digital Signal Processing: A Review Journal},
keywords = {Activation maximization,Deep neural networks,Layer-wise relevance propagation,Sensitivity analysis,Taylor decomposition},
pages = {1--15},
pmid = {23749},
title = {{Methods for interpreting and understanding deep neural networks}},
volume = {73},
year = {2018}
}
@article{Yildiz2010,
author = {Yildiz, Zeynep Cipiloglu},
number = {Figure 2},
pages = {1--6},
title = {{A Short Fuzzy Logic Tutorial}},
url = {http://cs.bilkent.edu.tr/{~}zeynep/files/short{\_}fuzzy{\_}logic{\_}tutorial.pdf},
year = {2010}
}
@inproceedings{Kok2010,
abstract = {Markov logic networks (MLNs) use first-order formulas to define features of Markov networks. Current MLN structure learners can only learn short clauses (4-5 literals) due to extreme computational costs, and thus
are unable to represent complex regularities in data. To address this problem, we present LSM, the first MLN structure learner capable of efficiently and accurately learning long clauses. LSM is based on the observation that
relational data typically contains patterns that are variations of the same structural motifs. By constraining the search for clauses to occur within motifs, LSM can greatly speed up the search and thereby reduce the cost of
finding long clauses. LSM uses random walks to identify densely connected objects in data, and groups them and their associated relations into a motif. Our experiments on three real-world datasets show that our approach is
2-5 orders of magnitude faster than the state-of-the-art ones, while achieving the same or better predictive performance.},
author = {Kok, Stanley and Domingos, Pedro},
booktitle = {International Conference on Machine Learning},
doi = {10.1145/1553374.1553440},
isbn = {9781605585161},
issn = {0031-9007},
pages = {551--558},
title = {{Learning Markov Logic Networks Using Structural Motifs}},
url = {http://www.icml2010.org/papers/502.pdf},
year = {2010}
}
@article{Kuo2016,
abstract = {This work attempts to address two fundamental questions about the structure of the convolutional neural networks (CNN): (1) why a nonlinear activation function is essential at the filter output of all intermediate
layers? (2) what is the advantage of the two-layer cascade system over the one-layer system? A mathematical model called the “REctified-COrrelations on a Sphere” (RECOS) is proposed to answer these two questions. After the
CNN training process, the converged filter weights define a set of anchor vectors in the RECOS model. Anchor vectors represent the frequently occurring patterns (or the spectral components). The necessity of rectification is
explained using the RECOS model. Then, the behavior of a two-layer RECOS system is analyzed and compared with its one-layer counterpart. The LeNet-5 and the MNIST dataset are used to illustrate discussion points. Finally, the
RECOS model is generalized to a multilayer system with the AlexNet as an example.},
archivePrefix = {arXiv},
arxivId = {1609.04112},
author = {Kuo, C. C.Jay},
doi = {10.1016/j.jvcir.2016.11.003},
eprint = {1609.04112},
isbn = {1532-4435},
issn = {10959076},
journal = {Journal of Visual Communication and Image Representation},
keywords = {Convolutional neural network (CNN),MNIST dataset,Nonlinear activation,RECOS model,Rectified linear unit (ReLU)},
pages = {406--413},
pmid = {21280716},
publisher = {Elsevier Inc.},
title = {{Understanding convolutional neural networks with a mathematical model}},
url = {http://dx.doi.org/10.1016/j.jvcir.2016.11.003},
volume = {41},
year = {2016}
}
@article{Mairal2016,
author = {Mairal, Julien},
title = {{Towards Deep Kernel Machines Part I : Scientific Context}},
year = {2016}
}
@inproceedings{Reddi,
abstract = {Several recently proposed stochastic optimization methods that have been suc-cessfully used in training deep networks such as RMSPROP, ADAM, ADADELTA, NADAM are based on using gradient updates scaled by square
roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or
a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimiza-tion setting where ADAM
does not converge to the optimal solution, and describe the precise problems with the previous analysis of ADAM algorithm. Our anal-ysis suggests that the convergence issues can be fixed by endowing such algo-rithms with "
long-term memory " of past gradients, and propose new variants of the ADAM algorithm which not only fix the convergence issues but often also lead to improved empirical performance.},
author = {Reddi, Sashank J and Kale, Satyen and Kumar, Sanjiv},
title = {{ON THE CONVERGENCE OF ADAM AND BEYOND}}
}
@book{Dutle2018,
abstract = {Agda is a dependently typed functional programming language and a proof assistant in which developing programs and proving their correctness is one activity. We show how this process can be enhanced by integrating
external automated theorem provers, provide a prototypical integration of the equational theorem prover Waldmeister, and give examples of how this proof automation works in practice.},
author = {Dutle, Aaron and Mu{\~{n}}oz, C{\'{e}}sar and Narkawicz, Anthony},
doi = {10.1007/978-3-642-38088-4},
isbn = {978-3-642-38087-7},
issn = {03029743},
title = {{NASA Formal Methods}},
url = {http://link.springer.com/10.1007/978-3-642-38088-4},
volume = {7871},
year = {2018}
}
@article{Vitor2015,
archivePrefix = {arXiv},
arxivId = {arXiv:0706.3384v2},
author = {Vitor, Manoel and Franc, Macedo and Artur, S and Garcez, Avila},
doi = {10.1007/978-3-540-25939-8},
eprint = {arXiv:0706.3384v2},
isbn = {1412920388},
journal = {2015 AAAI Spring Symposium in Knowledge Representation and Reasoning: Integrating Symbolic and Neural Approaches},
pages = {53--56},
title = {{Neural Relational Learning Through Semi-Propositionalization of Bottom Clauses}},
year = {2015}
}
@article{Fermi2018,
abstract = {The paper introduces the use of machine learning with rule generation to validate collision avoidance in vehicle platooning. Cooperative Adaptive Cruise Control is under test over a range of system parameters
including speed and distance of the vehicles as well as packet error rate of the communication channel. Safety regions are evidenced on test data with statistical error very close to zero. {\textcopyright} 2018 IEEE.},
author = {Fermi, Alessandro and Mongelli, Maurizio and Muselli, Marco and Ferrari, Enrico},
doi = {10.1109/WFCS.2018.8402372},
isbn = {9781538610664},
journal = {IEEE International Workshop on Factory Communication Systems - Proceedings, WFCS},
number = {0},
pages = {1--4},
publisher = {IEEE},
title = {{Identification of safety regions in vehicle platooning via machine learning}},
volume = {2018-June},
year = {2018}
}
@article{Landecker2013,
author = {Landecker, Will and {David Thomure}, Michael and Bettencourt, Luis MA and Mitchell, Melanie and Kenyon, Garrett T and David, Michael},
isbn = {9781467358958},
pages = {32--38},
title = {{Interpreting Individual Classifications of Hierarchical Networks Citation Details}},
url = {http://pdxscholar.library.pdx.edu/compsci{\_}fac{\%}0Ahttp://pdxscholar.library.pdx.edu/compsci{\_}fac/165},
volume = {165},
year = {2013}
}
@article{Williams2015,
author = {Williams, Steve},
doi = {10.1016/j.jnnfm.2015.01.003},
journal = {WHELF/HEWIT Colloquium 2014,
{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}},
number = {10},
pages = {1063--1080},
title = {{Cronfa - Swansea University Open Access Repository
{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}}},
volume = {2014},
year = {2015}
}
@article{Cousot,
author = {Cousot, Patrick},
title = {{ATutorialOnAbstractInterpretation}}
}
@article{Nguyen2015,
abstract = {Healthcare plays an important role in promoting the general health and well-being of people around the world. The difficulty in healthcare data classification arises from the uncertainty and the high-dimensional
nature of the medical data collected. This paper proposes an integration of fuzzy standard additive model (SAM) with genetic algorithm (GA), called GSAM, to deal with uncertainty and computational challenges. GSAM learning
process comprises three continual steps: rule initialization by unsupervised learning using the adaptive vector quantization clustering, evolutionary rule optimization by GA and parameter tuning by the gradient descent
supervised learning. Wavelet transformation is employed to extract discriminative features for high-dimensional datasets. GSAM becomes highly capable when deployed with small number of wavelet features as its computational
burden is remarkably reduced. The proposed method is evaluated using two frequently-used medical datasets: the Wisconsin breast cancer and Cleveland heart disease from the UCI Repository for machine learning. Experiments are
organized with a five-fold cross validation and performance of classification techniques are measured by a number of important metrics: accuracy, F-measure, mutual information and area under the receiver operating
characteristic curve. Results demonstrate the superiority of the GSAM compared to other machine learning methods including probabilistic neural network, support vector machine, fuzzy ARTMAP, and adaptive neuro-fuzzy inference
system. The proposed approach is thus helpful as a decision support system for medical practitioners in the healthcare practice.},
author = {Nguyen, Thanh and Khosravi, Abbas and Creighton, Douglas and Nahavandi, Saeid},
doi = {10.1016/j.eswa.2014.10.027},
issn = {09574174},
journal = {Expert Systems with Applications},
keywords = {Breast cancer,Fuzzy standard additive model,Genetic algorithm,Healthcare data classification,Heart disease,Wavelet transformation},
number = {4},
pages = {2184--2197},
publisher = {Elsevier Ltd},
title = {{Classification of healthcare data using genetic fuzzy logic system and wavelets}},
url = {http://dx.doi.org/10.1016/j.eswa.2014.10.027},
volume = {42},
year = {2015}
}
@article{Baehrens2009,
abstract = {After building a classifier with modern tools of machine learning we typically have a black box at hand that is able to predict well for unseen data. Thus, we get an answer to the question what is the most likely
label of a given unseen data point. However, most methods will provide no answer why the model predicted the particular label for a single instance and what features were most influential for that particular instance. The
only method that is currently able to provide such explanations are decision trees. This paper proposes a procedure which (based on a set of assumptions) allows to explain the decisions of any classification method.},
archivePrefix = {arXiv},
arxivId = {0912.1128},
author = {Baehrens, David and Schroeter, Timon and Harmeling, Stefan and Kawanabe, Motoaki and Hansen, Katja and Mueller, Klaus-Robert},
eprint = {0912.1128},
isbn = {1532-4435},
issn = {1532-4435},
keywords = {ames mutagenicity,black box model,explaining,kernel methods,nonlinear},
pages = {1803--1831},
title = {{How to Explain Individual Classification Decisions}},
url = {http://arxiv.org/abs/0912.1128},
volume = {11},
year = {2009}
}
@article{Muggleton2018,
abstract = {During the 1980s Michie defined Machine Learning in terms of two orthogonal axes of performance: predictive accuracy and comprehensibility of generated hypotheses. Since predictive accuracy was readily measurable
and comprehensibility not so, later definitions in the 1990s, such as Mitchell's, tended to use a one-dimensional approach to Machine Learning based solely on predictive accuracy, ultimately favouring statistical over
symbolic Machine Learning approaches. In this paper we provide a definition of comprehensibility of hypotheses which can be estimated using human participant trials. We present two sets of experiments testing human
comprehensibility of logic programs. In the first experiment we test human comprehensibility with and without predicate invention. Results indicate comprehensibility is affected not only by the complexity of the presented
program but also by the existence of anonymous predicate symbols. In the second experiment we directly test whether any state-of-the-art ILP systems are ultra-strong learners in Michie's sense, and select the Metagol system
for use in humans trials. Results show participants were not able to learn the relational concept on their own from a set of examples but they were able to apply the relational definition provided by the ILP system correctly.
This implies the existence of a class of relational concepts which are hard to acquire for humans, though easy to understand given an abstract explanation. We believe improved understanding of this class could have potential
relevance to contexts involving human learning, teaching and verbal interaction. {\textcopyright} 2018, The Author(s).},
author = {Muggleton, Stephen H. and Schmid, Ute and Zeller, Christina and Tamaddoni-Nezhad, Alireza and Besold, Tarek},
doi = {10.1007/s10994-018-5707-3},
issn = {15730565},
journal = {Machine Learning},
keywords = {Comprehensibility,Inductive logic programming,Ultra-strong machine learning},
number = {7},
pages = {1119--1140},
publisher = {Springer US},
title = {{Ultra-Strong Machine Learning: comprehensibility of programs learned with ILP}},
url = {https://doi.org/10.1007/s10994-018-5707-3},
volume = {107},
year = {2018}
}
@article{Elderhalli2018,
abstract = {Dynamic Fault Trees (DFTs) is a widely used failure modeling technique that allows capturing the dynamic failure characteristics of systems in a very effective manner. Simulation and model checking have been
traditionally used for the probabilistic analysis of DFTs. Simulation is usually based on sampling and thus its results are not guaranteed to be complete, whereas model checking employs computer arithmetic and numerical
algorithms to compute the exact values of probabilities, which contain many round-off errors. Leveraging upon the expressive and sound nature of higher-order-logic (HOL) theorem proving, we propose, in this work, a
formalization of DFT gates and their probabilistic behavior as well as some of their simplification properties in HOL. This formalization would allow us to conduct the probabilistic analysis of DFTs by verifying generic
mathematical expressions about their behavior in HOL. In particular, we formalize the AND, OR, Priority-AND, Functional DEPendency, Hot SPare, Cold SPare and the Warm SPare gates and also verify their corresponding
probabilistic expressions in HOL. Moreover, we formally verify an important property, Pr(X{\textless}Y), using the Lebesgue integral as this relationship allows us to reason about the probabilistic properties of Priority-AND
gate and the Before operator. We also formalize the notion of conditional densities in order to formally verify the probabilistic expressions of the Cold SPare and the Warm SPare gates. For illustrating the usefulness of our
formalization, we use it to formally analyze the DFT of a Cardiac Assist System.},
archivePrefix = {arXiv},
arxivId = {1807.11576},
author = {Elderhalli, Yassmeen and Ahmad, Waqar and Hasan, Osman and Tahar, Sofiene},
eprint = {1807.11576},
number = {July},
title = {{Formal Probabilistic Analysis of Dynamic Fault Trees in HOL4}},
url = {http://arxiv.org/abs/1807.11576},
year = {2018}
}
@article{Kazakov2001,
author = {Kazakov, Dimitar and Kudenko, Daniel},
title = {{Machine learning and inductive logic programming for multi-agent systems}},
year = {2001}
}
@article{Bach2015,
abstract = {Understanding and interpreting classification decisions of automated image classification systems is of high value in many applications, as it allows to verify the reasoning of the system and provides additional
information to the human expert. Although machine learning methods are solving very successfully a plethora of tasks, they have in most cases the disadvantage of acting as a black box, not providing any information about what
made them arrive at a particular decision. This work proposes a general solution to the problem of understanding classification decisions by pixel-wise decomposition of nonlinear classifiers. We introduce a methodology that
allows to visualize the contributions of single pixels to predictions for kernel-based classifiers over Bag of Words features and for multilayered neural networks. These pixel contributions can be visualized as heatmaps and
are provided to a human expert who can intuitively not only verify the validity of the classification decision, but also focus further analysis on regions of potential interest. We evaluate our method for classifiers trained
on PASCAL VOC 2009 images, synthetic image data containing geometric shapes, the MNIST handwritten digits data set and for the pre-trained ImageNet model available as part of the Caffe open source package.},
archivePrefix = {arXiv},
arxivId = {1606.04155},
author = {Bach, Sebastian and Binder, Alexander and Montavon, Gr{\'{e}}goire and Klauschen, Frederick and M{\"{u}}ller, Klaus Robert and Samek, Wojciech},
doi = {10.1371/journal.pone.0130140},
eprint = {1606.04155},
isbn = {10.1371/journal.pone.0130140},
issn = {19326203},
journal = {PLoS ONE},
number = {7},
pages = {1--46},
pmid = {26161953},
title = {{On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation}},
volume = {10},
year = {2015}
}
@article{Aziz2010,
author = {Aziz, Rehan Abdul},
pages = {1--24},
title = {{Distributed Model Checking Using Hadoop}},
year = {2010}
}
@article{Lehmann1982,
author = {Lehmann, D and Shelah, S},
journal = {Information and Control},
number = {3},
pages = {165--198},
title = {{Reasoning About Time and Chance}},
volume = {53},
year = {1982}
}
@article{Muselli2003,
author = {Muselli, Marco and Quarati, Alfonso},
keywords = {boolean lattice,clustering,logical synthesis,lower shadow,monotone boolean function,shadow},
pages = {1--22},
title = {{Shadow Clustering : A Method for Monotone Boolean Function Synthesis}},
year = {2003}
}
@inproceedings{Ehlers2017,
abstract = {We present an approach for the verification of feed-forward neural networks in which all nodes have a piece-wise linear activation function. Such networks are often used in deep learning and have been shown to be
hard to verify for modern satisfiability modulo theory (SMT) and integer linear programming (ILP) solvers. The starting point of our approach is the addition of a global linear approximation of the overall network behavior to
the verification problem that helps with SMT-like reasoning over the network behavior. We present a specialized verification algorithm that employs this approximation in a search process in which it infers additional node
phases for the non-linear nodes in the network from partial node phase assignments, similar to unit propagation in classical SAT solving. We also show how to infer additional conflict clauses and safe node fixtures from the
results of the analysis steps performed during the search. The resulting approach is evaluated on collision avoidance and handwritten digit recognition case studies.},
archivePrefix = {arXiv},
arxivId = {1705.01320},
author = {Ehlers, R{\"{u}}diger},
booktitle = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
doi = {10.1007/978-3-319-68167-2_19},
eprint = {1705.01320},
isbn = {9783319681665},
issn = {16113349},
pages = {269--286},
pmid = {21552100},
title = {{Formal verification of piece-wise linear feed-forward neural networks}},
volume = {10482 LNCS},
year = {2017}
}
@article{Ribeiro2016,
abstract = {Despite widespread adoption, machine learning models remain mostly black boxes. Understanding the reasons behind predictions is, however, quite important in assessing trust, which is fundamental if one plans to
take action based on a prediction, or when choosing whether to deploy a new model. Such understanding also provides insights into the model, which can be used to transform an untrustworthy model or prediction into a
trustworthy one. In this work, we propose LIME, a novel explanation technique that explains the predictions of any classifier in an interpretable and faithful manner, by learning an interpretable model locally around the
prediction. We also propose a method to explain models by presenting representative individual predictions and their explanations in a non-redundant way, framing the task as a submodular optimization problem. We demonstrate
the flexibility of these methods by explaining different models for text (e.g. random forests) and image classification (e.g. neural networks). We show the utility of explanations via novel experiments, both simulated and
with human subjects, on various scenarios that require trust: deciding if one should trust a prediction, choosing between models, improving an untrustworthy classifier, and identifying why a classifier should not be
trusted.},
archivePrefix = {arXiv},
arxivId = {1602.04938},
author = {Ribeiro, Marco Tulio and Singh, Sameer and Guestrin, Carlos},
doi = {10.18653/v1/N16-3020},
eprint = {1602.04938},
isbn = {9781450321389},
issn = {9781450321389},
pmid = {214160309},
title = {{"Why Should I Trust You?": Explaining the Predictions of Any Classifier}},
url = {http://arxiv.org/abs/1602.04938},
year = {2016}
}
@article{Kruger2014,
abstract = {The performance characteristics of 2 activated carbons and 8 synthetic adsorbents are compared. The comparison addresses both the equilibrium capacities and mass transport properties of the adsorbents. A new
empirical adsorption isotherm is employed for description of adsorption equilibrium data. A predictive mathematical model is used for forecasting adsorption bed performance from independent batch reactor system data.},
author = {Kr{\"{u}}ger, Frank and Nyolt, Martin and Yordanova, Kristina and Hein, Albert and Kirste, Thomas},
doi = {10.1371/journal.pone.0109381},
isbn = {1932-6203 (Electronic)$\backslash$r1932-6203 (Linking)},
issn = {19326203},
journal = {PLoS ONE},
number = {11},
pmid = {25372138},
title = {{Computational state space models for activity and intention recognition. A feasibility study}},
volume = {9},
year = {2014}
}
@article{Dutta2018,
abstract = {Deep neural networks (NN) are extensively used for machine learning tasks such as image classification, perception and control of autonomous systems. Increasingly, these deep NNs are also been deployed in
high-assurance applications. Thus, there is a pressing need for developing techniques to verify neural networks to check whether certain user-expected properties are satisfied. In this paper, we study a specific verification
problem of computing a guaranteed range for the output of a deep neural network given a set of inputs represented as a convex polyhedron. Range estimation is a key primitive for verifying deep NNs. We present an efficient
range estimation algorithm that uses a combination of local search and linear programming problems to efficiently find the maximum and minimum values taken by the outputs of the NN over the given input set. In contrast to
recently proposed "monolithic" optimization approaches, we use local gradient descent to repeatedly find and eliminate local minima of the function. The final global optimum is certified using a mixed integer programming
instance. We implement our approach and compare it with Reluplex, a recently proposed solver for deep neural networks. We demonstrate the effectiveness of the proposed approach for verification of NNs used in automated
control as well as those used in classification.},
archivePrefix = {arXiv},
arxivId = {1709.09130},
author = {Dutta, Souradeep and Jha, Susmit and Sankaranarayanan, Sriram and Tiwari, Ashish},
doi = {10.1007/978-3-319-77935-5_9},
eprint = {1709.09130},
isbn = {9783319779348},
issn = {16113349},
journal = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
pages = {121--138},
title = {{Output range analysis for deep feedforward neural networks}},
volume = {10811 LNCS},
year = {2018}
}
@article{Shrikumar2017,
abstract = {The purported "black box"' nature of neural networks is a barrier to adoption in applications where interpretability is essential. Here we present DeepLIFT (Deep Learning Important FeaTures), a method for
decomposing the output prediction of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. DeepLIFT compares the activation of each neuron to
its 'reference activation' and assigns contribution scores according to the difference. By optionally giving separate consideration to positive and negative contributions, DeepLIFT can also reveal dependencies which are
missed by other approaches. Scores can be computed efficiently in a single backward pass. We apply DeepLIFT to models trained on MNIST and simulated genomic data, and show significant advantages over gradient-based methods. A
detailed video tutorial on the method is at http://goo.gl/qKb7pL and code is at http://goo.gl/RM8jvH.},
archivePrefix = {arXiv},
arxivId = {1704.02685},
author = {Shrikumar, Avanti and Greenside, Peyton and Kundaje, Anshul},
eprint = {1704.02685},
isbn = {9781510855144},
issn = {1938-7228},
title = {{Learning Important Features Through Propagating Activation Differences}},
url = {http://arxiv.org/abs/1704.02685},
volume = {1},
year = {2017}
}
@article{Hein2017,
abstract = {Recent work has shown that state-of-the-art classifiers are quite brittle, in the sense that a small adversarial change of an originally with high confidence correctly classified input leads to a wrong
classification again with high confidence. This raises concerns that such classifiers are vulnerable to attacks and calls into question their usage in safety-critical systems. We show in this paper for the first time formal
guarantees on the robustness of a classifier by giving instance-specific lower bounds on the norm of the input manipulation required to change the classifier decision. Based on this analysis we propose the Cross-Lipschitz
regularization functional. We show that using this form of regularization in kernel methods resp. neural networks improves the robustness of the classifier without any loss in prediction performance.},
archivePrefix = {arXiv},
arxivId = {1705.08475},
author = {Hein, Matthias and Andriushchenko, Maksym},
doi = {10.1038/363558a0},
eprint = {1705.08475},
isbn = {0028-0836},
issn = {10495258},
title = {{Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation}},
year = {2017}
}
@article{Cangelosi2013,
abstract = {BACKGROUND: Neuroblastoma is the most common pediatric solid tumor. About fifty percent of high risk patients die despite treatment making the exploration of new and more effective strategies for improving
stratification mandatory. Hypoxia is a condition of low oxygen tension occurring in poorly vascularized areas of the tumor associated with poor prognosis. We had previously defined a robust gene expression signature measuring
the hypoxic component of neuroblastoma tumors (NB-hypo) which is a molecular risk factor. We wanted to develop a prognostic classifier of neuroblastoma patients' outcome blending existing knowledge on clinical and molecular
risk factors with the prognostic NB-hypo signature. Furthermore, we were interested in classifiers outputting explicit rules that could be easily translated into the clinical setting.$\backslash$n$\backslash$nRESULTS: Shadow
Clustering (SC) technique, which leads to final models called Logic Learning Machine (LLM), exhibits a good accuracy and promises to fulfill the aims of the work. We utilized this algorithm to classify NB-patients on the
bases of the following risk factors: Age at diagnosis, INSS stage, MYCN amplification and NB-hypo. The algorithm generated explicit classification rules in good agreement with existing clinical knowledge. Through an iterative
procedure we identified and removed from the dataset those examples which caused instability in the rules. This workflow generated a stable classifier very accurate in predicting good and poor outcome patients. The good
performance of the classifier was validated in an independent dataset. NB-hypo was an important component of the rules with a strength similar to that of tumor staging.$\backslash$n$\backslash$nCONCLUSIONS: The novelty of our
work is to identify stability, explicit rules and blending of molecular and clinical risk factors as the key features to generate classification rules for NB patients to be conveyed to the clinic and to be used to design new
therapies. We derived, through LLM, a set of four stable rules identifying a new class of poor outcome patients that could benefit from new therapies potentially targeting tumor hypoxia or its consequences.},
author = {Cangelosi, Davide and Blengio, Fabiola and Versteeg, Rogier and Eggert, Angelika and Garaventa, Alberto and Gambini, Claudio and Conte, Massimo and Eva, Alessandra and Muselli, Marco and Varesio, Luigi},
doi = {10.1186/1471-2105-14-S7-S12},
issn = {14712105},
journal = {BMC Bioinformatics},
number = {SUPPL7},
pmid = {23815266},
title = {{Logic learning machine creates explicit and stable rules stratifying neuroblastoma patients}},
volume = {14},
year = {2013}
}
@article{Cheng2012,
abstract = {This article illustrates how to compute the probability in Markov logic network (MLN) using examples. Also we show how the Markov logic ”softens” the rigorous constraints in First-order logic but still keep its
powerful express ability},
author = {Cheng, Guangchun},
doi = {10.1007/s00779-014-0827-7},
isbn = {1617-4909},
issn = {16174909},
keywords = {Abnormality detection,Activity recognition,Dementia care,Hierarchical activity recognition,Hybrid approach,Markov Logic Network,Smart environment},
title = {{How to Compute Probability in MLN: An Tutorial in Examples}},
year = {2012}
}
@book{Leveson2011,
author = {Leveson, Nancy G.},
publisher = {MIT Press},
title = {{Engineering A Safer World}},
year = {2011}
}
@article{DeRaedt2008,
abstract = {Probabilistic inductive logic programming aka. statistical relational learning addresses one of the central questions of artificial intelligence: the integration of probabilistic reasoning with machine learning
and first order and relational logic representations. A rich variety of different formalisms and learning techniques have been developed. A unifying characterization of the underlying learning settings, however, is missing so
far.},
author = {{De Raedt}, Luc and Kersting, Kristian},
doi = {10.1007/978-3-540-78652-8_1},
isbn = {978-3-540-78651-1},
journal = {Probabilistic Inductive Logic Programming: Theory and Applications},
pages = {1--27},
title = {{Probabilistic Inductive Logic Programming}},
url = {http://link.springer.com/10.1007/978-3-540-78652-8{\%}0Ahttp://link.springer.com/10.1007/978-3-540-78652-8{\_}1},
volume = {4911},
year = {2008}
}
@article{Rodriguez-Garcia2018,
abstract = {Ontologies are representations of a conceptualization of a domain. Traditionally, ontologies in biology were represented as directed acyclic graphs (DAG) which represent the backbone taxonomy and additional
relations between classes. These graphs are widely exploited for data analysis in the form of ontology enrichment or computation of semantic similarity. More recently, ontologies are developed in a formal language such as the
Web Ontology Language (OWL) and consist of a set of axioms through which classes are defined or constrained. While the taxonomy of an ontology can be inferred directly from the axioms of an ontology as one of the standard OWL
reasoning tasks, creating general graph structures from OWL ontologies that exploit the ontologies' semantic content remains a challenge. We developed a method to transform ontologies into graphs using an automated reasoner
while taking into account all relations between classes. Searching for (existential) patterns in the deductive closure of ontologies, we can identify relations between classes that are implied but not asserted and generate
graph structures that encode for a large part of the ontologies' semantic content. We demonstrate the advantages of our method by applying it to inference of protein-protein interactions through semantic similarity over the
Gene Ontology and demonstrate that performance is increased when graph structures are inferred using deductive inference according to our method. Our software and experiment results are available at
                    http://github.com/bio-ontology-research-group/Onto2Graph

                  . Onto2Graph is a method to generate graph structures from OWL ontologies using automated reasoning. The resulting graphs can be used for improved ontology visualization and ontology-based data analysis.},
author = {Rodr{\'{i}}guez-Garc{\'{i}}a, Miguel {\'{A}}ngel and Hoehndorf, Robert},
doi = {10.1186/s12859-017-1999-8},
isbn = {1471-2105 (Electronic)},
issn = {1471-2105},
journal = {BMC Bioinformatics},
keywords = {automated reasoning,ontology,ontology visualization,owl,semantic similarity},
number = {1},
pages = {7},
pmid = {29304741},
publisher = {BMC Bioinformatics},
title = {{Inferring ontology graph structures using OWL reasoning}},
url = {https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-017-1999-8},
volume = {19},
year = {2018}
}
@article{Gayathri2015,
abstract = {Statistics reveal that globally, the aging population in different stages of dementia are struggling to cope with daily activities and are progressively becoming dependent on care takers thereby making dementia
care a challenging social problem. Healthcare systems in smart environments that aim to address this growing social need require the support of technology to recognize and respond in an ubiquitous manner. To incorporate an
efficient activity recognition and abnormality detection system in smart environments, the routine activities of the occupant are modeled and any deviation from the activity model is recognized as abnormality. Recognition
systems are generally designed using machine learning strategies and in this paper a novel hybrid, data and knowledge-driven approach is introduced. Markov Logic Network (MLN) used in our design is a suitable approach for
activity recognition as it integrates common sense knowledge with a probabilistic model that augments the recognition ability of the system. The proposed activity recognition system for dementia care uses a hierarchical
approach to detect abnormality in occupant behavior using MLN. The abnormality in the context of dementia care is identified in terms of factors associated with the activity such as objects, location, time and duration. The
task of recognition is done in a hierarchical manner based on the priority of the factor that is associated with each layer. The motivation for designing a hierarchical approach was to enable each layer to commence its
computation after inferring from the lower layers that the ongoing activity was normal with regard to the associated factors. This hierarchical feature enables quick decisions, as factors that require immediate attention are
processed first at the lowest layer. Experimental results indicate that the hierarchical approach has higher accuracy in recognition and efficient response time when compared to the existing approaches.},
author = {Gayathri, K. S. and Elias, Susan and Ravindran, Balaraman},
doi = {10.1007/s00779-014-0827-7},
isbn = {1617-4909},
issn = {16174909},
journal = {Personal and Ubiquitous Computing},
keywords = {Abnormality detection,Activity recognition,Dementia care,Hierarchical activity recognition,Hybrid approach,Markov Logic Network,Smart environment},
number = {2},
pages = {271--285},
title = {{Hierarchical activity recognition for dementia care using Markov Logic Network}},
volume = {19},
year = {2015}
}
@article{Klinken2006,
abstract = {A new research area, Inductive Logic Programming, is presently emerging. While inheriting various positive characteristics of the parent subjects of Logic Programming and Machine Learning, it is hoped that the new
area will overcome many of the limitations of its forebears. The background to present developments within this area is discussed and various goals and aspirations for the increasing body of researchers are identified.
Inductive Logic Programming needs to be based on sound principles from both Logic and Statistics. On the side of statistical justification of hypotheses we discuss the possible relationship be-tween Algorithmic Complexity
theory and Probably-Approximately-Correct (PAC) Learning. In terms of logic we provide a unifying framework for Muggleton and Buntine's Inverse Resolution (IR) and Plotkin's Relative Least General Generali-sation (RLGG) by
rederiving RLGG in terms of IR. This leads to a discussion of the feasibility of extending the RLGG framework to allow for the invention of new predicates, previously discussed only within the cor{\~{}}text of IR.},
author = {Klinken, Jan Bert Van},
doi = {10.1007/BFb0027303},
isbn = {3-540-64738-4},
issn = {0288-3635},
journal = {October},
keywords = {inductive logic,inductive logic programming,ma-,scienti c method},
number = {January 2016},
pages = {1--81},
title = {{Inductive Logic Programming}},
volume = {8},
year = {2006}
}
@article{Richardson2006,
abstract = {We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base with a weight attached to
each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible grounding of a first-order formula in the KB, with
the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query. Weights are efficiently learned from relational databases by iteratively
optimizing a pseudo-likelihood measure. Optionally, additional clauses are learned using inductive logic programming techniques. Experiments with a real-world database and knowledge base in a university domain illustrate the
promise of this approach.},
archivePrefix = {arXiv},
arxivId = {arXiv:1103.0398},
author = {Richardson, Matthew and Domingos, Pedro},
doi = {10.1007/s10994-006-5833-1},
eprint = {arXiv:1103.0398},
isbn = {0885-6125},
issn = {08856125},
journal = {Machine Learning},
keywords = {First-order logic,Graphical models,Inductive logic programming,Knowledge-based model construction,Log-linear models,Markov chain Monte Carlo,Markov networks,Markov random
fields,Pseudo-likelihood,Satisfiability,Statistical relational learning},
number = {1-2 SPEC. ISS.},
pages = {107--136},
pmid = {20958249},
title = {{Markov logic networks}},
volume = {62},
year = {2006}
}
@article{Khot2015,
abstract = {Our goal is to answer elementary-level science questions using knowledge extracted automatically from science textbooks, expressed in a subset of first-order logic. Given the incomplete and noisy nature of these
automatically extracted rules, Markov Logic Networks (MLNs) seem a natural model to use, but the exact way of leveraging MLNs is by no means obvious. We investigate three ways of applying MLNs to our task. In the first, we
simply use the extracted science rules directly as MLN clauses. Unlike typical MLN applications, our domain has long and complex rules, leading to an unmanageable number of groundings. We exploit the structure present in hard
constraints to improve tractability, but the formulation remains ineffective. In the second approach, we instead interpret science rules as describing prototypical entities, thus mapping rules directly to grounded MLN
assertions, whose constants are then clustered using existing entity resolution methods. This drastically simplifies the network, but still suffers from brittleness. Finally, our third approach, called Praline, uses MLNs to
align the lexical elements as well as define and control how inference should be performed in this task. Our experiments, demonstrating a 15$\backslash${\%} accuracy boost and a 10x reduction in runtime, suggest that the
flexibility and different inference semantics of Praline are a better fit for the natural language question answering task.},
archivePrefix = {arXiv},
arxivId = {1507.03045},
author = {Khot, Tushar and Balasubramanian, Niranjan and Gribkoff, Eric and Sabharwal, Ashish and Clark, Peter and Etzioni, Oren},
eprint = {1507.03045},
isbn = {9781941643327},
journal = {arXiv preprint arXiv:1507.03045},
title = {{Markov Logic Networks for Natural Language Question Answering}},
url = {http://arxiv.org/abs/1507.03045},
year = {2015}
}
@article{Mairal2012,
abstract = {The regularization path of the Lasso can be shown to be piecewise linear, making it possible to "follow" and explicitly compute the entire path. We analyze in this paper this popular strategy, and prove that its
worst case complexity is exponential in the number of variables. We then oppose this pessimistic result to an (optimistic) approximate analysis: We show that an approximate path with at most O(1/sqrt(epsilon)) linear segments
can always be obtained, where every point on the path is guaranteed to be optimal up to a relative epsilon-duality gap. We complete our theoretical analysis with a practical algorithm to compute these approximate paths.},
archivePrefix = {arXiv},
arxivId = {1205.0079},
author = {Mairal, Julien and Yu, Bin},
eprint = {1205.0079},
isbn = {978-1-4503-1285-1},
issn = {10495258},
keywords = {Lasso, regularization path, homotopy},
number = {1952},
title = {{Complexity Analysis of the Lasso Regularization Path}},
url = {http://arxiv.org/abs/1205.0079},
year = {2012}
}
@article{Etal.Ash2012,
author = {et al. Ash, Robert},
pages = {1--47},
title = {{VNR0: Mathematics for Machine Learning}},
year = {2012}
}
@article{Yurichev2018,
author = {Yurichev, Dennis},
title = {{SAT / SMT by example}},
year = {2018}
}
@article{Mansour2017,
author = {Mansour, Romany F. and Hosni, Samar},
doi = {10.17485/ijst/2017/v10i19/113384},
issn = {0974-5645},
journal = {Indian Journal of Science and Technology},
keywords = {evolutionary algorithms,fuzzy logic,machine learning,markov logic,soft computing},
number = {19},
pages = {1--7},
title = {{Recent Advances in Markov Logic Networks}},
url = {http://www.indjst.org/index.php/indjst/article/view/113384},
volume = {10},
year = {2017}
}
@article{Muggleton1994,
annote = {- Closed/Open Domain},
author = {Muggleton, Stephen and {De Raedt}, Luc},
journal = {Journal of Logic Programming 1994},
title = {{Inductive Logic Programming: Theory and Methods}},
year = {1994}
}
@article{Jong1990,
author = {Jong, De},
pages = {351--353},
title = {{No Title}},
volume = {3},
year = {1990}
}
@inproceedings{Sibai2017,
author = {Sibai, Hussein},
booktitle = {Neural Information Processing Systems},
title = {{Verification of Neural Nets : Towards Deterministic Guarantees}},
year = {2018}
}
@article{Muselli,
author = {Muselli, Marco},
pages = {56--58},
title = {{Extracting knowledge from biomedical data through Logic Learning}}
}
@article{Grenoble2017,
author = {Grenoble, Inria},
title = {{Foundations of Deep Learning from a Kernel Point of View Julien Mairal}},
year = {2017}
}
@misc{Franco2009,
author = {Franco, John and Martin, John},
booktitle = {Frontiers in Artificial Intelligence and Applications},
doi = {10.3233/978-1-58603-929-5-3},
isbn = {9781586039295},
issn = {09226389},
number = {1},
pages = {3--74},
title = {{A history of satisfiability}},
volume = {185},
year = {2009}
}
@article{Meijer2017,
abstract = {Agda is a dependently typed functional programming language and a proof assistant in which developing programs and proving their correctness is one activity. We show how this process can be enhanced by integrating
external automated theorem provers, provide a prototypical integration of the equational theorem prover Waldmeister, and give examples of how this proof automation works in practice.},
author = {Meijer, Jeroen and {Van de Pol}, Jaco},
doi = {10.1007/978-3-642-38088-4},
isbn = {978-3-642-38087-7},
issn = {03029743},
pages = {349--366},
publisher = {Springer International Publishing},
title = {{Sound Black-Box Checking in the LearnLib}},
url = {http://link.springer.com/10.1007/978-3-642-38088-4},
volume = {7871},
year = {2017}
}
@article{Lipton2016,
abstract = {Supervised machine learning models boast remarkable predictive capabilities. But can you trust your model? Will it work in deployment? What else can it tell you about the world? We want models to be not only good,
but interpretable. And yet the task of interpretation appears underspecified. Papers provide diverse and sometimes non-overlapping motivations for interpretability, and offer myriad notions of what attributes render models
interpretable. Despite this ambiguity, many papers proclaim interpretability axiomatically, absent further explanation. In this paper, we seek to refine the discourse on interpretability. First, we examine the motivations
underlying interest in interpretability, finding them to be diverse and occasionally discordant. Then, we address model properties and techniques thought to confer interpretability, identifying transparency to humans and
post-hoc explanations as competing notions. Throughout, we discuss the feasibility and desirability of different notions, and question the oft-made assertions that linear models are interpretable and that deep neural networks
are not.},
archivePrefix = {arXiv},
arxivId = {1606.03490},
author = {Lipton, Zachary C.},
doi = {10.1145/3236386.3241340},
eprint = {1606.03490},
issn = {1542-7730},
number = {Whi},
title = {{The Mythos of Model Interpretability}},
url = {http://arxiv.org/abs/1606.03490},
year = {2016}
}
@article{Morente-Molinera2017,
abstract = {—Obtaining good classification results using supervised learning methods is critical if we want to obtain a high level of pre-cision in the classification processes. The training data used for the learning process
play a very important role in achieving this objec-tive. Therefore, it is important to represent the data in a way that best expresses its meaning. For this purpose, we propose to apply linguistic modeling methods in order to
obtain a linguistic repre-sentation. With the help of multigranular linguistic modeling, data can be transformed and expressed using different (unbalanced) linguistic label sets. Expressing the data using linguistic
expres-sions instead of numbers increases the readability and reduces the complexity of the problem, and data recovering methods allow us to manually control the level of precision. In this paper, several datasets are
transformed and utilized for classification tasks us-ing several supervised learning algorithms. For each combination of datasets and algorithms, the data have been expressed using several linguistic label sets that have
different granularity values. After carrying out the testing processes, we can conclude that, in some cases, reducing data complexity leads to better classification results. Therefore, it is found that linguistic
representation of the training data with just the necessary and sufficient precision can improve the reliability of the classification process.},
author = {Morente-Molinera, Juan Antonio and Mezei, Jozsef and Carlsson, Christer and Herrera-Viedma, Enrique},
doi = {10.1109/TFUZZ.2016.2594275},
issn = {10636706},
journal = {IEEE Transactions on Fuzzy Systems},
keywords = {Classification,computing with words,granular computing,multigranular fuzzy linguistic modeling,supervised learning},
number = {5},
pages = {1078--1089},
title = {{Improving Supervised Learning Classification Methods Using Multigranular Linguistic Modeling and Fuzzy Entropy}},
volume = {25},
year = {2017}
}
@article{Segler2017,
abstract = {Reaction prediction and retrosynthesis are the cornerstones of organic chemistry. Rule-based expert systems have been the most widespread approach to computationally solve these two related challenges to date.
However, reaction rules often fail because they ignore the molecular context, which leads to reactivity conflicts. We now report that deep neural networks can learn overnight on a single machine which reaction rules can be
applied and to resolve reactivity conflicts. We show that by training our model on 3.5 million reactions taken from the collective published knowledge of the entire discipline of chemistry, our model exhibits a top10-accuracy
of 95{\%} in retrosynthesis and 97{\%} for reaction prediction on a validation set of almost 1 million reactions},
author = {Segler, Marwin H.S. and Waller, Mark P.},
doi = {10.1002/chem.201605499},
isbn = {1751-7915 (Electronic) 1751-7915 (Linking)},
issn = {15213765},
journal = {Chemistry - A European Journal},
keywords = {artificial intelligence,machine learning,retrosynthesis,synthesis design,total synthesis},
number = {25},
pages = {5966--5971},
pmid = {28134452},
title = {{Neural-Symbolic Machine Learning for Retrosynthesis and Reaction Prediction}},
volume = {23},
year = {2017}
}
@misc{Kuper2017,
abstract = {Neural networks are turning up everywhere these days, including in safety-critical systems, such as autonomous driving and flight control systems. When these systems fail, human lives are at risk. But it's hard to
provide formal guarantees about the behavior of neural networks — how can we know for sure that they won't steer us the wrong way?},
author = {Kuper, Lindsey},
booktitle = {http://composition.al/blog/2017/05/30/proving-that-safety-critical-neural-networks-do-what-theyre-supposed-to-where-we-are-where-were-going-part-1-of-2/},
pages = {2},
title = {{Proving that safety-critical neural networks do what they're supposed to: where we are, where we're going (part 1 of 2)}},
url = {http://composition.al/blog/2017/05/30/proving-that-safety-critical-neural-networks-do-what-theyre-supposed-to-where-we-are-where-were-going-part-1-of-2/},
urldate = {2018-11-01},
year = {2017}
}
@article{Hullermeier2015,
abstract = {This article is a short position paper in which the author outlines his (necessarily subjective) perception of current research in fuzzy machine learning, that is, the use of formal concepts and mathematical tools
from fuzzy sets and fuzzy logic in the field of machine learning. The paper starts with a critical appraisal of previous contributions to fuzzy machine learning and ends with a suggestion of some directions for future work.},
annote = {Inferring fuzzy rules from a normal ML model using the deffuzification process of the features in the data set?},
author = {H{\"{u}}llermeier, Eyke},
doi = {10.1016/j.fss.2015.09.001},
issn = {01650114},
journal = {Fuzzy Sets and Systems},
keywords = {Fuzzy logic,Fuzzy sets,Machine learning},
pages = {292--299},
publisher = {Elsevier B.V.},
title = {{Does machine learning need fuzzy logic?}},
url = {http://dx.doi.org/10.1016/j.fss.2015.09.001},
volume = {281},
year = {2015}
}
@misc{Sagiv,
abstract = {Z3 is a high performance theorem prover developed at Microsoft Research. Z3 is used in many applications such as: software/hardware verification and testing, constraint solving, analysis of hybrid systems,
security, biology (in silico analysis), and geometrical problems. This tutorial demonstrates the main capabilities of Z3Py: the Z3 API in Python. No Python background is needed to read this tutorial. However, it is useful to
learn Python (a fun language!) at some point, and there are many excellent free resources for doing so (Python Tutorial).},
author = {Sagiv, Mooly},
title = {{Z3Py Guide}},
url = {http://www.cs.tau.ac.il/{~}msagiv/courses/asv/z3py/guide-examples.htm},
urldate = {2018-11-01}
}
@article{Narizzano2017,
abstract = {Property Specification Patterns (PSPs) have been proposed to solve recurring specification needs, to ease the formalization of requirements, and enable automated verification thereof. In this paper, we extend PSPs
by considering Boolean as well as atomic assertions from a constraint system. This extension enables us to reason about functional requirements which could not be captured by basic PSPs. We contribute an encoding from
constrained PSPs to LTL formulas, and we show experimental results demonstrating that our approach scales on requirements of realistic size generated using an artificial probabilistic model. Finally, we show that our
extension enables us to prove (in)consistency of requirements about an embedded controller for a robotic manipulator.},
archivePrefix = {arXiv},
arxivId = {1712.04162},
author = {Narizzano, Massimo and Pulina, Luca and Tacchella, Armando and Vuotto, Simone},
eprint = {1712.04162},
journal = {NASA Formal Methods},
pages = {383--398},
title = {{Consistency of Property Specification Patterns with Boolean and Constrained Numerical Signals}},
url = {http://arxiv.org/abs/1712.04162},
volume = {10811},
year = {2018}
}
@book{Counsell2014,
abstract = {Visual sensor network design facilitates applications such as intelligent rooms, video surveillance, automatic multi-camera tracking, activity recognition etc. These applications require an efficient visual sensor
layout which provides a minimum level of image quality or image resolution. This paper addresses the practical problem of optimally placing the multiple PTZ cameras to ensure maximum coverage of user defined priority areas
with optimum values of parameters like pan, tilt, zoom and the locations of the cameras. The proposed algorithm works offline and does not require camera calibration. We mapped this problem as an optimization problem using
Genetic Algorithm, by defining, coverage matrix as a set of sensor parameters and the space model parameters like priority areas, obstacles and feasible locations of the sensors, and by modelling discrete spaces using
probabilistic frame work. We minimized the probability of occlusion due to randomly moving objects by covering each priority area using multiple cameras. The proposed method will be applicable for surveillance of large spaces
with discrete priority areas like a hall with more than one entrance or many events happening at different locations in a hall eg.Casino. As we are optimizing the parameters like pan, tilt, zoom and even the locations of the
cameras, the coverage provided by this approach will assure good resolution, which improves the QOS of the visual sensor network.},
annote = {Page 23 for Switching Neural Networks},
archivePrefix = {arXiv},
arxivId = {9780201398298},
author = {Counsell, Steve and N{\'{u}}nez, Manuel},
booktitle = {Software Engineering and Formal Methods},
doi = {10.1021/pr800044q},
eprint = {9780201398298},
isbn = {1057-7149},
issn = {15353893},
keywords = {FLOSS,OSS,Petri nets,behavioral type theory,complex systems,formal methods,knowledge management,logic and verification,open source software,program constructs,programming languages,railways,software creation and
management,software development methods,software extra-functional properties,software organization and properties,software system structures,software verification and validation,sustainability,sustainable development,type
structures},
pages = {432},
pmid = {20350857},
title = {{Lecture notes in computer science}},
url = {http://link.springer.com/book/10.1007/978-3-319-05032-4},
year = {2014}
}
@article{Nilashi2017,
abstract = {Breast cancer has become a common disease around the world. Expert systems are valuable tools that have been successful for the disease diagnosis. In this research, we accordingly develop a new knowledge-based
system for classification of breast cancer disease using clustering, noise removal, and classification techniques. Expectation Maximization (EM) is used as a clustering method to cluster the data in similar groups. We then
use Classification and Regression Trees (CART) to generate the fuzzy rules to be used for the classification of breast cancer disease in the knowledge-based system of fuzzy rule-based reasoning method. To overcome the
multi-collinearity issue, we incorporate Principal Component Analysis (PCA) in the proposed knowledge-based system. Experimental results on Wisconsin Diagnostic Breast Cancer and Mammographic mass datasets show that proposed
methods remarkably improves the prediction accuracy of breast cancer. The proposed knowledge-based system can be used as a clinical decision support system to assist medical practitioners in the healthcare practice.},
author = {Nilashi, Mehrbakhsh and Ibrahim, Othman and Ahmadi, Hossein and Shahmoradi, Leila},
doi = {10.1016/j.tele.2017.01.007},
issn = {07365853},
journal = {Telematics and Informatics},
keywords = {Breast cancer,CART,Classification,Fuzzy logic,Mammographic mass,Wisconsin Diagnostic Breast Cancer},
number = {4},
pages = {133--144},
publisher = {Elsevier Ltd},
title = {{A knowledge-based system for breast cancer classification using fuzzy logic method}},
url = {http://dx.doi.org/10.1016/j.tele.2017.01.007},
volume = {34},
year = {2017}
}
@article{Jain2011,
abstract = {Within the realm of statistical relational knowledge represen-tation formalisms, Markov logic is perhaps one of the most flexible and general languages, for it generalises both first-order logic (for finite
do-mains) and probabilistic graphical models. Knowledge engineering with Markov logic is, however, not a straightforward task. In particular, mod-elling approaches that are too firmly rooted in the principles of logic often
tend to produce unexpected results in practice. In this paper, I collect a number of issues that are relevant to knowledge engineering practice: I describe the fundamental semantics of Markov logic networks and explain how
simple probabilistic properties can be represented. Furthermore, I discuss fallacious modelling assumptions and summarise conditions un-der which generalisation across domains may fail. As a collection of fun-damental
insights, the paper is primarily directed at knowledge engineers who are new to Markov logic.},
author = {Jain, Dominik},
journal = {Evolving Knowledge in Theory and Applications. 3rd Workshop on Dynamics of Knowledge and Belief (DKB-2011) at the 34th Annual German Conference on Artificial Intelligence, KI-2011, Berlin, Germany, October 4, 2011.
Proceedings},
pages = {16--30},
title = {{Knowledge Engineering with Markov Logic Networks: A Review}},
url = {http://www.fernuniversitthagen.de/wbs/research/papers/res/BeierleKernIsberner2011InfBericht361.pdf{\#}page=24},
volume = {361},
year = {2011}
}
@article{Arras2017,
abstract = {Text documents can be described by a number of abstract concepts such as semantic category, writing style, or sentiment. Machine learning (ML) models have been trained to automatically map documents to these
abstract concepts, allowing to annotate very large text collections, more than could be processed by a human in a lifetime. Besides predicting the text's category very accurately, it is also highly desirable to understand how
and why the categorization process takes place. In this paper, we demonstrate that such understanding can be achieved by tracing the classification decision back to individual words using layer-wise relevance propagation
(LRP), a recently developed technique for explaining predictions of complex non-linear classifiers. We train two word-based ML models, a convolutional neural network (CNN) and a bag-of-words SVM classifier, on a topic
categorization task and adapt the LRP method to decompose the predictions of these models onto words. Resulting scores indicate how much individual words contribute to the overall classification decision. This enables one to
distill relevant information from text documents without an explicit semantic information extraction step. We further use the word-wise relevance scores for generating novel vector-based document representations which capture
semantic information. Based on these document vectors, we introduce a measure of model explanatory power and show that, although the SVM and CNN models perform similarly in terms of classification accuracy, the latter
exhibits a higher level of explainability which makes it more comprehensible for humans and potentially more useful for other applications.},
archivePrefix = {arXiv},
arxivId = {1612.07843},
author = {Arras, Leila and Horn, Franziska and Montavon, Gr{\'{e}}goire and M{\"{u}}ller, Klaus Robert and Samek, Wojciech},
doi = {10.1371/journal.pone.0181142},
eprint = {1612.07843},
isbn = {1111111111},
issn = {19326203},
journal = {PLoS ONE},
number = {8},
pages = {1--23},
pmid = {28800619},
title = {{"What is relevant in a text document?": An interpretable machine learning approach}},
volume = {12},
year = {2017}
}
@article{Kondor2003,
author = {Kondor, Risi},
journal = {October},
number = {3},
pages = {1--12},
title = {{A Short Introduction to Hilbert Space Methods in Machine Learning}},
volume = {1},
year = {2003}
}
@article{Romero2018,
abstract = {Spatial concurrent constraint programming (SCCP) is an algebraic model of spatial modalities in constrained-based process calculi; it can be used to reason about spatial information distributed among the agents of
a system. This work presents an executable rewriting logic semantics of SCCP with extrusion (i.e., process mobility) that uses rewriting modulo SMT, a novel technique that combines the power of term rewriting, matching
algorithms, and SMT-solving. In this setting, constraints are encoded as formulas in a theory with a satisfaction relation decided by an SMT solver, while the topology of the spatial hierarchy is encoded as part of the term
structure of symbolic states. By being executable, the rewriting logic specification offers support for the inherent symbolic and challenging task of reachability analysis in the constrained-based model. The approach is
illustrated with examples about the automatic verification of fault-tolerance, consistency, and privacy in distributed spatial and hierarchical systems.},
archivePrefix = {arXiv},
arxivId = {1805.07434},
author = {Romero, Miguel and Rocha, Camilo},
eprint = {1805.07434},
title = {{Reachability Analysis for Spatial Concurrent Constraint Systems with Extrusion}},
url = {http://arxiv.org/abs/1805.07434},
year = {2018}
}
@article{Selsam2017,
abstract = {Noisy data, non-convex objectives, model misspecification, and numerical instability can all cause undesired behaviors in machine learning systems. As a result, detecting actual implementation errors can be
extremely difficult. We demonstrate a methodology in which developers use an interactive proof assistant to both implement their system and to state a formal theorem defining what it means for their system to be correct. The
process of proving this theorem interactively in the proof assistant exposes all implementation errors since any error in the program would cause the proof to fail. As a case study, we implement a new system, Certigrad, for
optimizing over stochastic computation graphs, and we generate a formal (i.e. machine-checkable) proof that the gradients sampled by the system are unbiased estimates of the true mathematical gradients. We train a variational
autoencoder using Certigrad and find the performance comparable to training the same model in TensorFlow.},
archivePrefix = {arXiv},
arxivId = {1706.08605},
author = {Selsam, Daniel and Liang, Percy and Dill, David L.},
eprint = {1706.08605},
isbn = {9781510855144},
title = {{Developing Bug-Free Machine Learning Systems With Formal Mathematics}},
url = {http://arxiv.org/abs/1706.08605},
year = {2017}
}
@article{Yao2014,
author = {Yao, Bo and Hagras, Hani and Alhaddad, Mohammed J. and Alghazzawi, Daniyal},
doi = {10.1007/s00500-014-1270-4},
issn = {14337479},
journal = {Soft Computing},
keywords = {Action classification,Fuzzy logic system,Human behavior recognition},
number = {2},
pages = {499--506},
title = {{A fuzzy logic-based system for the automation of human behavior recognition using machine vision in intelligent environments}},
volume = {19},
year = {2014}
}
@article{Study2010,
author = {Study, A Comparative},
isbn = {9781424465422},
keywords = {first-order bayesian network,probabilistic languages},
pages = {571--575},
title = {{Modeling First-Order Bayesian Networks ( FOBN )}},
year = {2010}
}
@article{Seisenberger2008,
abstract = {This article generalises the refined A-translation method for extracting programs from classical proofs [U. Berger,W. Buchholz, H. Schwichtenberg, Refined program extraction from classical proofs, Annals of Pure
and Applied Logic 114 (2002) 3-25] to the scenario where additional assumptions such as choice principles are involved. In the case of choice principles, this is done by adding computational content to the 'translated'
assumptions, an idea which goes back to [S. Berardi, M. Bezem, T. Coquand, On the computational content of the axiom of choice, JSL 63 (2) (1998) 600-622] and has been further elaborated in [U. Berger, P. Oliva, Modified bar
recursion and classical dependent choice, in: M. Baaz, S.D. Friedman, J. Kraijcek (Eds.), Logic Colloquium '01, in: Lecture Notes in Logic, vol. 20, Springer, Berlin, 2005, pp. 89-107]. We further investigate the
applicability of this extension by means of a small but non-trivial example, and discuss the formalisation as well as some optimisations necessary in order to extract terminating programs. Both formalisation and term
extraction have been implemented in the interactive proof assistant Minlog. {\textcopyright} 2008 Elsevier B.V. All rights reserved.},
author = {Seisenberger, Monika},
doi = {10.1016/j.apal.2008.01.004},
issn = {01680072},
journal = {Annals of Pure and Applied Logic},
keywords = {A-translation,Classical dependent choice,Computational content of classical proofs,Infinite tape example,Minlog},
number = {1-3},
pages = {97--110},
title = {{Programs from proofs using classical dependent choice}},
volume = {153},
year = {2008}
}
@article{Natarajan2009,
abstract = {This paper introduces a new type of application for ILP called Bootstrapped Learning (BL). BL brings several challenges to ILP, including the need to (a) automate the “ILP setup” problem, (b) exploit the fact that
a well-meaning teacher is providing pedagogically chosen examples and may be offering hints, (c) deal with small numbers of training examples and sometimes no explicit negative examples; and (d) "bootstrap", i.e., to
automatically base learning in part on the results of earlier learning sessions.},
author = {Natarajan, S and Kunapuli, G and Reilly, C O and Maclin, R and Walker, T and Page, D and Shavlik, J},
journal = {Artificial Intelligence},
keywords = {helpful teachers,human intervention,running ilp systems without},
pages = {1--7},
title = {{ILP for Bootstrapped Learning : A Layered Approach to Automating the ILP Setup Problem}},
year = {2009}
}
@article{Hoffmann2017,
abstract = {Agda is a dependently typed functional programming language and a proof assistant in which developing programs and proving their correctness is one activity. We show how this process can be enhanced by integrating
external automated theorem provers, provide a prototypical integration of the equational theorem prover Waldmeister, and give examples of how this proof automation works in practice.},
author = {Hoffmann, Ruth and Ireland, Murray L and Prospect, Craft and Miller, Alice and Norman, Gethin},
doi = {10.1007/978-3-642-38088-4},
isbn = {978-3-642-38087-7},
issn = {03029743},
journal = {NASA Formal Methods},
number = {July},
title = {{Strategy Synthesis For Autonomous Agents Using PRISM}},
url = {http://link.springer.com/10.1007/978-3-642-38088-4},
volume = {7871},
year = {2017}
}
@article{Cangelosi2014,
abstract = {BACKGROUND: Cancer patient's outcome is written, in part, in the gene expression profile of the tumor. We previously identified a 62-probe sets signature (NB-hypo) to identify tissue hypoxia in neuroblastoma
tumors and showed that NB-hypo stratified neuroblastoma patients in good and poor outcome 1. It was important to develop a prognostic classifier to cluster patients into risk groups benefiting of defined therapeutic
approaches. Novel classification and data discretization approaches can be instrumental for the generation of accurate predictors and robust tools for clinical decision support. We explored the application to gene expression
data of Rulex, a novel software suite including the Attribute Driven Incremental Discretization technique for transforming continuous variables into simplified discrete ones and the Logic Learning Machine model for
intelligible rule generation.$\backslash$n$\backslash$nRESULTS: We applied Rulex components to the problem of predicting the outcome of neuroblastoma patients on the bases of 62 probe sets NB-hypo gene expression signature.
The resulting classifier consisted in 9 rules utilizing mainly two conditions of the relative expression of 11 probe sets. These rules were very effective predictors, as shown in an independent validation set, demonstrating
the validity of the LLM algorithm applied to microarray data and patients' classification. The LLM performed as efficiently as Prediction Analysis of Microarray and Support Vector Machine, and outperformed other learning
algorithms such as C4.5. Rulex carried out a feature selection by selecting a new signature (NB-hypo-II) of 11 probe sets that turned out to be the most relevant in predicting outcome among the 62 of the NB-hypo signature.
Rules are easily interpretable as they involve only few conditions.$\backslash$n$\backslash$nCONCLUSIONS: Our findings provided evidence that the application of Rulex to the expression values of NB-hypo signature created a
set of accurate, high quality, consistent and interpretable rules for the prediction of neuroblastoma patients' outcome. We identified the Rulex weighted classification as a flexible tool that can support clinical decisions.
For these reasons, we consider Rulex to be a useful tool for cancer classification from microarray gene expression data.},
author = {Cangelosi, Davide and Muselli, Marco and Parodi, Stefano and Blengio, Fabiola and Becherini, Pamela and Versteeg, Rogier and Conte, Massimo and Varesio, Luigi},
doi = {10.1186/1471-2105-15-S5-S4},
issn = {14712105},
journal = {BMC Bioinformatics},
number = {Suppl 5},
pages = {1--15},
pmid = {25078098},
title = {{Use of attribute driven incremental discretization and logic learning machine to build a prognostic classifier for neuroblastoma patients}},
volume = {15},
year = {2014}
}
@article{Berger2018,
abstract = {The European Rail Traffic Management System (ERTMS) is a state-of-the-art train control system designed as a standard for railways across Europe. It generalises traditional discrete interlocking systems to a world
in which trains hold on-board equipment for signalling, and trains and interlockings communicate via radio block processors. The ERTMS aims at improving performance and capacity of rail traffic systems without compromising
their safety. The ERTMS system is of hybrid nature, in contrast to classical railway signalling systems which deal with discrete data only. Consequently, the switch to ERTMS poses a number of research questions to the formal
methods community, most prominently: How can safety be guaranteed? In this paper we present the first formal modelling of ERTMS comprising all subsystems participating in its control cycle. We capture what safety means in
physical and in logical terms, and we demonstrate that it is feasible to prove safety of ERTMS systems utilising Real-Time Maude model-checking by considering a number of bi-directional track layouts. ERTMS is currently being
installed in many countries. It will be the main train control standard for the foreseeable future. The concepts presented in this paper offer applicable methods supporting the design of dependable ERTMS systems. We
demonstrate model-checking to be a viable option in the analysis of large and complex real-time systems. Furthermore, we establish Real-Time Maude as a modelling and verification tool applicable to the railway domain. The
approach given in this paper is a rigorous one. In order to avoid modelling errors, we follow a systematic approach: First, as a requirement specification, we identify the event-response structures present in the ERTMS. Then,
we model these structures in Real-Time Maude in a traceable way, i.e., specification text in Real-Time Maude can be directly mapped to requirements. We explore our models by checking if they have the desired behaviour, and
apply systematic model-exploration through error injection – both these steps are carried out using the formal method Real-Time Maude. Finally, we analyse ERTMS by model-checking, thus applying a formal method to the railway
domain, and we mathematically prove that our analysis of ERTMS by model-checking is complete, i.e., that it guarantees safety at all times.},
author = {Berger, Ulrich and James, Phillip and Lawrence, Andrew and Roggenbach, Markus and Seisenberger, Monika},
doi = {10.1016/j.scico.2017.10.011},
issn = {01676423},
journal = {Science of Computer Programming},
keywords = {ERTMS/ETCS,Hybrid systems,Model-checking,Railway signalling,Real-Time Maude},
pages = {61--88},
publisher = {Elsevier B.V.},
title = {{Verification of the European Rail Traffic Management System in Real-Time Maude}},
url = {https://doi.org/10.1016/j.scico.2017.10.011},
volume = {154},
year = {2018}
}
@article{Mairal2016a,
abstract = {In this paper, we introduce a new image representation based on a multilayer kernel machine. Unlike traditional kernel methods where data representation is decoupled from the prediction task, we learn how to shape
the kernel with supervision. We proceed by first proposing improvements of the recently-introduced convolutional kernel networks (CKNs) in the context of unsupervised learning; then, we derive backpropagation rules to take
advantage of labeled training data. The resulting model is a new type of convolutional neural network, where optimizing the filters at each layer is equivalent to learning a linear subspace in a reproducing kernel Hilbert
space (RKHS). We show that our method achieves reasonably competitive performance for image classification on some standard "deep learning" datasets such as CIFAR-10 and SVHN, and also for image super-resolution,
demonstrating the applicability of our approach to a large variety of image-related tasks.},
archivePrefix = {arXiv},
arxivId = {1605.06265},
author = {Mairal, Julien},
eprint = {1605.06265},
isbn = {9781510838819},
issn = {10495258},
number = {Nips},
pmid = {205822},
title = {{End-to-End Kernel Learning with Supervised Convolutional Kernel Networks}},
url = {http://arxiv.org/abs/1605.06265},
year = {2016}
}
@article{Bellodi2013,
abstract = {Learning probabilistic logic programming languages is receiving an increasing attention and systems are available for learning the parameters (PRISM, LeProbLog, LFI-ProbLog and EMBLEM) or both the structure and
the parameters (SEM-CP-logic and SLIPCASE) of these languages. In this paper we present the algorithm SLIPCOVER for "Structure LearnIng of Probabilistic logic programs by searChing OVER the clause space". It performs a beam
search in the space of probabilistic clauses and a greedy search in the space of theories, using the log likelihood of the data as the guiding heuristics. To estimate the log likelihood SLIPCOVER performs Expectation
Maximization with EMBLEM. The algorithm has been tested on five real world datasets and compared with SLIPCASE, SEM-CP-logic, Aleph and two algorithms for learning Markov Logic Networks (Learning using Structural Motifs (LSM)
and ALEPH++ExactL1). SLIPCOVER achieves higher areas under the precision-recall and ROC curves in most cases.},
archivePrefix = {arXiv},
arxivId = {1309.2080},
author = {Bellodi, Elena and Riguzzi, Fabrizio},
doi = {10.1017/S1471068413000689},
eprint = {1309.2080},
keywords = {parameter learning,probabilistic logic programming,structure},
number = {Ll},
pages = {1--12},
title = {{Structure Learning of Probabilistic Logic Programs by Searching the Clause Space}},
url = {http://arxiv.org/abs/1309.2080{\%}0Ahttp://dx.doi.org/10.1017/S1471068413000689},
year = {2013}
}
@article{Dobnik2017,
author = {Dobnik, Simon and Kelleher, John and Dobnik, Simon},
doi = {10.21427/D7041Z},
title = {{Back to the Future : Logic and Machine Learning Back to the Future : Logic and Machine Learning}},
year = {2017}
}
@article{Nyolt2015,
abstract = {Recognising everyday activities including information about the context requires to handle large state spaces. The usage of wearable sensors like six degree of freedom accelerometers increases complexity even
more. Common approaches are unable to maintain an accurate belief state within such complex domains. We show how marginal filtering can overcome limitations of standard particle filtering and efficiently infer the context of
actions. Symbolic models of human behaviour are used to recognise activities in two different settings with different state space sizes. Based on these scenarios we compare the marginal filter to the standard particle filter.
An evaluation shows that the marginal filter performs comparably in small state spaces but outperforms the particle filter in large state spaces},
author = {Nyolt, Martin and Kr{\"{u}}ger, Frank and Yordanova, Kristina and Hein, Albert and Kirste, Thomas},
doi = {10.1016/j.ijar.2015.04.003},
isbn = {9780199233410},
issn = {0888613X},
journal = {International Journal of Approximate Reasoning},
keywords = {Activity recognition,Bayesian inference,Marginal filter,Particle filter,Plan recognition,Symbolic models},
pages = {16--32},
publisher = {Elsevier Inc.},
title = {{Marginal filtering in large state spaces}},
url = {http://dx.doi.org/10.1016/j.ijar.2015.04.003},
volume = {61},
year = {2015}
}
@article{Garrabrant2016,
abstract = {We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it
assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations, and its own probabilities. We show that our algorithm, an instance of what we
call a logical inductor, satisfies a number of intuitive desiderata, including: (1) it learns to predict patterns of truth and falsehood in logical statements, often long before having the resources to evaluate the
statements, so long as the patterns can be written down in polynomial time; (2) it learns to use appropriate statistical summaries to predict sequences of statements whose truth values appear pseudorandom; and (3) it learns
to have accurate beliefs about its own current beliefs, in a manner that avoids the standard paradoxes of self-reference. For example, if a given computer program only ever produces outputs in a certain range, a logical
inductor learns this fact in a timely manner; and if late digits in the decimal expansion of {\$}\backslashpi{\$} are difficult to predict, then a logical inductor learns to assign {\$}\backslashapprox 10\backslash{\%}{\$}
probability to "the {\$}n{\$}th digit of {\$}\backslashpi{\$} is a 7" for large {\$}n{\$}. Logical inductors also learn to trust their future beliefs more than their current beliefs, and their beliefs are coherent in the
limit (whenever {\$}\backslashphi \backslashimplies \backslashpsi{\$}, {\$}\backslashmathbb{\{}P{\}}{\_}\backslashinfty(\backslashphi) \backslashle \backslashmathbb{\{}P{\}}{\_}\backslashinfty(\backslashpsi){\$}, and so on);
and logical inductors strictly dominate the universal semimeasure in the limit. These properties and many others all follow from a single logical induction criterion, which is motivated by a series of stock trading analogies.
Roughly speaking, each logical sentence {\$}\backslashphi{\$} is associated with a stock that is worth $\backslash${\$}1 per share if [...]},
archivePrefix = {arXiv},
arxivId = {1609.03543},
author = {Garrabrant, Scott and Benson-Tilsen, Tsvi and Critch, Andrew and Soares, Nate and Taylor, Jessica},
eprint = {1609.03543},
pages = {1--131},
title = {{Logical Induction}},
url = {http://arxiv.org/abs/1609.03543},
year = {2016}
}
@article{Hoehndorf2011,
abstract = {Researchers design ontologies as a means to accurately annotate and integrate experimental data across heterogeneous and disparate data- and knowledge bases. Formal ontologies make the semantics of terms and
relations explicit such that automated reasoning can be used to verify the consistency of knowledge. However, many biomedical ontologies do not sufficiently formalize the semantics of their relations and are therefore limited
with respect to automated reasoning for large scale data integration and knowledge discovery. We describe a method to improve automated reasoning over biomedical ontologies and identify several thousand contradictory class
definitions. Our approach aligns terms in biomedical ontologies with foundational classes in a top-level ontology and formalizes composite relations as class expressions. We describe the semi-automated repair of
contradictions and demonstrate expressive queries over interoperable ontologies. Our work forms an important cornerstone for data integration, automatic inference and knowledge discovery based on formal representations of
knowledge. Our results and analysis software are available at http://bioonto.de/pmwiki.php/Main/ReasonableOntologies.},
author = {Hoehndorf, Robert and Dumontier, Michel and Oellrich, Anika and Rebholz-Schuhmann, Dietrich and Schofield, Paul N. and Gkoutos, Georgios V.},
doi = {10.1371/journal.pone.0022006},
isbn = {1932-6203 (Electronic)$\backslash$n1932-6203 (Linking)},
issn = {19326203},
journal = {PLoS ONE},
number = {7},
pages = {1--9},
pmid = {21789201},
title = {{Interoperability between biomedical ontologies through relation expansion, upper-level ontologies and automatic reasoning}},
volume = {6},
year = {2011}
}
@inproceedings{Vechev2018,
author = {Vechev, Martin},
title = {{AI 2 : AI Safety and Robustness with Abstract Interpretation}},
url = {http://qav.cs.ox.ac.uk/FLoC/Vechev.pdf},
year = {2018}
}
@article{Goodfellow2015,
abstract = {Several machine learning models, including neural networks, consistently mis-classify adversarial examples—inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset,
such that the perturbed in-put results in the model outputting an incorrect answer with high confidence. Early attempts at explaining this phenomenon focused on nonlinearity and overfitting. We argue instead that the primary
cause of neural networks' vulnerability to ad-versarial perturbation is their linear nature. This explanation is supported by new quantitative results while giving the first explanation of the most intriguing fact about them:
their generalization across architectures and training sets. Moreover, this view yields a simple and fast method of generating adversarial examples. Us-ing this approach to provide examples for adversarial training, we reduce
the test set error of a maxout network on the MNIST dataset.},
author = {Goodfellow, Ian and Shlens, Jonathon and Christian, Szegedy},
journal = {arXiv preprint arXiv:1412.6572},
title = {{Explaining and Harnessing Adversarial Examples}},
year = {2014}
}
@article{Lin1991,
abstract = {A general neural-network (connectionist) model for fuzzy logic control and decision systems is proposed. This connectionist model, in the form of feedforward multilayer net, combines the idea of fuzzy logic
controller and neural-network structure and learning abilities into an integrated neural-network-based fuzzy logic control and decision system. A fuzzy logic control decision network is constructed automatically by learning
the training examples itself. By combining both unsupervised (self-organized) and supervised learning schemes, the learning speed converges much faster than the original backpropagation learning algorithm. The connectionist
structure avoids the rule-matching time of the inference engine in the traditional fuzzy logic system. Two examples are presented to illustrate the performance and applicability of the proposed model},
author = {Lin, Chin Teng and Lee, George C S},
doi = {10.1109/12.106218},
isbn = {0018-9340},
issn = {00189340},
journal = {IEEE Transactions on Computers},
keywords = {Connectionist,defuzzifier,distributed representation,fuzzifier,fuzzy logic rule,membership function,receptive field,supervised and unsupervised learning},
number = {12},
pages = {1320--1336},
title = {{Neural-Network-Based Fuzzy Logic Control and Decision System}},
volume = {40},
year = {1991}
}
@article{Nakade2018,
abstract = {Data-race detection is the problem of determining if a concurrent program has a data-race in some execution and input; it has been long studied and often solved. The research in this paper reprises the problem in
the context of task parallel programs with the intent to prove, via model checking, the absence of data-race on any feasible schedule for a given input. Many of the correctness properties afforded by task parallel programming
models such as OpenMP, Cilk, X10, Chapel, Habanero, etc. rely on data-race freedom. Model checking for data-race, presented here, is in contrast to recent work using run-time monitoring, log analysis, or static analysis which
are complete or sound but never both. The model checking algorithm builds a happens-before relation from the program execution and uses that relation to detect data-race similar to many solutions that reason over a single
observed execution. Unlike those solutions, model checking generates additional program schedules sufficient to prove data-race freedom over all schedules on the given input. The approach is evaluated in a Java implementation
of Habanero using the JavaPathfinder model checker. The results, when compared to existing data-race detectors in Java Pathfinder, show a significant reduction in the time required for proving data race freedom.},
author = {Nakade, Radha and Mercer, Eric and Aldous, Peter and McCarthy, Jay},
doi = {10.1007/978-3-319-77935-5_25},
isbn = {9783319779348},
issn = {16113349},
journal = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
pages = {367--382},
title = {{Model-checking task parallel programs for data-race}},
volume = {10811 LNCS},
year = {2018}
}
@article{VanWesel2017,
abstract = {Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important
role. The NASA STI program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for archiving, and disseminates NASA's STI. The NASA STI program provides access to the NTRS
Registered and its public interface, the NASA Technical Reports Server, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by
NASA in the NASA STI Report Series, which includes the following report types: • TECHNICAL PUBLICATION. Reports of completed research or a major significant phase of research that present the results of NASA Programs and
include extensive data or theoretical analysis. Includes compilations of significant scientific and technical data and information deemed to be of continuing reference value. NASA counter-part of peer-reviewed formal
professional papers but has less stringent limitations on manuscript length and extent of graphic presentations.},
author = {{Van Wesel}, Perry and Goodloe, Alwyn E},
number = {June 2017},
pages = {2017--219628},
title = {{Challenges in the Verification of Reinforcement Learning Algorithms NASA STI Program . . . in Profile}},
url = {https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20170007190.pdf},
year = {2017}
}
@article{MohamadHekarl2007,
abstract = {Tens of thousands of students have learned to be more discerning at$\backslash$nconstructing and evaluating arguments with the help of Patrick J.$\backslash$nHurley. Hurley's lucid, friendly, yet thorough
presentation has made$\backslash$nA CONCISE INTRODUCTION TO LOGIC the most widely used logic text in$\backslash$nNorth America. In addition, the book's accompanying technological$\backslash$nresources, such as CengageNOW and
Learning Logic, include interactive$\backslash$nexercises as well as video and audio clips to reinforce what you$\backslash$nread in the book and hear in class. In short, you'll have all the$\backslash$nassistance you need to
become a more logical thinker and communicator.},
author = {{Mohamad Hekarl}, Uzir and Mashitah, Mat Don},
doi = {10.2307/3007606},
isbn = {ISBN-13: 978-0-495-50383-5 ISBN-10: 0-495-50383-5},
issn = {14732858},
journal = {Biochemical Engineering},
pages = {1--109},
pmid = {9102197},
title = {{Linear Programming: A Concise Introduction}},
url = {http://chemical.eng.usm.my/notes/HEKARL/notes/ekc471{\_}notes.pdf},
year = {2007}
}
@article{Fahey2016,
abstract = {This article argues that Transatlantic Trade and Investment Partnership (TTIP) negotiations have a positive legal story worth recounting as to the effects of the negotiations upon the EU legal order. The paper
explores the negotiation of EU international agreements as a specific field of law and considers how the TTIP negotiations overall contribute to the politicisation of the EU through shifts in legal practice on the part of all
major institutional actors to EU international relations in changes to approach, procedure and actions. It uses the metric of responsiveness to measure or chart the former, considering the responses as actually practiced by
institutional actors in law to normative concerns arising. It considers these practices across a range of actors, specifically the European Commission, the European Parliament, the Ombudsman, the Committee of the Regions and
the TTIP Advisory Group. Introduction},
author = {Fahey, Elaine},
doi = {10.1007/s11245-006-9005-5},
isbn = {1124500690055},
issn = {18756433},
journal = {Legal Issues of Economic Integration},
keywords = {abduction {\'{a}} abductive logic,connectionist modal logic {\'{a}},neural networks,programming {\'{a}}},
number = {4},
pages = {327--340},
title = {{On the benefits of the transatlantic trade and investment partnership (TTIP) Negotiations for the EU legal order: A legal perspective}},
volume = {43},
year = {2016}
}
@article{Pulina2011,
abstract = {The adoption of Artificial Neural Networks (ANNs) in safety-related applications is often avoided because it is difficult to rule out possible misbehaviors with traditional analytical or probabilistic techniques.
In this paper we present NEVER, our tool for checking safety of ANNs. NEVER encodes the problem of verifying safety of ANNs into the problem of satisfying corresponding Boolean combinations of linear arithmetic constraints.
We describe the main verification algorithm and the structure of NEVER. We present also empirical results confirming the effectiveness of NEVER on realistic case studies. {\textcopyright} 2012 Springer Science+Business Media
B.V.},
author = {Pulina, Luca and Tacchella, Armando},
doi = {10.1007/s10472-011-9243-0},
isbn = {1047201192430},
issn = {10122443},
journal = {Annals of Mathematics and Artificial Intelligence},
keywords = {Abstraction techniques,Algorithms and tools for verification,Formal methods for adaptive systems},
number = {3-4},
pages = {403--425},
title = {{NEVER: A tool for artificial neural networks verification}},
volume = {62},
year = {2011}
}
@article{Fawzi2018,
abstract = {The goal of this paper is to analyze an intriguing phenomenon recently discovered in deep networks, namely their instability to adversarial perturbations (Szegedy et. al., 2014). We provide a theoretical framework
for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers
to adversarial perturbations, and then illustrate the obtained upper bound on the families of linear and quadratic classifiers. In both cases, our upper bound depends on a distinguishability measure that captures the notion
of difficulty of the classification task. Our results for both classes imply that in tasks involving small distinguishability, no classifier in the considered set will be robust to adversarial perturbations, even if a good
accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured by
the distinguishability). Moreover, we show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be
larger than the latter by a factor that is proportional to $\backslash$sqrt{\{}d{\}} (with d being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two
robustness properties in high dimensional problems, which was empirically observed in the context of neural networks. To the best of our knowledge, our results provide the first theoretical work that addresses the phenomenon
of adversarial instability recently observed for deep networks. Our analysis is complemented by experimental results on controlled and real-world data.},
archivePrefix = {arXiv},
arxivId = {1502.02590},
author = {Fawzi, Alhussein and Fawzi, Omar and Frossard, Pascal},
doi = {10.1007/s10994-017-5663-3},
eprint = {1502.02590},
isbn = {1502.02590},
issn = {15730565},
journal = {Machine Learning},
keywords = {Adversarial examples,Classification robustness,Deep networks,Instability,Random noise},
number = {3},
pages = {481--508},
title = {{Analysis of classifiers' robustness to adversarial perturbations}},
volume = {107},
year = {2018}
}
@article{Pulina2010,
abstract = {A key problem in the adoption of artificial neural networks in safety-related applications is that misbehaviors can be hardly ruled out with traditional analytical or probabilistic techniques. In this paper we
focus on specific networks known as Multi-Layer Perceptrons (MLPs), and we propose a solution to verify their safety using abstractions to Boolean combinations of linear arithmetic constraints. We show that our abstractions
are consistent, i.e., whenever the abstract MLP is declared to be safe, the same holds for the concrete one. Spurious counterexamples, on the other hand, trigger refinements and can be leveraged to automate the correction of
misbehaviors. We describe an implementation of our approach based on the HYSAT solver, detailing the abstraction-refinement process and the automated correction strategy. Finally, we present experimental results confirming
the feasibility of our approach on a realistic case study. {\textcopyright} Springer-Verlag Berlin Heidelberg 2010.},
archivePrefix = {arXiv},
arxivId = {1301.4779},
author = {Pulina, Luca and Tacchella, Armando},
doi = {10.1007/978-3-642-14295-6_24},
eprint = {1301.4779},
isbn = {364214294X},
issn = {03029743},
journal = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
pages = {243--257},
pmid = {4520227},
title = {{An abstraction-refinement approach to verification of artificial neural networks}},
volume = {6174 LNCS},
year = {2010}
}
@article{Yosinski2015,
archivePrefix = {arXiv},
arxivId = {arXiv:1412.1897v4},
author = {Yosinski, Jason and Clune, Jeff},
eprint = {arXiv:1412.1897v4},
title = {{Deep Neural Networks are Easily Fooled : High Confidence Predictions for Unrecognizable Images}},
year = {2015}
}
@article{Jong1990a,
author = {Jong, De},
pages = {351--353},
title = {{No Title}},
volume = {3},
year = {1990}
}

@inproceedings{Lindenbaum2018GeometryGeneration,
    title = {{Geometry based data generation}},
    year = {2018},
    booktitle = {Advances in Neural Information Processing Systems},
    author = {Lindenbaum, Ofir and Stanley, Jay S. and Wolf, Guy and Krishnaswamy, Smita},
    pages = {1400--1411}
}

@article{morgan2021adaptive,
      title={Adaptive Neighbourhoods for the Discovery of Adversarial Examples},
      author={Jay Morgan and Adeline Paiement and Arno Pauly and Monika Seisenberger},
      year={2021},
      journal={arXiv preprint arXiv:2101.09108}
}
@article{hearst1998support,
  title={Support vector machines},
  author={Hearst, Marti A. and Dumais, Susan T and Osuna, Edgar and Platt, John and Scholkopf, Bernhard},
  journal={IEEE Intelligent Systems and their applications},
  volume={13},
  number={4},
  pages={18--28},
  year={1998},
  publisher={IEEE}
}

 @InProceedings{pmlr-v80-athalye18b, title = {Synthesizing Robust Adversarial Examples}, author = {Athalye, Anish and Engstrom, Logan and Ilyas, Andrew and Kwok, Kevin}, booktitle = {Proceedings of the 35th International
 Conference on Machine Learning}, pages = {284--293}, volume = {80}, year={2018} }
