Extended Differential Aggregations in Process Algebra for Performance and Biology
Max Tschaikowski \institute{University of Southampton
United Kingdom
Max Tschaikowski
Mirco Tribastone
We study aggregations for ordinary differential equations induced by fluid semantics for Markovian process algebra which can capture the dynamics of performance models and chemical reaction networks. Whilst previous work has required perfect symmetry for exact aggregation, we present approximate fluid lumpability, which makes nearby processes perfectly symmetric after a perturbation of their parameters. We prove that small perturbations yield nearby differential trajectories. Numerically, we show that many heterogeneous processes can be aggregated with negligible errors.
eptcsdata.tex
\providecommand{\firstpage}{34}
\providecommand{\publicationstatus}{N. Bertand and L. Bortolussi (Eds.): QAPL 2014\\
EPTCS 154, 2014, pp.\ \pageref*{FirstPage}--\pageref{LastPage},
doi:\href{http://dx.doi.org/10.4204/EPTCS.154.3}{10.4204/EPTCS.154.3}}
\pdfoutput=1

qapl14.tex
\documentclass[copyright,creativecommons]{eptcs}
\providecommand{\event}{QAPL 2014} % Name of the event you are submitting to
\usepackage{breakurl}              % Not needed if you use pdflatex only.


\usepackage{array}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[crop=off, off]{auto-pst-pdf}
%\usepackage{booktabs}

\usepackage{caption}
\usepackage{subfig}
\usepackage{pepa}
\usepackage{extarrows}
%\usepackage{soul}
\usepackage{enumerate}
\usepackage{dsfont}
\usepackage{todonotes}
\usepackage{graphicx}
%\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps}
%\usepackage{tikz}
%\usetikzlibrary{arrows,automata,fit,matrix}
\usepackage{url}


\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}

 \newcommand{\m}{{\bf m}}


 \newcommand{\xe}{\mathrm{e}}            % Exponential
 \newcommand{\IsDef}{\triangleq}			% "Is Defined" symbol
 \newcommand{\xR}{\mathbb{R}}
 \newcommand{\abs}[1]{%					% Absolute value
	\left\lvert#1\right\rvert%
 }
 \newcommand{\norm}[1]{%
   \left\lVert #1 \right\rVert%
    }


\newcommand{\pepa}[1]{\ensuremath{\mathit{#1}}}
\newcommand{\dsl}{\displaystyle}
\newcommand{\crate}[1][\alpha]{\mathcal{R}_{#1}}    % compeonent rate
\newcommand{\arate}[1][\alpha]{r_{#1}}              % apparent rate
\newcommand{\act}[1]{\xlongrightarrow{#1}}          % transition of with action type [1]
\newcommand{\slab}{\small{\textcircled{s}}}         % small label
\newcommand{\lab}{\textcircled{s}}                  % label
\newcommand{\teq}[1]{\stackrel{#1}=}

\begin{document}


\title{Extended Differential Aggregations in Process Algebra for Performance and Biology}
\author{Max Tschaikowski
\institute{University of Southampton \\ United Kingdom}
\email{m.tschaikowski@soton.ac.uk}
\and
Mirco Tribastone
\institute{University of Southampton \\ United Kingdom}
\email{m.tribastone@soton.ac.uk}
}

\def\titlerunning{Extended Differential Aggregations in Process Algebra for Performance and Biology}
\def\authorrunning{M. Tschaikowski \& M. Tribastone}


%\title{Extended Differential Aggregations in Process Algebra for Performance and Biology}
%
%\author{Max Tschaikowski \and Mirco Tribastone}
%
%\institute{Electronics and Computer Science\\
%University of Southampton, United Kingdom\\
%\url{{m.tschaikowski,m.tribastone}@soton.ac.uk}}

\allowdisplaybreaks[3]
\maketitle


\begin{abstract}
We study aggregations for ordinary differential equations induced by fluid semantics for Markovian process algebra which can capture the dynamics of performance models and chemical reaction networks. Whilst previous work has required perfect symmetry for exact aggregation, we present \emph{approximate fluid lumpability}, which makes nearby processes perfectly symmetric after a perturbation of their parameters. We prove that small perturbations yield nearby differential trajectories. Numerically, we show that many heterogeneous processes can be aggregated with negligible errors.
\end{abstract}

\section{Introduction}\label{sec:intro}

%The problem of space explosion arises whenever population dynamics known from biology ?, chemistry ? or the area of massively parallel computer systems ? are modelled by continuous-time Markov chains (CTMCs). This problem is usually addressed by the stochastic lumping where a (potentially) much smaller lumped CTMC is defined in such a way that the original stochastic behaviour remains unchanged, cf. \cite{Buchholz94}. Another popular approach, known as fluid approximation, is to approximate the original CTMC by a system of ordinary differential equations (ODEs). Given that the CTMC under study is very large, the error of this technique is negligible. Also, the computational amount needed is much less compared to the stochastic lumping.
%
%Compositional Markov models of reactive systems such as process algebra exploit symmetries between the interacting processes, inducing an aggregated Markov chain that can be exactly related to the original one~(e.g., \cite{canonical}). {\bfseries link!}

Fluid semantics for process algebra describe the dynamics of a model in terms of a system of ordinary differential equations (ODEs), which can be interpreted as a deterministic approximation to the expectation of the stochastic process underlying the classical Markovian semantics (e.g., \cite{Hillston_QEST05,Cardelli_TCS_08,Ciocchetta_Hillston_09,HaydenLipschitz,DBLP:journals/tse/TribastoneGH12}). When the model under consideration consists of many copies of processes in parallel, the ODE system size is independent from the multiplicities of such copies, unlike the Markovian representation that is well-known to suffer from state explosion.

Unfortunately, not every process algebra model enjoys a compact ODE description.
A possible solution to this problem is to exploit symmetries, captured by appropriate behavioural relations, that give rise to an \emph{aggregated} ODE system whose solution can be related to that of the original, potentially massive, one. Arguably, in the process algebra literature this topic has so far received less attention than its stochastic counterpart, concerned with developing behavioural equivalences that that induce state-space aggregations in the underlying Markov chain (e.g.,~\cite{DBLP:conf/esm/HermannsR98,canonical,bucholz94papm}). Indeed, in~\cite{concur12,efl_tcs} we studied a notion of behavioural equivalence for ODEs in the context of the stochastic process algebra PEPA~\cite{pepa}; this has found application to reducing the ODE system sizes of hierarchical models exhibiting replicas of massively parallel composite processes with arbitrary levels of nesting~\cite{Tschaikowski2013a}. ODE aggregations have also been investigated in~\cite{DBLP:conf/lics/DanosFFHK10} for rule-based models such as Kappa~\cite{Danos200469} and BioNetGen~\cite{Faeder:2009aa}, for the modelling and analysis of biomolecular networks.

The goal of this paper is to extend the toolkit of ODE aggregations available for stochastic process algebra, by making the following contributions.

\paragraph{\bf A unified framework for performance and biological modelling.}   Let us first observe that PEPA and rule-based models have complementary domain-specific semantics of synchronisation that inevitably circumscribe the scope of validity of the results on ODE aggregations. Indeed, while PEPA can be particularly useful for the performance evaluation of computing systems, rule-based models use the \emph{law of mass action}. This is well known to be at the basis of biochemical reaction networks, though it has also been employed in epidemiological models (e.g.,~\cite{Watson1972}) as well as in certain wireless networks (e.g.~\cite{Wang22052009}).

Our initial starting point is to consider a unified framework that encompasses both kinds of interaction.  Specifically, we introduce and study \emph{Fluid Extended Process Algebra} (FEPA), a lightweight and conservative extension of \emph{Fluid Process Algebra}  presented in \cite{concur12}, featuring a more general parallel operator that can capture both dynamics.  We will first show that the notion of \emph{exact fluid lumpability} (EFL) of~\cite{concur12} carries over to FEPA\@. This will be used to set the stage for approximate notions of ODE aggregations, discussed later in the paper, which are defined in terms of their exact counterparts.

For an informal overview of our results, let us consider the process
\begin{equation}\label{eq:example}
\Big( P_1[N_1] \parallel_K P_2[N_2] \parallel_K \cdots \parallel_K P_D[N_D] \Big) \parallel_{L} Q[M]
\end{equation}
where, for all $1 \leq i \leq D$, $P_i$ is some sequential component that is replicated $N_i$ times, and $\parallel_{K}$ is the (generic) parallel operator, parameterised by an action set $K$, in a CSP-like fashion. EFL may essentially reduce the analysis of such a model by considering the fluid trajectory of a \emph{representative} $P_i$, which is shown to be equal to that of any other $P_j$ if $N_i = N_j$ and  $P_i$ and $P_j$ are isomorphic. Thus, symmetry is required both at the level of the sequential component and at the compositional level, by ensuring that all populations have the same size.

%This paper significantly extends the toolkit of process-algebraic differential aggregations by making the following contributions.
%\begin{enumerate}[i)]
%
%\item
%
%\item
\paragraph{\bf Ordinary fluid lumpability.} We relax the symmetry requirements of EFL by introducing the notion of \emph{ordinary fluid lumpability} (OFL). Similarly to EFL, it considers symmetry through isomorphism at the sequential level. However, it allows \emph{heterogeneity} at the compositional level: in the example above, it may yield an exactly aggregated ODE system even if $N_i \neq N_j$.  However, unlike EFL, where all the trajectories of the original ODE system can be obtained from the solution of the aggregate, in OFL the aggregate gives the exact sum of the solutions of its parts, but their individual trajectories cannot be recovered. In this sense, it corresponds to the aggregation presented in~\cite{DBLP:conf/lics/DanosFFHK10}, but with some differences. In~\cite{DBLP:conf/lics/DanosFFHK10} aggregates can be obtained by collapsing non-isomorphic biochemical species. This is useful in that application domain, where species are modelled as non-atomic entities with an internal state characterised by the simplest agents of which they are formed. This is in contrast to the modelling scenario envisaged in our process algebra, where atomic entities do not combine, but only interact with other atomic entities. More specifically, a typical modelling pattern amenable to our notions of aggregation is that of \emph{multi-class} systems, where analogous processes (e.g., two or more kinds of infections~\cite{Watson1972}) exhibit similar behaviour, but with different rates and/or with different multiplicites. Finally, unlike~\cite{DBLP:conf/lics/DanosFFHK10}, we also focus  on compositionality properties of our aggregations.

%studied in a process-algebraic setting, whereby its compositionality properties are also analysed; secondly, it is extended to the minimum-based semantics.
%On passing, let us notice that the difference between EFL and OFL parallels that between exact and ordinary lumpability for Markov chains~\cite{BuchholzOrdinaryExact}.
%
%\item

\paragraph{\bf Approximate aggregations.}
We aim to go beyond \cite{concur12}, \cite{DBLP:conf/lics/DanosFFHK10}, as well as OFL, in that we also relax the requirement on the \emph{exactness} of the aggregation. This fact partly stems from the use of strong symmetry at the sequential level. However, there is evidence of criticism on this assumption when models confront real systems, where the difficulty in measuring rates may induce estimation errors that numerically tell apart apparently  identical agents (e.g.,~\cite{springerlink:10.1007/BFb0039071,1029849}). This has motivated work on approximate reasoning with probabilistic and stochastic models (e.g.,~\cite{worrell01:approximate,DBLP:conf/concur/PierroHW03}).
In this paper, we study $\varepsilon$-variants of both EFL and OFL as a means to relaxing symmetries at the sequential level. These variants allow non-isomorphic processes to be aggregated if there exists a \emph{perturbation} in the rates make them isomorphic. For instance, let us take
$P_i \xrightarrow{(\alpha, r)} P_k$ and $P_j \xrightarrow{(\alpha, r + \varepsilon)} P_k$,
   for some $P_k$, where the edges give the usual action/rate pairs, with $r > 0$ and $\varepsilon > 0$. Then, whilst these processes cannot be aggregated with either EFL or OFL, it holds for example that they are $\varepsilon$-ordinarily fluid lumpable for any $N_i$ and $N_j$. Clearly, the aggregated system will not be in exact correspondence with the original one. However, we provide a theoretical bound that shows that the aggregation error  depends \emph{linearly} in the intensity of the perturbation $| \varepsilon |$.

Exhibiting such near-symmetries may appear quite limiting for practical applications; however, there are models in the literature that do exhibit this characteristic  (see, e.g.,~\cite{dsn2013} and references therein). When such a condition is satisfied, it is possible to systematically construct a reduced model independently from the abundances of the species involved, or from the speed at which certain interactions occur, as is required for approximate aggregation methods based on quasi steady-state or quasi-equilibrium (see, e.g.,~\cite{10.3389/fgene.2012.00131}). In this respect, this paper is more closely related to~\cite{dsn2013}, which studies how perturbations of model parameters can lead to ODE aggregations. However, while \cite{dsn2013} is more general in that the ODEs are not necessarily derived from a process algebra, it is  restricted to aggregations of OFL type. Furthermore, \cite{dsn2013} does not address compositionality issues, i.e., how to reuse ODE aggregations of a model in another context.
   %
    %\item

 \paragraph{\bf Characterisation of ODE aggregations.} We characterise the nature of such aggregations in two main ways. (i) Firstly, we show that all our ODE aggregations can be induced by suitable notions of behavioural equivalence, which turn out to be congruences for FEPA\@.
 %(ii) Secondly, as in~\cite{concur12}, we consider the property of model \emph{well-posedness}. This condition implies that processes which can be aggregated are related by \emph{semi-isomorphism}, an extension of graph isomorphism to labelled transition systems with transition multi-sets, which does not distinguish between the multiplicity of arcs connecting two nodes whenever the total rate is the same. Interestingly, we prove that the very same characterisation applies also to OFL. Furthermore, under well-posedness we show that $\varepsilon$-EFL and $\varepsilon$-OFL imply the behavioural notion of $\varepsilon$-semi-isomorphism, the natural extension of semi-isomorphism which relates graphs up to changes in the transition rates.
 (ii) Secondly, we provide some numerical evidence on the usefulness of the approximate versions of EFL and OFL, presenting model examples where even significant perturbation in the rates may yield negligible error in practice.

%\end{enumerate}

%\paragraph{Note to reviewers} All proofs may be found in the appendix, also published in a technical report at \ldots
%{\bfseries Discuss that EFL does not imply stochastic lumping} these authors introduced the notion of \emph{exact fluid lumpability} (EFL), based on strong symmetries for the process algebra PEPA~\cite{pepa}. In order to provide an intuitive and informal overview of the contributions of this paper, and relate it to our previous work, let us consider the illustrative example
%\begin{equation}\label{eq:example}
%\Big( P_1[N_1] \sync{K} P_2[N_2] \sync{K} \cdots \sync{K} P_D[N_D] \Big) \sync{L} Q[M]
%\end{equation}
%where, for all $1 \leq i \leq D$, $P_i$ is some sequential component that is replicated $N_i$ times, and $\sync{L}$ is the parallel operator, parametrised by an action set~$L$. EFL may essentially reduce the analysis of such a model by considering the fluid trajectory of a \emph{representative} $P_i$, which is shown to be equal to that of any other $P_j$ if $N_i = N_j$ and the labelled transition systems of $P_i$ and $P_j$ are isomorphic. Thus, symmetry is required both at the level of the sequential component and at the compositional level by ensuring that all populations have the same size.

%In some cases, however, even the ODE systems may be too large for an efficient analysis. This stimulated the work \cite{concur12} where the authors presented the notion of \emph{exactly fluid lumpability} (EFL) which allowed to relate the original ODE system to a (potentially) much smaller lumped ODE system. The ODE systems considered in \cite{concur12} are induced by a fluid process algebra (FEPA) and apply for compositional reasoning by means of a congruence. The authors were also able to bridge the gap between the fluid (ODEs) and discrete world (graphs), in that they have shown that EFL implies semi-isomorphism, a behavioural notion which is closely related to the classical (graph) isomorphism. In this follow-up paper the theory is extended by the following contributions:

%Here we significantly extend~\cite{concur12} in the following ways.
%\begin{enumerate}
%    \item
 %
%\end{enumerate}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fluid Extended Process Algebra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To define FEPA, we exploit the fact that fluid semantics reason about representatives of replicated sequential components, which we will also call \emph{fluid atoms}. For instance, in~(\ref{eq:example}), $P_1[N_1]$ will be represented in FEPA using a single fluid atom $P_1$ together with the information about the multiplicity of replicas, encoded in a \emph{population function}. Let us first define the grammar for fluid atoms.
\begin{definition}\label{def:fepa}
The syntax of a FEPA fluid atom is given by
\begin{align*}
S & ::= (\alpha, r).S \mid S + S \mid A  & \text{with} \quad A \rmdef S \text{\quad(constant)},
\end{align*}
where $\alpha$ is an action in the \emph{action set} $\mathcal{A}$  and $r \in \mathds{R}_{>0}$.
\end{definition}
The structured operational semantics is given by the following four rules:
\[
\frac{\phantom{P \act{(\alpha, r)} P'}}{(\alpha,r).P \act{(\alpha, r)} P} \ ,
\ \
\frac{P \act{(\alpha, r)} P'}{P + Q \act{(\alpha, r)} P'} \ ,
\ \
\frac{Q \act{(\alpha, r)} Q'}{P + Q \act{(\alpha, r)} Q'} \ ,
\ \
\frac{P \act{(\alpha, r)} P'}{A \act{(\alpha, r)} P'} \ A \rmdef P .
\]
Given a fluid atom $P$, these rules induce a labelled transition system, denoted by $dg(P)$ (the derivation graph), with nodes denoted by $ds(P)$ (the \emph{derivative set}), and with a transition multi-set  where transitions have a  multiplicity equal to the number of distinct derivations between any two fluid atoms.
%\begin{definition}[Derivative Set]\label{def_ds} The derivative set of a PEPA component $P$, denoted by $ds(P)$, is defined as the smallest set such that
%$P \in ds(P)$; and if $P' \in ds(P)$ and $P' \xrightarrow{(\alpha, r)} P''$ then $P'' \in ds(P)$.
%\end{definition}
%
%\begin{definition}[Derivation Graph]\label{def_dg}
%Let $\mathcal{A}ct := \mathcal{A} \times \mathbb{R}_{> 0}$ denote the set of all activities of PEPA. The derivation graph $dg(P)$ of a PEPA component $P$ has $ds(P)$ as the set of nodes. The multiset of transitions $\mathcal{T} \subseteq ds(P) \times \mathcal{A}ct \times ds(P)$ is such that
%\[ P_0 \act{(\alpha,r)} P_1 \Leftrightarrow (P_0, (\alpha,r), P_1) \in \mathcal{T}, \]
%with multiplicity equal to the number of distinct derivations of $P_0 \act{(\alpha,r)} P_1$.
%\end{definition}


%\subsection{Semi-Isomorphism}\label{sec:semi}
Semi-isomorphism, at the basis of our characterisation results, relates two fluid atoms whenever their derivation graphs are isomorphic up to replacing multiple equally-labelled transitions between two states with a single transition with the same action type and the rate sum across all such transitions.

\begin{definition}[Semi-Isomorphism]\label{def_semi-iso}
Two FEPA sequential components $P$ and $Q$ are semi-isomorphic if there is a bijection $\sigma : ds(P) \rightarrow ds(Q)$ which satisfies $\sum_{P_i \xrightarrow{(\alpha, r)} P_j} r $ $= \sum_{\sigma(P_i) \xrightarrow{(\alpha, r)} \sigma(P_j)} r$ for all $P_i, P_j \in ds(P)$ and $\alpha \in \mathcal{A}$. We shall call such a bijection $\sigma$ a semi-isomorphism.
\end{definition}
For instance $P \rmdef (\alpha, r_1).P + (\alpha, r_2).P$ is semi-isomorphic to $Q \rmdef (\alpha, r_1 + r_2).Q$.


A FEPA model is a composition of fluid atoms, using the parallel operator $\parallel_L$. As in~\cite{concur12}, FEPA does not feature the hiding operator.
\begin{definition}[FEPA Model]
A FEPA model $M$ is given by the grammar
\[ M ::= M \parallel_L M \mid P \]
where $L \subseteq \mathcal{A}$ and $P$ is a fluid atom. For any two distinct fluid atoms $P$ and $P'$ in $M$, we require that $ds(P) \cap ds(P') = \emptyset$.
\end{definition}
In comparing derivative sets, equality is intended to be syntactical. The requirement on pairwise disjoint derivative sets is without loss of generality: If two distinct fluid atoms have a common derivative, it is always possible to relabel one atom with appropriate fresh constants so as to satisfy the above condition.

\begin{example}\label{ex:first}
Let us consider the FEPA process
\begin{equation}\label{eq_running_example_fpa}
\pepa{Sys} := (P_1 \parallel_\emptyset \ldots \parallel_\emptyset P_D) \parallel_{\{\alpha\}} Q \ ,
\end{equation}
with fluid atoms given by
\begin{align*}
P_d & \rmdef (\alpha, r).P'_d, & P'_d & \rmdef (\beta, s).P_d,& Q & \rmdef (\alpha, u).Q', & Q' & \rmdef (\delta, w).Q, & 1 \leq d \leq D.
\end{align*}
Intuitively, the above model considers a situation where $D$ \emph{independent} groups of agents, recognisable by the empty cooperation sets, synchronise with a common group of agents, of type $Q$, through action $\alpha$.
\end{example}

%The next definitions will be needed for the setup of the fluid framework.

\begin{definition}
Let $M$ be a FEPA model. We define then
$\mathcal{G}(M)$ as the set of all fluid atoms in $M$; $\mathcal{B}(M)$ as the set of all atoms' derivatives, i.e., $\mathcal{B}(M) = \bigcup \{ ds(P) \ | \ P \in \mathcal{G}(M) \}$; a \emph{population function} $V : X \rightarrow \mathbb{R}_{\geq 0}$ with $\mathcal{B}(M) \subseteq X$; and an \emph{initial} population function $V(0) : X \rightarrow \mathbb{N}_0$.
\end{definition}
For instance, $\mathcal{G}(\pepa{Sys}) = \{P_1,\ldots,P_D,Q\}$, $\mathcal{B}(\pepa{Sys}) = \{ P_d, P'_d \mid 1 \leq d \leq D \} \cup \{ Q, Q' \}$. Function $V(0)$ encodes the size of the system at time $t = 0$. For instance,
\begin{align}\label{eq:v0}
V_{P_d}(0) & = N_d, & V_{P'_d}(0) & = 0, & V_Q(0) & = N_Q, & V_{Q'}(0) & = 0,
\end{align}
states that at $t = 0$ there are $N_d$ agents in the state $P_d$, no agents in the state $P'_d$, $N_Q$ agents in the state $Q$ and no agents in the state $Q'$.

We are now ready to provide the semantics for interaction in FEPA. We consider two instances, distinguished by the choice of a (binary) \emph{synchronisation function} that is hereafter denoted by $\rho$. Choosing $\rho = \min$ yields the minimum-based semantics of PEPA; the law of mass action is instead recovered by choosing $\rho = \cdot$ (intended as multiplication).  With this latter choice, FEPA can be seen as the fluid counterpart of Markovian process algebra such as~\cite{bucholz94papm}, or as an alternative to process algebra for biological networks such as Bio-PEPA~\cite{Ciocchetta_Hillston_09}.

\begin{definition}[Apparent Rate]\label{def:app.rate}
The apparent rate of action $\alpha$ in a FEPA fluid atom $P$, denoted by $r_\alpha(P)$, is defined as follows:
\[
r_\alpha((\beta,r).S)  :=
\left \{
\begin{array}{lllrl}
r & , \ \beta = \alpha, & \qquad \qquad \qquad \qquad & r_\alpha(A)  & := r_\alpha(S), \  A \rmdef S ,\\
0 & , \ \text{else,} & \qquad \qquad \qquad  \qquad & r_\alpha(P + Q)  & := r_\alpha(P) + r_\alpha(Q).
\end{array}
\right.
 \]
\end{definition}

\begin{definition}[Parameterised Apparent Rate]\label{def:app.rate.v}
Let $M$ be a FEPA model, $\alpha \in \mathcal{A}$, $V$ a population function, and $\rho$ the synchronisation function.
The apparent rate of $M$ with respect to $V$ is defined as
\[
\begin{split}
\arate(M_0 \parallel_L M_1, V) & :=
    \begin{cases}
        \min(\arate(M_0, V), \arate(M_1, V))           & , \ \alpha \in L \ \land \ \rho = \min , \\
        \arate(M_0, V) \cdot \arate(M_1, V)            & , \ \alpha \in L \ \land \ \rho = \cdot , \\
        \arate(M_0, V) + \arate(M_1, V)                & , \ \alpha \notin L.
    \end{cases} \\
\arate(P, V) & := \sum_{P_i \in ds(P)} V_{P_i} \arate(P_i),
\end{split}
\]
where $\arate(P_i)$ is the apparent rate of a FEPA fluid atom $P_i$, by Definition~\ref{def:app.rate}.
\end{definition}
For instance, in (\ref{eq_running_example_fpa}) it holds that $\arate(P_d, V) = r V_{P_d}$, which gives the apparent rate at which a population of $V_{P_d}$ $P_d$-components exhibits action $\alpha$. Let us assume that $D = 1$ in (\ref{eq_running_example_fpa}). Then $r_\alpha(P_1 \parallel_{\{\alpha\}} Q, V) = \min (r V_{P_1}, u V_Q)$ if $\rho = \min$; for $\rho = \cdot$, instead, we have that $r_\alpha(P_1 \parallel_{\{\alpha\}} Q, V) = r\cdot u \cdot V_{P_1} \cdot V_Q$. In this case, the model corresponds to a chemical reaction network which may be expressed, using standard notation, by $P_1 + Q \rightarrow P_1' + Q'$, with rate constant equal to $r \cdot u$, and two monomolecular reactions, $P_1' \rightarrow P_1$ and $Q' \rightarrow Q$, with rate constants $s$ and $w$, respectively.

The following quantities are used to define the vector field of the ODE system to be analysed.

\begin{definition}[Parameterised Component Rate]\label{def:rate.v}
Let $M$ be a FEPA model, $\alpha \in \mathcal{A}$ and $V$ a population function. The component rate of $P' \in \mathcal{B}(M)$ is parameterised by $V$ in the following manner.
\begin{itemize}
\item $M = M_0 \parallel_L M_1$: if $P' \in \mathcal{B}(M_i)$, $i = 0,1$, and $\alpha \in L$ then
\[
 \crate(M_0 \parallel_L M_1, V, P') :=
    \dsl \frac{\crate(M_i, V, P')}{\arate(M_i, V)} \arate(M_0 \parallel_L M_1, V).
\]
\item $M = M_0 \parallel_L M_1$: if $P' \in \mathcal{B}(M_i),$ $i = 0,1$, and $\alpha \notin L$ then
\[
 \crate(M_0 \parallel_L M_1, V, P') := \crate(M_i, V, P').
\]
\item $M = P$: then
\[
\crate(P, V, P') := V_{P'} \arate(P') .
\]
\end{itemize}
\end{definition}

\paragraph{Notation.}  We use Newton's dot notation $\dot{V}_P$ for the derivative of $V_P$. To enhance readability, time $t$ will be suppressed, e.g., $\dot{V}_P$ denotes $\dot{V}_P(t)$. %and $V_P$ denotes $V_P(t)$.

\begin{definition}\label{def_35}
Let $M$ be a FEPA model. The initial value problem for $M$ is given by $\dot{V} = F(V)$ with initial condition $V(0)$, where
\[ F_P(V) := \sum_{\alpha \in \mathcal{A}} \bigg( \Big( \sum_{P' \in \mathcal{B}(M)} p_\alpha(P', P) \crate(M, V, P') \Big) - \crate(M, V, P) \bigg) \]
and
\[
p_\alpha(P,P') = \frac{1}{\arate(P)} \sum_{P \xrightarrow{(\alpha, r)} P'} r
\]
for all $\alpha \in \mathcal{A}$ and $P,P' \in \mathcal{B}(M)$.
\end{definition}


For instance, the initial value problem of (\ref{eq_running_example_fpa}) and (\ref{eq:v0}) in the case of $\rho = \cdot$ is given by the initial condition (\ref{eq:v0}) and the ODE system
\begin{align}\label{eq:ode}
\dot{V}_{P_d} & = - r u V_{P_d} \cdot V_{Q} + s V_{P'_d} & \dot{V}_{P'_d} & = - s V_{P'_d} + r u V_{P_d} \cdot V_{Q} \nonumber \\
\dot{V}_{Q} & = - u V_{Q} \cdot \sum_{1 \leq d \leq D} r V_{P_d} + w V_{Q'} & \dot{V}_{Q'} & = - w V_{Q'} + u V_{Q} \cdot \sum_{1 \leq d \leq D} r V_{P_d}
\end{align}


%{
%\bfseries
%\begin{remark}
%\textnormal{For any fluid atom $P \in \mathcal{G}(M)$ it holds that $\sum_{P' \in ds(P)} \dot{V}_{P'} = 0$. For instance, in (\ref{eq_running_example_fpa}) we have $\dot{V}_{P_d} + \dot{V}_{P'_d} = 0$. In the remainder, for brevity we will write one of the ODEs for a fluid atom label in terms of the derivatives of its other states, e.g., $\dot{V}_{P'_d} = - \dot{V}_{P_d}$ instead of its explicit form as in (\ref{eq:ode}).
%}\end{remark}
%}


The notion of well-posedness given below is needed to characterise ODE aggregations with respect to the structure of the fluid atoms. We wish to point out, however, that this is without loss of generality, since each non well-posed model can be transformed into a well-posed one \emph{without} changing the underlying ODE system, see \cite{efl_tcs}.


\begin{definition}[Well-posedness]\label{def_well_posedness}
A FEPA model $M$ is said to be well-posed if and only if for all occurrences $M_1 \parallel_{L} M_2$ in $M$ it holds that $\exists V_1 \big( \arate(M_1,V_1) > 0 \big) \ \land \ \exists V_2 \big( \arate(M_2,V_2) > 0 \big)$ for all $\alpha \in L$.
\end{definition}
%
In essence, a model is well-posed whenever any synchronised action may be performed by both operands, for some population function. Thus, $\pepa{Sys}$ is well-posed, but $P_1 \parallel_{\{\beta\}} Q$ is not since $Q$ does not do $\beta$-actions.

%\begin{definition}[Merged Derivation Graph]\label{def_merged}
%The merged derivation graph $dg_m(P)$ of $P$ arises from $dg(P)$, if, for all $\alpha \in \mathcal{A}$, all $\alpha$-transitions between any two states whose rate-sum across all transitions is equal to $q$ are replaced by a single transition $(\alpha,q)$.
%\end{definition}
%
%\ldots {\bfseries Perhaps remove it}
%\begin{theorem}\label{thm_semi_iso}
%Let $\sigma : ds(P) \rightarrow ds(Q)$ be a semi-isomorphism between the PEPA processes $P$ and $Q$. Then it holds that $dg_m(P)$ and $dg_m(Q)$ stand in a one-to-one correspondence.
%\end{theorem}
%


%\section{Modelling Language}\label{sec:background}
%
%\subsection{Graphs and Semi-Isomorphism}\label{sec:semi}
%
%A the base of the fluid framework stand the notions of a graph and label.
%
%\begin{definition}[Graph]\label{def_dg}
%For a fixed set of action types $\mathcal{A}$, let $\mathcal{A}ct = \mathcal{A} \times \mathbb{R}_{> 0}$ denote the set of all transitions. A graph is given by a triple $G = (ds(P),\mathcal{T}(P),P)$, where $ds(P)$ denotes the set of nodes, $\mathcal{T} \subseteq ds(P) \times \mathcal{A}ct \times ds(P)$ the multiset of transitions and $P \in ds(P)$ is a initial state/label of $G$, meaning that all $P' \in ds(P)$ can be reached from $P$.
%\end{definition}
%
%Applying the usual notation from graph theory, $(P',(\alpha,r),P'') \in \mathcal{T}(P)$ is abbreviated to $P' \xrightarrow{(\alpha, r)} P''$. For instance, we will use later the graphs
%\begin{center}
%    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,semithick]
%    \matrix[row sep=0.6cm, column sep=1cm]
%    {
%        \node (1) {$\rightarrow P_d$}; \pgfmatrixnextcell \node (2) {$P'_d$}; \pgfmatrixnextcell \node (3) {$\rightarrow R_d$}; \pgfmatrixnextcell \node (4) {$R'_d$};
%        \pgfmatrixnextcell \node (5) {$\rightarrow Q$}; \pgfmatrixnextcell \node (6) {$Q'$}; \\
%    };
%
%    \path (1) edge [bend left=12] node {$(\alpha, r)$} (2)
%        (2) edge [bend left=12] node {$(\beta, s)$} (1)
%        (3) edge [bend left=12] node {$(\alpha, \tilde{r})$} (4)
%        (4) edge [bend left=12] node {$(\gamma, \tilde{s})$} (3)
%        (5) edge [bend left=12] node {$(\alpha, u)$} (6)
%        (6) edge [bend left=12] node {$(\delta, w)$} (5);
%    \end{tikzpicture}
%\end{center}
%where $\rightarrow P$ denotes the label/initial state of a graph. We wish to stress that $\mathcal{T}(P)$ may contain, as a \emph{multiset}, more than one triple $(P',(\alpha,r),P'')$. In essence, this allows the modeler to define graphs where a transition from $P'$ to $P''$ is present several times. The next definition will be needed later in the setup of the fluid framework.
%
%\begin{definition}[Apparent Rate]\label{def:app.rate}
%Let $\alpha \in \mathcal{A}$ denote an arbitrary action type.
%\begin{enumerate}
%    \item The $\alpha-$apparent rate of a node $P' \in ds(P)$, $\arate(P')$, is the sum of all outgoing $\alpha$-transitions of $P'$, i.e. $\arate(P') := \sum_{P' \xrightarrow{(\alpha, r)} \bullet} r$.
%    \item For $P',P'' \in ds(P)$, the $\alpha-$percentage from $P'$ to $P''$ is given by $p_\alpha(P', P'') := (1/\arate(P')) \sum_{P' \xrightarrow{(\alpha, r)} P''} r$.
%\end{enumerate}
%\end{definition}
%
%It will be shown later that fluid lumpability implies semi-isomorphism, a weaker version of the classic isomorphism.
%
%\begin{definition}[Semi-Isomorphism]\label{def_semi-iso}
%Two labels $P$ and $Q$ are called
%\begin{enumerate}
%    \item isomorphic if there is a bijection $\sigma : ds(P) \rightarrow ds(Q)$ which satisfies
%    \[ \{| x \in \mathcal{T}(P) \mid x = (P',(\alpha,r),P'') |\} = \{| x \in \mathcal{T}(Q) \mid x = \big(\sigma(P'),(\alpha,r),\sigma(P'')\big) |\} \]
%    for all $(\alpha,r) \in \mathcal{A}ct$ and $P',P'' \in ds(P)$, where $\{| \ldots |\}$ denotes a multiset.
%    \item semi-isomorphic if there is a bijection $\sigma : ds(P) \rightarrow ds(Q)$ which satisfies $\sum_{P' \xrightarrow{(\alpha, r)} P''} r $ $= \sum_{\sigma(P') \xrightarrow{(\alpha, r)} \sigma(P'')} r$ for all $P', P'' \in ds(P)$ and $\alpha \in \mathcal{A}$. We shall call such a $\sigma$ semi-isomorphism.
%\end{enumerate}
%\end{definition}
%
%Clearly, isomorphism implies semi-isomorphism, but not vice versa.
%
%
%\subsection{Fluid Process Algebra}
%
%In this section, we define a process algebra which allows for a compositional construction of ODE systems. Two types of synchronisation are covered: the first one covers the classic PEPA style synchronisation via the minimum function, whereas the second one incorporates the law of mass action via products. In that respect, the discussed process algebra is a generalised and simplified version of the language from \cite{concur12} which dealt solely with the fluid approximation of PEPA models.
%
%\begin{definition}[FEPA Model]
%An FEPA model $M$ is given by the grammar
%\[ M ::= M \parallel_L M \mid P, \]
%where $L \subseteq \mathcal{A}$ and $P$ denotes the label of a graph $(ds(P),\mathcal{T}(P),P)$. For two arbitrary subprocesses $P$ and $P'$ of $M$, we require that $ds(P) \cap ds(P') = \emptyset$ if $P \neq P'$.
%\end{definition}
%
%We call sometimes a subprocess $P$ of an FEPA model $M$ a \emph{fluid atom}. The size of the ODE system will be as large as the sum of the sizes of all graphs in a model. Intuitively, a label $P$ stands for a group of agents and $\parallel_L$ is a binary synchronisation operator. In order to build on intuition, we shall stick with the running example
%\begin{equation}\label{eq_running_example_fpa}
%\pepa{Sys} := (P_1 \parallel_\emptyset \ldots \parallel_\emptyset P_D) \parallel_{\{\alpha\}} Q ,
%\end{equation}
%where $P_d$ and $Q$ are given in Section \ref{sec:semi}. Intuitively, the above model considers a situation where $D$ \emph{independent} groups of agents, recognisable by the empty cooperation sets, share a common group of agents via the action type $\alpha$. The next definitions will be needed for the setup of the fluid framework.
%
%\begin{definition}
%Let $M$ be an FEPA model. We define then
%\begin{itemize}
%\item $\mathcal{G}(M)$ as the set of all labels of $M$, e.g., $\mathcal{G}(\pepa{Sys}) = \{P_1,\ldots,P_D,Q\}$.
%\item $\mathcal{B}(M)$ as $\bigcup \{ ds(P) \ | \ P \in \mathcal{G}(M) \}$.
%\item A population function $V : X \rightarrow \mathbb{R}_{\geq 0}$ with $\mathcal{B}(M) \subseteq X$.
%\item An \emph{initial} population function $V(0) : X \rightarrow \mathbb{N}_0$.
%\end{itemize}
%\end{definition}
%In essence, function $V(0)$ encodes the size of the system at time $t = 0$. For instance, the choice
%\begin{align}\label{eq:v0}
%V_{P_d}(0) & = N_d, & V_{P'_d}(0) & = 0, & V_Q(0) & = N_Q, & V_{Q'}(0) & = 0
%\end{align}
%states that at $t = 0$ there are $N_d$ agents in the state $P_d$, no agents in the state $P'_d$, $N_Q$ agents in the state $Q$ and no agents in the state $Q'$.
%
%\begin{definition}[Parameterised Apparent Rate]\label{def:app.rate.v}
%Let $M$ be an FEPA model, $\alpha \in \mathcal{A}$ and $V$ a population function.
%The apparent rate of $M$ with respect to $V$ is defined as
%\[
%\begin{split}
%\arate(M_0 \parallel_L M_1, V) & :=
%    \begin{cases}
%        \min(\arate(M_0, V), \arate(M_1, V))           & , \ \alpha \in L \ \land \ \rho = \min , \\
%        \arate(M_0, V) \cdot \arate(M_1, V)            & , \ \alpha \in L \ \land \ \rho = \cdot , \\
%        \arate(M_0, V) + \arate(M_1, V)                & , \ \alpha \notin L.
%    \end{cases} \\
%\arate(P, V) & := \sum_{P_i \in ds(P)} V_{P_i} \arate(P_i),
%\end{split}
%\]
%where $\arate(P_i)$ denotes the apparent rate of a PEPA component $P_i$ according to Definition~\ref{def:app.rate}.
%\end{definition}
%For instance, in (\ref{eq_running_example_fpa}) it holds that $\arate(P_d, V) = r V_{P_d}$, which gives the apparent rate at which a population of $V_{P_d}$ $P_d$-components exhibits action $\alpha$.
%
%\begin{definition}[Parameterised Component Rate]\label{def:rate.v}
%Let $M$ be an FEPA model, $\alpha \in \mathcal{A}$ and $V$ a population function. The component rate of $P' \in \mathcal{B}(M)$ is parameterised by $V$ in the following manner.
%\begin{itemize}
%\item $M = M_0 \parallel_L M_1$: if $P' \in \mathcal{B}(M_i)$, $i = 0,1$, and $\alpha \in L$ then
%\[
% \crate(M_0 \parallel_L M_1, V, P') :=
%    \dsl \frac{\crate(M_i, V, P')}{\arate(M_i, V)} \arate(M_0 \parallel_L M_1, V).
%\]
%\item $M = M_0 \parallel_L M_1$: if $P' \in \mathcal{B}(M_i),$ $i = 0,1$, and $\alpha \notin L$ then
%\[
% \crate(M_0 \parallel_L M_1, V, P') := \crate(M_i, V, P').
%\]
%\item $M = P$: then
%\[
%\crate(P, V, P') := V_{P'} \arate(P') .
%\]
%\end{itemize}
%\end{definition}
%These quantities are used to define the ODE system to be analysed.
%
%\paragraph{Notation} The derivative of $V_P$ is denoted by Newton's dot notation $\dot{V}_P$. To enhance readability, time $t$ will be suppressed in the representation of ODEs, i.e. $\dot{V}_P$ denotes $\dot{V}_P(t)$ and $V_P$ denotes $V_P(t)$.
%
%\begin{definition}\label{def_35}
%Let $M$ be an FEPA model. The initial value problem for $M$ is given by $\dot{V} = F(V)$ with initial condition $V(0)$, where for all $P \in \mathcal{B}(M)$
%\[ F_P := \sum_{\alpha \in \mathcal{A}} \bigg( \Big( \sum_{P' \in \mathcal{B}(M)} p_\alpha(P', P) \crate(M, V, P') \Big) - \crate(M, V, P) \bigg) . \]
%\end{definition}
%
%
%For instance, the initial value problem of (\ref{eq_running_example_fpa}) and (\ref{eq:v0}) in the case of $\rho = \cdot$ is given by the initial condition (\ref{eq:v0}) and the ODE system
%\begin{align}\label{eq:ode}
%\dot{V}_{P_d} & = - r u P_d \cdot Q + s P'_d & \dot{V}_{P'_d} & = - s P'_d + r u P_d \cdot Q \nonumber \\
%\dot{V}_{Q} & = - u Q \cdot \sum_{1 \leq d \leq D} r P_d + w Q' & \dot{V}_{Q'} & = - w Q' + u Q \cdot \sum_{1 \leq d \leq D} r P_d
%\end{align}
%
%
%{
%\bfseries
%\begin{remark}
%\textnormal{It can be shown that for any fluid atom $P \in \mathcal{G}(M)$ it holds that $\sum_{P' \in ds(P)} \dot{V}_{P'} = 0$. For instance, in (\ref{eq_running_example_fpa}) we have that $\dot{V}_{P_d} + \dot{V}_{P'_d} = 0$. Throughout the paper, for the sake of brevity we will write one of the ODEs for a label in terms of the derivatives of its other states, e.g., $\dot{V}_{P'_d} = - \dot{V}_{P_d}$ instead of its explicit form as in (\ref{eq:ode}).
%}\end{remark}
%}
%
%
%The following definition will be used to characterise ODE aggregations with respect to the structure of the fluid atoms.
%
%\begin{definition}[Well-Posedness]\label{def_well_posedness}
%An FEPA model $M$ is well-posed if for all occurrences $M_1 \parallel_{L} M_2$ in $M$ it holds $\exists V_1 \big( \arate(M_1,V_1) > 0 \big) \ \land \ \exists V_2 \big( \arate(M_2,V_2) > 0 \big)$ for all $\alpha \in L$.
%\end{definition}
%
%In essence, a model is well-posed whenever any synchronised action may be performed by both operands, for some population function. Thus, $\pepa{Sys}$ is well-posed, but $P_1 \parallel_{\{\beta\}} R_1$ is not since $R_1$ does not do $\beta$-actions.



\section{Exact Aggregations}

\paragraph{Exact Fluid Lumpability.}
As discussed in Section~\ref{sec:intro}, EFL reduces the ODE system size by exploiting the fact that distinct fluid atoms with the same initial population function may have undistinguishable ODE solutions.
%In Section~\ref{sec:exact.fluid.lumpability}, we extend EFL to FEPA in order to cover the semantics base don the low of mass action.

\begin{definition}[Exact Fluid Lumpability (EFL)]\label{def_exact_fluid_lumpability}
Let $M$ be a FEPA model and $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$, where $\mathcal{P}^i = \{ P^i_j \mid 1 \leq j \leq k_i \}$, be a partition of $\mathcal{G}(M)$. The partition is called \emph{exactly fluid lumpable} if there exist bijections
\[ \sigma_{P^i_j} : ds(P^i_1) \rightarrow ds(P^i_j), \qquad 1 \leq i \leq n, \ 1 \leq j \leq k_i, \]
where $\sigma_{P^i_1} \equiv \text{id}_{ds(P^i_1)}$, such that for all initial populations $V(0)$ which satisfy
\[ V_P(0) = V_{\sigma_{P^i_j}(P)}(0), \qquad \forall 1 \leq i \leq n \forall P \in ds(P^i_1) \forall 1 \leq j \leq k_i \]
the same holds for all $t \geq 0$ in the corresponding ODE solution $V$, i.e.
\[ V_P(t) = V_{\sigma_{P^i_j}(P)}(t), \qquad \forall 1 \leq i \leq n \forall P \in ds(P^i_1) \forall 1 \leq j \leq k_i \forall t \geq 0 . \]
\end{definition}
%
Exact fluid lumpability of a partition is induced by the notion of label equivalence, established between tuples of fluid atoms. Intuitively, relating two tuples (of the same length) $(S_1, T_1)$ and $(S_2, T_2)$ means that, component-wise, the fluid atoms have the same trajectories; that is, $S_1$ (resp., $T_1$), has the same ODE solution as $S_2$ (resp., $T_2$). Fluid atoms within the same tuple are coupled in the sense that matching fluid atoms have to be provided for each element of a tuple.

\begin{definition}[Label Equivalence]\label{def_label_eq}
Let $M$ be a FEPA model and let $\mathcal{P} = (\vec{P}^1, \ldots, \vec{P}^N)$, $\vec{P}^i = (P^i_1, \ldots, P^i_{K_i})$, be a \emph{tuple partition} on $\mathcal{G}(M)$, that is, for each $P \in \mathcal{G}(M)$ there exist unique $1 \leq i \leq N$ and $1 \leq k \leq K_i$ with $P = P^i_k$. $\vec{P}^i$ and $\vec{P}^j$ are said to be \emph{label equivalent}, written $\vec{P}^i \sim_\mathcal{P} \vec{P}^j$, if $K_i = K_j$ and there exist bijections $\sigma_{k}: ds(P^i_k) \rightarrow ds(P^j_k)$, where $1 \leq k \leq K_i$, such that for all population functions $V$ of $M$ and
\[
V^\sigma_P :=
\begin{cases}
    V_{\sigma_k(P)} & \ , \exists 1 \leq k \leq K_i ( P \in ds(P^i_k) ) \\
    V_{\sigma_k^{-1}(P)} & \ , \exists 1 \leq k \leq K_i ( P \in ds(P^j_k) ) \\
    V_{P} & \ , \text{otherwise}
\end{cases}
\]
it holds that
\begin{enumerate}[a)]
    \item $\crate(M, V, P) = \crate(M, V^\sigma, \sigma_k(P))$
   	\item $\dsl \sum_{P'} p_\alpha(P', P) \crate(M, V, P') = \sum_{P'} p_\alpha(P', \sigma_{k}(P)) \crate(M, V^\sigma, P')$
   	\item $\crate(M, V, P) = \crate(M, V^\sigma, P)$ for all $P \in ds(P^l_k)$ with $P^l_k \notin \vec{P}^i,\vec{P}^j$
	\item $\arate(M,V) = \arate(M,V^\sigma)$ and $\arate(P^i_k) = \arate(P^j_k)$ for all $1 \leq k \leq K$.
\end{enumerate}
\end{definition}

\begin{example}\label{ex:second}
EFL has been used to simplify replicas of composite processes~\cite{Tschaikowski2013a}. For instance, let us consider the fluid atoms in Example~\ref{ex:first}, $R_d \rmdef (\alpha, \tilde{r}).R'_d$, and $R'_d \rmdef (\gamma, \tilde{s}).R_d$. Further, let us take the FEPA model
$$\pepa{Sys}_\mathcal{E} := \big((P_1 \parallel_{\{\alpha\}} R_1) \parallel_\emptyset \ldots \parallel_\emptyset (P_D \parallel_{\{\alpha\}} R_D)\big) \parallel_{\{\alpha\}} Q \ ,$$
which features $D$ replicas of composite processes of type $P_d \parallel_{\{\alpha\}} R_d$.  Let us consider now the tuple partition $\mathcal{P} := \{(P_1,R_1), \ldots,  (P_D,R_D), (Q)\}$. Then, it can be shown that $(P_i,R_i) \sim_\mathcal{P} (P_j,R_j)$, thus formalising the intuitive idea that each replica has the same solution (if initialised with identical conditions).
\end{example}

Using label equivalence, which acts on \emph{tuples} of labels, we define the following notion which allows to relate single labels.

\begin{definition}[Projected Label Equivalence]\label{def_proj_label_eq}
Fix a FEPA model $M$ and a tuple partition $\mathcal{P}$ of $\mathcal{G}(M)$. Two fluid atoms $P_1,P_2 \in \mathcal{G}(M)$ are projected label equivalent, $P_1 \approx_\mathcal{P} P_2$, if $\vec{P}^i \sim_\mathcal{P} \vec{P}^j$ and $k_i = k_j$ in the unique assignment $P_1 = P_{k_i}^i$, $P_2 = P_{k_j}^j$.
\end{definition}
Therefore, we have that $P_1 \approx_{\cal P} P_2$, $R_1 \approx_{\cal P} R_2$, and so on.

Following~\cite{concur12}, we are finally ready to extend EFL to FEPA, showing that the following is valid also for semantics based on the law of mass action, $\rho = \cdot$.%
%\footnote{{\bfseries Note to reviewers.} All proofs are given in the appendix, which will be turned into a technical report should the paper be accepted.}

\begin{theorem}\label{thm_proj_label_eq}
Fix a FEPA model $M$ and a tuple partition $\mathcal{P}$ of $\mathcal{G}(M)$. Then, $\sim_\mathcal{P}$ is a congruence relation with respect to $\parallel_L$, $\approx_\mathcal{P}$ is an equivalence relation on $\mathcal{G}(M)$ and $\mathcal{G}(M) / \approx_\mathcal{P}$ is exactly fluid lumpable.
\end{theorem}
For instance, it holds that $\mathcal{G}(\pepa{Sys}_\mathcal{E}) / \approx_\mathcal{P}$ yields the exactly fluid lumpable partition
%\begin{equation}\label{eq_efl_partition}
$\big\{ \{P_1, \ldots, P_D\}, $ $ \{R_1, \ldots, R_D\}, \{Q\} \big\}$.
%\end{equation}
Let us notice that this result gives us a tool which aggregates ODE systems to smaller ones. In Example~\ref{ex:second}, for instance, this allows one to recover the solution of an ODE system of size $4D + 2$ by solving an aggregated ODE system of size $4 + 2$.

Furthermore, the characterisation of EFL can be shown also when $\rho = \cdot$.

\begin{theorem}\label{thm_char_efl}
Fix a well-posed FEPA model $M$, a tuple partition $\mathcal{P} = \{\vec{P}^1, \ldots$ $\ldots, \vec{P}^N\}$ on $\mathcal{G}(M)$ and assume that $\vec{P}^i \sim_\mathcal{P} \vec{P}^j$. Then, $P^i_k$ is semi-isomorphic to $P^j_k$ for all $1 \leq k \leq K_i$.
\end{theorem}

Using the last theorem, one can show that, under the condition of well-posedness, different exactly fluid lumpable partitions can be merged.

\begin{theorem}\label{thm_merge}
Fix a well-posed FEPA model $M$ and a set of tuple partitions $S = \{\mathcal{P}_1, \ldots, \mathcal{P}_m\}$ of $\mathcal{G}(M)$. Then, the partition $\mathcal{G}(M) / (\approx_{\mathcal{P}_1} \cup \ldots \cup \approx_{\mathcal{P}_m})^*$ is exactly fluid lumpable.
\end{theorem}

{
\bfseries
\begin{remark}
\textnormal{In that what follows, we will assume that an EFL partition is established as in Theorem \ref{thm_proj_label_eq}.
}\end{remark}
}


\paragraph{Ordinary Fluid Lumpability.} EFL considers a partition of fluid atoms such that elements in the same part have the \emph{same solution}. Instead, in ordinary fluid lumpability (OFL) \emph{the sum of the solutions} of elements within the same part are fully recovered from the solution of a (smaller) ODE system consisting of one single ODE for a representative element of each part.

\begin{definition}[Ordinary Fluid Lumpability (OFL)]\label{def_ordfluidlump}
Let $M$ be a FEPA model and let $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ be a partition of $\mathcal{G}(M)$. The partition is called \emph{ordinarily fluid lumpable} if there exist bijections
\[ \sigma_{P^i_j} : ds(P^i_1) \rightarrow ds(P^i_j), \qquad 1 \leq i \leq n, \ 1 \leq j \leq k_i \]
such that $\sigma_{P^i_1} \equiv \text{id}_{ds(P^i_1)}$ and for all $\alpha \in \mathcal{A}$, $1 \leq i \leq n$ and $V$, it holds that
\begin{enumerate}[i)]
	\item $\dsl \sum_{1 \leq j \leq k_i} \crate(M,V,\sigma_{P^i_j}(P)) = \crate(M,V^\sigma,P), \quad \forall P \in ds(P^i_1)$
    \item $\dsl \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \crate(M,V,\sigma_{P^i_j}(P'))$ \vspace{-0.25cm}
    \[ \quad = \sum_{P' \in ds(P^i_1)} p_\alpha(P', P) \crate(M,V^\sigma,P'), \quad \forall P \in ds(P^i_1) \]
	\item $\arate(M,V) = \arate(M,V^\sigma)$ and $\arate(P^i_1) = \arate(P^i_j)$ for all $1 \leq j \leq k_i$,
\end{enumerate}
where $V^\sigma_P :=
\begin{cases}
    \dsl \sum_{1 \leq j \leq k_i} V_{\sigma_{P^i_j}(P)} & \ , \exists 1 \leq i \leq n \big( P \in ds(P^i_1) \big) \\
    0 & \ , \text{otherwise}.
\end{cases}$
\end{definition}

We can now define and relate the lumped ODE system to the original one.

\begin{theorem}[ODE Lumping]\label{thm_ode_lump}
Let $M$ be a FEPA model, $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ an ordinarily fluid lumpable partition of $\mathcal{G}(M)$, and $V$ the ODE solution of $M$ for a given initial condition $V(0)$. Define
\[ W_P := \sum_{1 \leq j \leq k_i} V_{\sigma_{P^i_j}(P)}, \qquad 1 \leq i \leq n, \quad P \in ds(P^i_1)   \]
and
\[ \overline{W}_P :=
\begin{cases}
W_P & , \ \exists 1 \leq i \leq n \big( P \in ds(P^i_1) \big) \\
0 & , \ \text{otherwise}
\end{cases} \]
for all $P \in \mathcal{B}(M)$. Then, $W$ is the unique solution of the ODE system
\begin{align}\label{thm_ord_lump_general}
\dot{W}_P & = \sum_{\alpha \in \mathcal{A}} \Big(\sum_{P' \in ds(P^i_1)} p_\alpha(P',P) \crate(M,\overline{W},P') - \crate(M,\overline{W},P) \Big), \nonumber \\
W_P(0) & = \sum_{1 \leq j \leq k_i} V_{\sigma_{P^i_j}(P)}(0),
\end{align}
where $1 \leq i \leq n$ and $P \in ds(P^i_1)$. Hence,
\[ \sum_{1 \leq j \leq k_i} V_{\sigma_{P^i_j}(P)}, \qquad 1 \leq i \leq n, \quad P \in ds(P^i_1), \]
can be recovered by solving the lumped ODE system (\ref{thm_ord_lump_general}).
\end{theorem}
%
For instance, let us consider again Example~\ref{ex:first}. It can be shown that the partition $\big\{ \{P_1, \ldots, P_D\}, \{Q\} \big\}$ is an OFL partition of $\pepa{Sys}$. According to the above theorem, the aggregated ODE system, of size 4, is given by
\begin{align}
\dot{W}_{P} & = - ru W_P \cdot W_Q + s W_{P'} &
\dot{W}_{P'} & = + ru W_P \cdot W_Q - s W_{P'} \nonumber \\
\dot{W}_{Q} & = - ru W_P \cdot W_Q + w W_{Q'} &
\dot{W}_{Q'} & = + ru W_P \cdot W_Q - w W_{Q'} \nonumber %\label{eq_ofl_ode}
\end{align}
with initial condition $W_P(0) = \sum_{d = 1}^D V_{P_d}(0) =\sum_{d = 1}^D N_d$, $W_{P'}(0) = \sum_{d = 1}^D V_{P'_d}(0)$  $= 0$, $W_Q(0) = V_Q(0) = M$, and $W_{Q'}(0) = V_{Q'}(0) = 0$. It holds that $W_P(t) = \sum_{d = 1}^D V_{P_d}(t)$, but each individual solution, $V_{P_d}(t)$,  cannot be recovered.
%Let us remark that the same partition can be shown to be EFL, leading however to two distinct aggregated ODE systems. This is not always the case. For instance, it can be easily seen that (\ref{eq_efl_partition}) it is not an OFL partition.

The next theorem states the congruence property of OFL with respect to the parallel composition of FEPA.

\begin{theorem}[Congruence]\label{thm_cong}
Let us fix two FEPA models $M_1, M_2$ and assume that $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ and $\{\mathcal{P}^{n+1}, \ldots, \mathcal{P}^{n+m+1}\}$ are ordinarily fluid lumpable partitions of $\mathcal{G}(M_1)$ and $\mathcal{G}(M_2)$, respectively. Then, thanks to the set of bijections
\[ \sigma_{P^i_j} : ds(P^i_1) \rightarrow ds(P^i_j), \qquad 1 \leq i \leq n+m+1, \quad 1 \leq j \leq k_i , \]
the partition $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\} \cup \{\mathcal{P}^{n+1}, \ldots, \mathcal{P}^{n+m+1}\}$ of $\mathcal{G}(M_1 \parallel_{L} M_2)$ is also ordinarily fluid lumpable.
\end{theorem}


Finally, similarly to EFL, the next theorem characterises OFL with respect to semi-isomorphism.

\begin{theorem}\label{thm_char_ofl}
Fix a well-posed FEPA model $M$ and assume that the partition $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ of $\mathcal{G}(M)$ is ordinarily fluid lumpable. Then, $P^i_1$ is semi-isomorphic to $P^i_j$ for all $2 \leq j \leq k_i$ and $1 \leq i \leq n$.
\end{theorem}
Well-posedness is needed in Theorem \ref{thm_char_efl} and \ref{thm_char_ofl}. Let $\tilde{P}_d \rmdef (\alpha,r).\tilde{P}'_d + (\gamma,r/d).\tilde{P}_d + (\gamma,r - r/d).\tilde{P}_d$ and $\tilde{P}_d \rmdef (\beta,s).\tilde{P}'_d$, i.e. $\tilde{P}_i$ and $\tilde{P}_j$ are not semi-isomorphic. Then, $\big((\tilde{P}_1 \parallel_{\{\alpha\}} R_1) \parallel_\emptyset \ldots \parallel_\emptyset (\tilde{P}_D \parallel_{\{\alpha\}} R_D)\big) \parallel_{\{\alpha,\gamma\}} Q$ and $(\tilde{P}_1 \parallel_{\emptyset} \ldots \parallel_{\emptyset} \tilde{P}_D) \parallel_{\{\alpha,\gamma\}} Q$ are ill-posed, while $(\tilde{P}_i,R_i) \sim_{\mathcal{P}} (\tilde{P}_j,R_j)$ and $\big\{ \{ \tilde{P}_1, \ldots, \tilde{P}_D \}, \{Q\} \big\}$ is lumpable.


%Note that well-posedness is crucial for Theorem \ref{thm_char_efl} and \ref{thm_char_ofl}. To see this, define $\tilde{P}_d \rmdef (\alpha,r).\tilde{P}'_d + (\gamma,r/d).\tilde{P}_d + (\gamma,r - r/d).\tilde{P}'_d$, $\tilde{P}_d \rmdef (\beta,s).\tilde{P}'_d$ and observe that $\tilde{P}_d$ and $\tilde{P}_{d'}$ are not semi-isomorphic. Then,
%replacing $P_d$ with $\tilde{P}_d$ in $\pepa{Sys}_{\mathcal{E}}$ yields an ill-posed model with $(\tilde{P}_i,R_i) \sim_{\mathcal{P}} (\tilde{P}_j,R_j)$. Similarly, replacing $P_d$ with $\tilde{P}_d$ in $\pepa{Sys}$ yields also ill-posed model, while $\big\{ \{ \tilde{P}_1, \ldots, \tilde{P}_D \}, \{Q\} \big\}$ is an OFL partition.


%As for EFL (see~\cite{concur12}),  well-posedness cannot be dropped in the above theorem. For instance, given $\tilde{P} \rmdef (\alpha, r).\tilde{P}' + (\gamma,u).\tilde{P}'$ and  $\tilde{P}' \rmdef (\beta, s).\tilde{P}$,
%the model $(\tilde{P} \parallel_{\emptyset} P_1 \parallel_{\emptyset} \ldots \parallel_{\emptyset} P_D) \parallel_{\{\alpha,\gamma\}} Q$ is ill-posed and has the ordinary fluid lumpable partition
%$\big\{ \{ \tilde{P}, P_1, \ldots, P_D \}, \{Q\} \big\}$. %(Note that the ill-posedness arises from the fact that $\tilde{P}_1$ is hindered in performing its $\gamma$ action.)
%However, $\tilde{P}$ is not semi-isomorphic to any $P_d$.





\section{Fluid $\varepsilon$-Lumpability}\label{sec:theory:epsilon}

We now study aggregations for models where certain fluid atoms can be made elements of the same partition block after a suitable perturbation of their parameters. In the case of EFL, we allow different rates and initial populations. In the case of OFL, instead, we only consider the former because there is no requirement on identical initial populations for aggregated fluid atoms.


At the basis of our investigation is the following comparison theorem which relates two ODE systems of the same size, where the vector field is made dependent on a vector of parameters, here denoted by $\zeta$ and $\xi$. Thus, for some norm $\norm{\cdot}$, we interpret $\varepsilon = \norm{\xi - \zeta}$ as the intensity of the perturbation on the rates of the same model. The two ODE systems may also have different initial conditions $\underline{x}_\zeta$ and $\underline{x}_\xi$, and we let $\delta  = \norm{\underline{x}_\xi - \underline{x}_\zeta}$. This will be used to define our approximate version of EFL. The theorem states that, on a fixed time interval $[0;t]$, the distance between the two solutions depends \emph{linearly} on both $\varepsilon$ and $\delta$.
\begin{theorem}\label{thm_peturbation_0}
Consider the ODE systems
\begin{align*}
\begin{cases}
\dot x_\zeta = f(\zeta,x_\zeta) \\
x_\zeta(0)= \underline{x}_\zeta
\end{cases}
\qquad\qquad\qquad
\begin{cases}
\dot x_\xi = f(\xi,x_\xi)\\
x_\xi(0)= \underline{x}_\xi
\end{cases}
\end{align*}
where $f$ is assumed to be Lipschitz continuous in some domain $\mathcal{D} \subseteq \xR^{n+m}$, both with respect to $x$ as with respect
to $\zeta$ with Lipschitz constant $L_\zeta$ and $K_x$ respectively, that is
\begin{align*}
\norm{f(\zeta,x)-f(\zeta,x')} & \leq L_\zeta \norm{x-x'} , \qquad (\zeta,x), (\zeta,x')\in \mathcal{D}, \\
\norm{f(\zeta,x)-f(\zeta',x)} & \leq K_x \norm{\zeta-\zeta'} , \qquad (\zeta,x), (\zeta',x)\in \mathcal{D}.
\end{align*}
Let us assume further that both ODE systems have a solution on $[0;t]$, where $t > 0$, which remains in $\mathcal{D}$, and that $K := \sup_{0 \leq s \leq t} K_{x_\xi(s)} < \infinity$. Then
\[
\norm{x_\zeta(t)-x_\xi(t)}\leq \left(\frac{\varepsilon K}{L_\zeta} + \delta\right)\xe^{L_\zeta t} - \frac{\varepsilon K}{L_\zeta}
\]
if $\varepsilon = \norm{\xi-\zeta}$ and $\delta = \norm{\underline{x}_\zeta - \underline{x}_\xi}$.
\end{theorem}

Next, we formally introduce the notion of perturbation on rates.

% Max's old definition
%\begin{definition}
%For an FEPA model $M$, let $\nu(M) \geq 1$ denote the number of rate variables in $M$, i.e. the vector of rate variables of $M$ may be written as $(x_1, \ldots, x_{\nu(M)})$. Then, for a $\xi \in \mathbb{R}_{>0}^{\nu(M)}$, the model $M(\xi)$ arises from $M$ by replacing each occurrence of $x_i$ by $\xi_i$.
%\end{definition}

\begin{definition}
For a FEPA model $M$, let $\nu(M)$ denote the vector of distinct occurrences of action rates in $M$, written $\nu(M) = (x_1, \ldots, x_i, \ldots, x_{\lvert \nu(M) \rvert})$. Then, for a $\xi \in \mathbb{R}_{>0}^{|\nu(M)|}$, the model $M(\xi)$ arises from $M$ by replacing each $x_i$ with $\xi_i$.
\end{definition}
%

Theorem~\ref{thm_peturbation_0} can be applied to FEPA  by proving that FEPA models induce globally Lipschitz ODE systems and have bounded ODE solutions.
%{\bfseries Mirco changed next statement slightly}
%\begin{theorem}\label{thm_perturbation}
%Fix an FEPA model $M$, a $\xi, \zeta  \in \mathbb{R}_{>0}^{\lvert \nu(M) \rvert}$, population functions $V^\xi(0), V^\zeta(0)$, and some $c, T > 0$. Then, there exist $C_1, C_2 > 0$ such that $\| V^\xi(0) - V^\zeta(0) \| \leq c$ implies
%\[ \max_{0 \leq t \leq T} \| V^\xi(t) - V^\zeta(t) \| \leq C_1 \| \xi - \zeta \| + C_2 \| V^\xi(0) - V^\zeta(0) \| , \]
%where $V^\xi$ and $V^\zeta$ are the ODE solutions of $M(\xi)$ and $M(\zeta)$, respectively.
%\end{theorem}

\begin{theorem}\label{thm_perturbation}
Fix a FEPA model $M$, a $\zeta \in \mathbb{R}_{>0}^{|\nu(M)|}$, a population function $V^\zeta(0)$ and $c, t > 0$. Then, there exist $C_1, C_2 > 0$ such that $\| V^\xi(0) \|, \| \xi \| \leq c$ implies
\[ \max_{0 \leq s \leq t} \| V^\xi(s) - V^\zeta(s) \| \leq C_1 \| \xi - \zeta \| + C_2 \| V^\xi(0) - V^\zeta(0) \| , \]
where $V^\xi$ and $V^\zeta$ refer to the ODE solutions of $M(\xi)$ and $M(\zeta)$, respectively.
\end{theorem}
Let us remark that the above result states that the perturbations on the rate parameters and on the initial conditions yield separate additive contributions to the aggregation error. While the former kind of perturbation can be related to analogous approximate aggregations for Markov chains, e.g.,~\cite{BuchholzOrdinaryExact}, the latter does not have a stochastic counterpart to the best of our knowledge. This is because altering the initial population of agents leads to a generator matrix of different size, while a perturbation on the rates preserves the matrix structure. In the fluid semantics, instead, both cases do not alter the structure of the ODE system, but only its parameters.
Let us also notice that, in the above theorem, $M(\zeta)$ is arbitrary but fixed, whereas $M(\xi)$ varies. We now focus  on the situation where $M(\zeta)$ has either an exactly or an ordinarily fluid lumpable partition.

\begin{definition}[Fluid $\varepsilon$-Lumpability]
Fix a FEPA model $M$ and $\xi \in \mathbb{R}_{>0}^{|\nu(M)|}$. If $M(\zeta)$ has an exactly/ordinarily fluid lumpable partition $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ for some $\zeta \in \mathbb{R}_{>0}^{|\nu(M)|}$, $M(\xi)$ is said to be $\norm{\xi - \zeta}$-exactly/ordinarily fluid lumpable with respect to some norm $\norm{\cdot}$.
\end{definition}
%
For instance, let us take $D = 2$ and, with obvious ordering of the rate occurrences, $\zeta = (r,s,r,s, u,w)$, for which (\ref{eq_running_example_fpa}) admits EFL whenever $V_{P_1} = V_{P_2}$ and $V_{P'_1} = V_{P'_2}$. Consider now the same model, where the rates are replaced with $\xi = (r + \varepsilon_1, s + \varepsilon_2, r, s, u, w)$. This model is $\varepsilon$-exactly fluid lumpable with $\varepsilon = \norm{(\varepsilon_1, \varepsilon_2, 0, 0, 0, 0)}$.
%The partition $\{ \mathcal{P}^1, \mathcal{P}^2 \} = \big\{ \{ P_1, \ldots, P_D \}, \{Q\} \big\}$ of $\pepa{Sys}(\zeta)$ is ordinarily fluid lumpable if $\zeta = (r, s, r,s,u,w)$. Then, it holds that $M(\xi)$ is $\varepsilon$-ordinarily fluid lumpable, where $\varepsilon = \lVert (\xi_1 - r,\xi_2 - s, \ldots,\xi_{2D-1} - r,\xi_{2D}-s,\xi_{2D+1}-r,\xi_{2D+2}-w) \rVert$.
In general, an exactly/ordinarily fluid lumpable partition admits an infinity of $\varepsilon$-lumpable partitions; $\varepsilon$ gives the measure of how close a given model is to error-free lumping.

Both $\varepsilon$-EFL and $\varepsilon$-OFL enjoy congruence.
\begin{theorem}[Congruence]\label{thm_eps_cong}
Fix two FEPA models $M_1, M_2$ and assume that $\{\mathcal{P}^1_k, \ldots, \mathcal{P}^{n_k}_k\}$ is $\norm{\xi_k - \zeta_k}$-exactly/ordinarily fluid lumpable in $\mathcal{G}(M_k(\xi_k))$ for some $\xi_k,\zeta_k \in \mathbb{R}_{>0}^{|\nu(M_k)|}$, and $k = 1,2$. Then, for any $L \subseteq \mathcal{A}$, $\bigcup_{k=1}^2 \{\mathcal{P}^1_k, \ldots, \mathcal{P}^{n_k}_k\}$ is $\norm{(\xi_1,\xi_2) - (\zeta_1,\zeta_2)}$-exactly/ordinarily fluid lumpable in $\mathcal{G}(M_1 \parallel_{L} M_2)$.
\end{theorem}
Clearly, as an OFL partition does not depend on the initial values, a perturbation of initial values is interesting only in the case of EFL.

We turn now to a characterisation of $\varepsilon$-OFL and $\varepsilon$-EFL. It is natural consider an $\varepsilon$-extension of semi-isomorphism to relate fluid atoms that are isomorphic up to an appropriate perturbation of their rates.


\begin{definition}[$\varepsilon$-Semi-Isomorphism]
Two fluid atoms $P$ and $Q$ are $\varepsilon$-semi-isomorphic for some $\varepsilon > 0$, if there is a bijection $\sigma : ds(P) \rightarrow ds(Q)$ which satisfies
\[ \Big| \sum_{P_i \xrightarrow{(\alpha, r)} P_j} r - \sum_{\sigma(P_i) \xrightarrow{(\alpha, r)} \sigma(P_j)} r \Big| \leq \varepsilon\]
for all $P_i, P_j \in ds(P)$ and $\alpha \in \mathcal{A}$. Such $\sigma$ is called $\varepsilon$-semi-isomorphism.
\end{definition}

Analogously to EFL and OFL, the following characterises $\varepsilon$-lumpability with respect to $\varepsilon$-semi-isomorphism.
%\begin{theorem}\label{thm_eps_char}
%For any well-posed FEPA model $M$ and norm $\norm{\cdot}$, there exists a real $C > 0$ such that: assume that $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ is an $\norm{\xi-\zeta}$-exactly/ordinarily fluid lumpable partition  of $M(\xi)$, where $\xi, \zeta \in \mathbb{R}_{>0}^{|\nu(M)|}$; in the case of exact lumpability, assume further that the exact lumpability of $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ in $M(\zeta)$ is induced by label equivalence. Then, any two fluid atoms $P^i_j, P^i_{j'}$ of $M(\xi)$ are $C \norm{\xi-\zeta}$-semi-isomorphic.
%\end{theorem}
%For instance, let us assume that $\norm{\cdot} = \norm{\cdot}_2$ and, for any given $\alpha \in \mathcal{A}$, each $P \in \mathcal{B}(M)$ performs at most one $\alpha$-transition. Then, the proof of Theorem \ref{thm_eps_char} and $\norm{\cdot}_\infinity \leq \norm{\cdot}_2$ show that $P^i_j, P^i_{j'} \in \mathcal{B}(M(\xi))$ are $\norm{\xi-\zeta}_2$-semi-isomorphic for all $\xi \in \mathbb{R}_{>0}^{|\nu(M)|}$. Specifically, by noting that $\pepa{Sys}$ satisfies the assumption, we can infer that $P_d$, $P_{d'}$ are $\norm{\xi-\zeta}_2$-semi-isomorphic in $\pepa{Sys}(\xi)$ for all $\xi \in \mathbb{R}_{>0}^{|\nu(\pepa{Sys})|}$.
\begin{theorem}\label{thm_eps_char}
For any well-posed FEPA model $M$ and norm $\norm{\cdot}$, there exists a $C > 0$ such that if $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ is an $\norm{\xi-\zeta}$-exactly/ordinarily fluid lumpable partition of $M(\xi)$, where $\xi, \zeta \in \mathbb{R}_{>0}^{|\nu(M)|}$, then:
\begin{enumerate}[1)]
   \item Any two fluid atoms $P^i_j, P^i_{j'}$ of $M(\xi)$ are $C \norm{\xi-\zeta}$-semi-isomorphic;
   \item In the special case where for all $\alpha \in \mathcal{A}$ and $P,P' \in \mathcal{B}(M)$ there is at most one $\alpha$-transition from $P$ to $P'$ and $\norm{\cdot} = \norm{\cdot}_\infinity$, $1)$ holds for $C = 1$.
\end{enumerate}
\end{theorem}
%
For instance, the above theorem ensures that $P_d$, $P_{d'}$ are $\norm{\xi-\zeta}_\infinity$-semi-isomorphic in $\pepa{Sys}(\xi)$, cf. (\ref{eq_running_example_fpa}), for all $\xi \in \mathbb{R}_{>0}^{|\nu(\pepa{Sys})|}$, if $\zeta = (r, s, \ldots,r,s,r,w)$.

%\begin{table}[t]
%\begin{center}
%\begin{tabular}{ccccccc}
%\toprule
%$\Delta$ & \multicolumn{1}{c}{$\phantom{000}0.00\phantom{000}$} & $\phantom{000}0.01\phantom{000}$ &  $\phantom{000}0.02\phantom{000}$  &   $\phantom{000}0.03\phantom{000}$  & $\phantom{000}0.04\phantom{000}$  &  $\phantom{000}0.05\phantom{000}$ \\
%\midrule
%$D = 2$ & 0.41\%  & 0.70\%  & 1.02\% & 1.33\% & 1.64\% & 1.94\% \\
%$D = 3$ & 0.82\%  & 1.24\%  & 1.71\% & 2.17\% & 2.62\% & 3.07\% \\
%$D = 4$ & 1.23\%  & 1.78\%  & 2.39\% &  2.99\% & 3.59\% & 4.17\% \\
%$D = 5$ & 1.63\%  & 2.31\%  & 3.06\% &  3.81\% & 4.53\% & 5.25\% \\
%\bottomrule
%\end{tabular}
%\end{center}
%\caption{Approximation errors for $\varepsilon$-EFL, by perturbations of initial conditions and rates.}\label{tab:elf}
%\end{table}
%%
%\begin{table}[t]
%\begin{center}
%\begin{tabular}{cccccc}
%\toprule
%$\Delta$ &  $\phantom{000}0.01\phantom{000}$  &  $\phantom{000}0.02\phantom{000}$  &   $\phantom{000}0.03\phantom{000}$  & $\phantom{000}0.04\phantom{000}$  &  $\phantom{000}0.05\phantom{000}$ \\
%\midrule
%$D = 2,3,4,5$ & 0.16\%  & 0.32\%  & 0.47\% & 0.63\% & 0.78\% \\
%\bottomrule
%\end{tabular}
%\end{center}
%\caption{Approximation errors for OFL, by perturbations of rates (as OFL does not depend on the initial conditions).}\label{tab:ofl}
%\end{table}
%
\section{Numerical Evaluation}

\begin{figure}[t]
%\centering
\subfloat[$\varepsilon$-EFL for $\rho = \min$]{
\label{fig:efl.min}
\centering
\includegraphics[scale=0.64]{img/efl_min}
}
%\,
\subfloat[$\varepsilon$-EFL for $\rho = \cdot$]{
\label{fig:efl.prod}
\centering
\includegraphics[scale=0.64]{img/efl_prod}
}
%\,
\subfloat[$\varepsilon$-OFL, for $D = 12$]{
\label{fig:ofl}
\centering
\includegraphics[scale=0.64]{img/ofl}
}
\caption{Numerical evaluation of $\varepsilon$-lumpability.}\label{fig:elf}
\end{figure}
In this section we provide some numerical evidence of the aggregation error introduced by $\varepsilon$-EFL and $\varepsilon$-OFL\@. We considered the FEPA model (\ref{eq_running_example_fpa}) with
\begin{equation}\label{eq:original}
P_d \rmdef (\alpha, r_d).P'_d \qquad P'_d \rmdef (\beta, s).P_d \qquad Q = (\alpha, r).Q' \qquad Q' = (\gamma, w).Q \ .
\end{equation}
We fixed $s = 0.5$, $w = 15.0$. To obtain non-isomorphic fluid atoms, we set $r_d =  1.0 + (d-1)\Delta$, where $\Delta$ is a parameter that was varied  between 0.0005 and 0.1000 at 0.005 steps in our tests. In this way, $\Delta$ is related to the intensity of the perturbation.
%For instance, in a model with $D = 12$ and $\Delta = 0.1$, we have  $r_{10}/r_1 = 2.1$, showing a significant difference between the rate parameters of fluid atom $P_1$ and those of $P_{10}$.
The initial population function was set as
$V_{P_d}(0) = 200 + (d - 1)$, $V_{P'_d}(0) = 0$, $V_Q(0) = 400$, and $V_{Q'}(0) = 0$; thus, the $P_d$ fluid atoms  have initial populations separated by a few percent.
For evaluating both $\varepsilon$-EFL and $\varepsilon$-OFL, we considered a perturbed model where $r_d$ in (\ref{eq:original}) was made independent of $d$ and set equal to the average value in the original model, i.e., $1.0 + (\Delta/D) \sum_{d = 1}^D (d-1)$. In such a perturbed model, all $P_d$ fluid atoms are now isomorphic.
%%%%
\paragraph*{Assessment of $\varepsilon$-EFL\@.}  We considered different values of $D$ to numerically evaluate the impact of different initial conditions on the quality of the aggregation of $\varepsilon$-EFL. Specifically, we set $D = 3,6,9,12$. Let us recall that (\ref{eq_running_example_fpa}) has $2D + 2$ ODEs. For each value of $D$ and $\Delta$, the model solution was compared against that of the perturbed model with the initial population function set as follows:
$V^\varepsilon_{P_d}(0) = 200 + (1/D) \sum_{d=1}^D (d-1)$, $V^\varepsilon_{P'_d}(0) = 0$, $V^\varepsilon_Q(0) = 400$, and $V^\varepsilon_{Q'} = 0$.
In this way, the initial population of $P_d$ fluid atoms is made independent from $d$ and is set equal to the average initial population across $d$, similarly to what done for the perturbation on $r_d$. It follows that, in the perturbed model, $\{ \{ P_1, \ldots, P_D \}, \{ Q \} \}$ is an exactly fluid lumpable partition. Hence, the original model and the perturbed one are related by $\varepsilon$-EFL.
Both models were solved over the time interval $[0;100]$ (ensuring convergence to equilibrium in all cases), with solutions registered at 0.2 time steps. The approximation relative error for $\varepsilon$-EFL is as:
\[
100 \times \max_{t \in \{ 0,0.02,\ldots, 100 \}} \max_{S \in \{ P_1, \ldots, P_d, Q \} }  \frac{\left \lvert V_S(t) - V_S^\varepsilon(t)\right \rvert}{V_{S}(0)},
\]
where $V_S(t)$ is the solution of the original model and $V_S^\varepsilon(t)$ is the corresponding solution in the  perturbed one. The absolute difference is normalised with respect to the total population of the fluid atom.

The results are presented in Figures~\ref{fig:efl.min} and \ref{fig:efl.prod}, for  $\rho = \min$ and $\rho = \cdot$, respectively. In both cases, we observe a linear growth of the error as a function of the perturbation $\Delta$. For any fixed $D$, the case $\rho = \cdot$ yields more accurate aggregates than $\rho = \min$, with particularly small errors for $D = 3,6,9$. These tests show that even non-negligible perturbations (i.e., up to $\Delta$ ca 0.04) can produce acceptable errors (i.e., less than 10\%) in practice.
%%%%
\paragraph*{Assessment of $\varepsilon$-OFL\@.} A similar setting was used for the assessment of  $\varepsilon$-OFL, since in the perturbed model $\{ \{ P_1, \ldots, P_D \}, \{ Q \} \}$ is also an ordinarily fluid lumpable partition. However, unlike $\varepsilon$-EFL,  as discussed, $\varepsilon$-OFL \emph{does not} depend on the initial population function. Therefore, in our tests the initial conditions were not changed in the perturbed model. Furthermore, we analysed only the case $D = 12$, which yielded the worst accuracy in $\varepsilon$-EFL; the other cases showed the same errors (up to numerical precision of the ODE solver). A different error metric was used, to reflect the fact that OFL involves sums of ODE solutions of the unaggregated model. The approximation relative error is defined as:
\[
100 \times  \max_{t \in \{ 0,0.02,\ldots, 100 \}} \max \left \{ \frac{\left \lvert \sum_{d = 1}^D V_{P_d}(t) - V_P^\varepsilon(t)\right \rvert}{\sum_{d = 1}^D V_{P_d}(0)}, \frac{\lvert V_Q(t) - V^\varepsilon_Q(t) \rvert}{V_Q(0)} \right \} ,
\]
where $V_P^\varepsilon(t)$ and $V_Q^\varepsilon(t)$ are the solutions in the OFL model corresponding to the sum of the $P_d$ derivatives and to the $Q$ derivative, respectively.
The results are shown in Figure~\ref{fig:ofl}. Overall, both for $\rho = \min$ and $\rho = \cdot$, the $\varepsilon$-OFL appears to be much more robust, with negligible errors across all values of $\Delta$.






\section{Conclusion}
This paper has studied ODE aggregations for a process algebra that uniformly treats two different dynamics of interactions, for capturing models of performance and chemical reaction networks. Our approximate aggregations allow models with heterogeneous processes to be treated as homogeneous models by appropriate perturbations of rate parameters and initial populations. Although the numerical results suggest that this aggregation can be robust,  tightening of the theoretical error bound is part of future work for increasing the a-priori confidence on the practical usefulness of these techniques.



\paragraph{Acknowledgement.} This work has been partially supported by the EU project QUANTICOL, 600708, and by the DFG project FEMPA.



\bibliographystyle{eptcs}
\bibliography{fluid_equivalences}

\newpage



%\appendix
%
%
%\section{Appendix}
%
%\paragraph*{Note to the reviewers} This appendix contains technical details (e.g., proofs) which could not be included in the main paper due to lack of space. If the paper is accepted, the appendix will be made publicly available in the form of a technical report.
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_proj_label_eq}]
%It is sufficient to note that the proofs of Theorem 2,3 and 4 from~\cite{concur12} carry over in the case of $\rho = \cdot$.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%The following lemmas will be needed in the proof of Theorems \ref{thm_char_efl} and \ref{thm_char_ofl}.
%
%\begin{lemma}\label{lemma_nonzero}
%Fix a well-posed FEPA model $M$, one of its fluid atoms $P^0$, a $P' \in ds(P^0)$ and assume that $\arate(P') > 0$. Then $\crate(M,V,P') > 0$ for the population function
%\[ V_P :=
%\begin{cases}
%1 & , \ P \notin ds(P^0) \\
%1 & , \ P \in ds(P^0) \land P = P' \\
%0 & , \ P \in ds(P^0) \land P \neq P'
%\end{cases}\]
%\end{lemma}
%
%\begin{proof}
%We prove this by structural induction on $M$.
%\begin{itemize}
%\item $M = P$: Since $M$ has then only one fluid atom, i.e. $P^0 = P$, the claim follows with $\arate(P') > 0$.
%\item $M = M_1 \parallel_L M_2$: We assume w.l.o.g. that $P^0 \in \mathcal{G}(M_1)$. Since the case $\alpha \notin L$ follows directly from the I.H., we focus on the case $\alpha \in L$. Let us denote by $V_i$ the $M_i$-part of $V$. Since $\crate(M_1,V_1,P') > 0$ by I.H., we infer $\arate(M_1,V_1) > 0$. Moreover, the well-posedness of $M$ yields $\arate(M_2,V_2) > 0$. (Note that well-posedness guarantees only the existence of some $V'_2$ such that $\arate(M_2,V'_2) > 0$. However, this implies $\arate(M_2,V_2) > 0$, since all populations of $V_2$ are positive.) This yields  $\rho(\arate(M_1,V_1),\arate(M_2,V_2)) > 0$, implying $\crate(M,V,P') > 0$.
%\end{itemize}
%\end{proof}
%
%
%
%\begin{lemma}\label{lem_hat_0}
%Let $M$ be a FEPA model, $P^0 \in \mathcal{G}(M)$, $P' \in ds(P^0)$ satisfying $\arate(P') > 0$ and $V$ a population function of $M$ such that $\arate(M,V) > 0$. Then, it holds that $\arate(M,\widehat{V}) > 0$, where
%\[ \widehat{V}_{P} =
%\begin{cases}
%1 & , \  P\in ds(P^0) \land P = P' \\
%0 & , \ P\in ds(P^0) \land P \neq P' \\
%V_{P} & , \ P\notin ds(P^0)
%\end{cases}\]
%\end{lemma}
%
%\begin{proof}
%We prove the claim using structural induction.
%\begin{itemize}
%    \item $M = P^0$: the claim follows with $\arate(P') > 0$.
%    \item $M = M_1 \parallel_L M_2$: We assume without loss of generality that $P^0 \in \mathcal{G}(M_1)$. Then, it holds that $\arate(M_2,V) = \arate(M_2,\widehat{V})$.
%        \begin{itemize}
%            \item $\alpha \in L$: Note that $0 < \arate(M_1 \parallel_L M_2,V) = \rho(\arate(M_1,V),\arate(M_2,V))$ implies $\arate(M_1,V) > 0$ and $\arate(M_2,V) > 0$. As $\arate(M_1,\widehat{V}) > 0$ by I.H., it holds that $\rho(\arate(M_1,\widehat{V}),\arate(M_2,\widehat{V})) > 0$.
%            \item $\alpha \notin L$: As the claim is trivial in the case of $\arate(M_2,V) > 0$, we may assume that $\arate(M_2,V) = 0$. Then, $\arate(M_1 \parallel_L M_2, \widehat{V}) = \arate(M_1,\widehat{V})$ and the claim follows with the induction hypothesis.
%        \end{itemize}
%\end{itemize}
%\end{proof}
%
%
%
%\begin{lemma}\label{lem_hat}
%Let $M$ be a FEPA model, $P^0 \in \mathcal{G}(M)$, $P' \in ds(P^0)$ satisfying $\arate(P') > 0$ and $V$ a population function of $M$ such that $\crate(M,V,P') > 0$. Then, it holds that $\crate(M,\widehat{V},P') > 0$, where
%\[ \widehat{V}_{P} =
%\begin{cases}
%1 & , \ P\in ds(P^0) \land P = P' \\
%0 & , \ P\in ds(P^0) \land P \neq P' \\
%V_{P} & , \ P \notin ds(P^0)
%\end{cases}\]
%\end{lemma}
%
%\begin{proof}
%We prove the claim using structural induction.
%\begin{itemize}
%    \item $M = P^0$: The claim follows with $\arate(P') > 0$.
%    \item $M = M_1 \parallel_L M_2$: We assume without loss of generality that $P^0 \in \mathcal{G}(M_1)$. Then, it holds that $\arate(M_2,V) = \arate(M_2,\widehat{V})$.
%        \begin{itemize}
%            \item $\alpha \in L$: The I.H., Lemma \ref{lem_hat_0} and
%            \begin{align*}
%                0 < \crate(M,V,P') = \frac{\crate(M_1,V,P')}{\arate(M_1,V)} \rho(\arate(M_1,V),\arate(M_2,V))
%            \end{align*}
%            imply then
%            \begin{align*}
%                0 < \frac{\crate(M_1,\widehat{V},P')}{\arate(M_1,\widehat{V})} \rho(\arate(M_1,\widehat{V}),\arate(M_2,\widehat{V})) =
% \crate(M,\widehat{V},P')
%            \end{align*}
%            \item $\alpha \notin L$: It holds that $0 < \crate(M,V,P') = \crate(M_1,V,P')$ and $\crate(M,\widehat{V},P') = \crate(M_1,\widehat{V},P')$. Hence, the I.H. yields the claim.
%        \end{itemize}
%\end{itemize}
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_char_efl}]
%Let us fix a set of bijections $\sigma_l : ds(P^i_l) \rightarrow ds(P^j_l)$, $1 \leq l \leq K_i$, which establishes $\vec{P}^i \sim_{\mathcal{P}} \vec{P}^j$ and some $1 \leq k \leq K_i$. We will show that $\sigma_k$ is a semi-isomorphism between $P^i_k$ and $P^j_k$. For this, we fix some $P',P'' \in \mathit{ds}(P^i_k)$, an $\alpha \in \mathcal{A}$, define
%\[ V_P :=
%\begin{cases}
%1 & , \ P \in \mathcal{B}(M) \land P \notin ds(P^i_k) \\
%1 & , \ P = P' \\
%0 & , \ P \in ds(P^i_k) \setminus \{P'\}
%\end{cases}\]
%and assume first that $\arate(P') > 0$. Property $b)$ of Definition \ref{def_label_eq} yields then
%\[ \sum_{P \in ds(P^i_k)} p_\alpha(P, P'') \crate(M, V, P) = \sum_{P \in ds(P^i_k)} p_\alpha(\sigma_k(P), \sigma_k(P'')) \crate(M, V^\sigma, \sigma_k(P)) \]
%which implies (together with the definition of $V$)
%\[ p_\alpha(P', P'') \crate(M, V, P') = p_\alpha(\sigma_k(P'), \sigma_k(P'')) \crate(M, V^\sigma, \sigma_k(P')) . \]
%Since $a)$ implies $\crate(M, V, P') = \crate(M, V^\sigma, \sigma_k(P'))$ and Lemma \ref{lemma_nonzero} induces $\crate(M, V, P') > 0$, we infer $p_\alpha(P', P'') = p_\alpha(\sigma_k(P'), \sigma_k(P''))$. This and $d)$ show then the desired equality
%\[ \sum_{P' \xrightarrow{(\alpha, r)} P''} r = \sum_{\sigma_k(P') \xrightarrow{(\alpha, r)} \sigma_k(P'')} r . \]
%Instead, if $\arate(P') = 0$, property $d)$ implies $\arate(\sigma_k(P')) = 0$ and the above equality trivially holds.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_merge}]
%It is sufficient to note that the proof of Theorem 5 from~\cite{concur12} applies when $\rho = \cdot$.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_ode_lump}]
%It can be shown that the ODE system is Lipschitz, cf. Propositions \ref{prop_lip_min} and \ref{prop_lip_prod}. Consequently, the lumped ODE system (\ref{thm_ord_lump_general}) is Lipschitz too and the theorem of Picard-Lindel\"{o}f asserts then that (\ref{thm_ord_lump_general}) has a unique solution. Moreover, for all $1 \leq i \leq n$ and $P \in ds(P^i_1)$, it holds that
%\begin{align*}
%\dot{W}_{P} & = \sum_{1 \leq j \leq k_i} \dot{V}_{\sigma_{P^i_j}(P)} = \sum_{1 \leq j \leq k_i} \sum_{\alpha \in \mathcal{A}} \bigg( - \crate(M,V,\sigma_{P^i_j}(P)) \\
%& \qquad + \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'),\sigma_{P^i_j}(P)) \crate(M,V,\sigma_{P^i_j}(P')) \bigg) \\
%& \teq{i),ii)} \sum_{\alpha \in \mathcal{A}} \Big( \sum_{P' \in ds(P^i_1)} p_\alpha(P',P) \crate(M,V^\sigma,P') - \crate(M,V^\sigma,P) \Big) \\
%& = \sum_{\alpha \in \mathcal{A}} \Big( \sum_{P' \in ds(P^i_1)} p_\alpha(P',P) \crate(M,\overline{W},P') - \crate(M,\overline{W},P) \Big)
%\end{align*}
%and $\dsl W_{P}(0) = \sum_{1 \leq j \leq k_i} V_{\sigma_{P^i_j}(P))}(0)$. This yields the claim.
%\end{proof}
%
%
%\begin{proof}[Theorem \ref{thm_cong}]
%Let us fix an arbitrary $V$ of $M = M_1  \parallel_{L} M_2$. Then, in the case of $\alpha \in L$ it holds that
%\[
%\begin{split}
%\arate(M_1  \parallel_{L} M_2,V) & = \rho( \arate(M_1,V), \arate(M_2,V) ) \\
%& \teq{iii)} \rho( \arate(M_1,V^\sigma), \arate(M_2,V^\sigma) ) = \arate(M_1 \parallel_{L} M_2,V^\sigma),
%\end{split}
%\]
%whereas, in the case of $\alpha \notin L$
%\[
%\begin{split}
%\arate(M_1 \parallel_{L} M_2,V) & = \arate(M_1,V) + \arate(M_2,V) \\
%& \teq{iii)} \arate(M_1,V^\sigma) + \arate(M_2,V^\sigma) = \arate(M_1 \parallel_{L} M_2,V^\sigma)\;.
%\end{split}
%\]
%Let us assume without loss of generality that $P^i_1 \in \mathcal{G}(M_1)$. Then, if $\alpha \in L$, for all $P \in ds(P^i_1)$ it holds that
%\begin{align*}
%\sum_{1 \leq j \leq k_i} \crate(M,V,\sigma_{P^i_j}(P)) & = \sum_{1 \leq j \leq k_i} \frac{\crate(M_1,V,\sigma_{P^i_j}(P))}{\arate(M_1,V)} \arate(M_1 \parallel_{L} M_2 ,V) \\
%& \teq{iii)} \sum_{1 \leq j \leq k_i} \frac{\crate(M_1,V,\sigma_{P^i_j}(P))}{\arate(M_1,V^\sigma)} \arate(M_1 \parallel_{L} M_2 ,V^\sigma) \\
%& \teq{ii)} \sum_{1 \leq j \leq k_i} \frac{\crate(M_1,V^\sigma,\sigma_{P^i_j}(P))}{\arate(M_1,V^\sigma)} \arate(M_1 \parallel_{L} M_2 ,V^\sigma) \\
%& = \sum_{1 \leq j \leq k_i} \crate(M,V^\sigma,\sigma_{P^i_j}(P))
%\end{align*}
%and
%\begin{align*}
%& \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \crate(M,V,\sigma_{P^i_j}(P')) \\
%& = \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \frac{\crate(M_1,V,\sigma_{P^i_j}(P'))}{\arate(M_1,V)} \arate(M,V) \\
%& \teq{iii)} \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \frac{\crate(M_1,V,\sigma_{P^i_j}(P'))}{\arate(M_1,V^\sigma)} \arate(M,V^\sigma) \\
%& \teq{ii)} \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \frac{\crate(M_1,V^\sigma,\sigma_{P^i_j}(P'))}{\arate(M_1,V^\sigma)} \arate(M,V^\sigma) \\
%& = \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \crate(M,V^\sigma,\sigma_{P^i_j}(P')) .
%\end{align*}
%As similar calculations show that for $P^i_1\in \mathcal{G}(M_1)$ and for all $P\in ds(P^i_1)$
%\[ \sum_{1 \leq j \leq k_i} \crate(M,V,\sigma_{P^i_j}(P)) = \sum_{1 \leq j \leq k_i} \crate(M,V^\sigma,\sigma_{P^i_j}(P)) \]
%and
%\begin{multline*}
%\sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \crate(M,V,\sigma_{P^i_j}(P')) \\
%= \sum_{1 \leq j \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P)) \crate(M,V^\sigma,\sigma_{P^i_j}(P'))
%\end{multline*}
%in the case of $\alpha \notin L$, the proof is complete.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_char_ofl}]
%Let us fix arbitrary $P_0, P_1 \in ds(P^i_1)$ and $\alpha \in \mathcal{A}$. Let us first assume that $\arate(\sigma_{P^i_j}(P_0)) > 0$. Then, Lemma \ref{lemma_nonzero} asserts that the population function
%\[ W_{P} =
%\begin{cases}
%1 & , \ P\in ds(P^i_j)\land P = \sigma_{P^i_j}(P_0) \\
%0 & , \  P\in ds(P^i_j)\land P \neq \sigma_{P^i_j}(P_0) \\
%1 & , \ P\notin ds(P^i_j)
%\end{cases} \]
%satisfies $\crate(M,W,\sigma_{P^i_j}(P_0)) > 0$ and $i)$ yields
%\[ 0 < \sum_{1 \leq l \leq k_i} \crate(M,W,\sigma_{P^i_l}(P_0)) \teq{i)}
% \crate(M,W^\sigma,P_0) . \]
%Since $\arate(P_0) = \arate(\sigma_{P^i_j}(P_0)) > 0$ because of property $iii)$, Lemma \ref{lem_hat} ensures that $\crate(M,\widehat{V},P_0) > 0$ for
%\[ \widehat{V}_{P} :=
%\begin{cases}
%1 & , \ P\in ds(P^i_1) \land P = P_0 \\
%0 & , \ P\in ds(P^i_1)  \land P \neq P_0 \\
%W^\sigma_{P} & , \ P\notin ds(P^i_1)
%\end{cases} \]
%Using
%\[ V_{P} :=
%\begin{cases}
%1 & , \ P\in ds(P^i_j)  \land P = \sigma_{P^i_j}(P_0) \\
%0 & , \ P\in ds(P^i_j)  \land P \neq \sigma_{P^i_j}(P_0) \\
%0 & , \ P\in ds(P^i_1) \\
%W^\sigma_{P} & , \ P\notin ds(P^i_1)  \land P\notin ds(P^i_j)
%\end{cases} \]
%we infer
%\[ \crate(M,V,\sigma_{P^i_j}(P_0)) = \sum_{1 \leq l \leq k_i} \crate(M,V,\sigma_{P^i_l}(P_0))
%\teq{i)} \crate(M,V^\sigma,P_0) \]
%and
%\begin{align*}
%& p_\alpha(\sigma_{P^i_j}(P_0), \sigma_{P^i_j}(P_1)) \crate(M,V,\sigma_{P^i_j}(P_0)) = \\
%& \qquad = \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_j}(P'), \sigma_{P^i_j}(P_1))
% \crate(M,V,\sigma_{P^i_j}(P')) \\
%& \qquad = \sum_{1 \leq l \leq k_i} \sum_{P' \in ds(P^i_1)} p_\alpha(\sigma_{P^i_l}(P'), \sigma_{P^i_l}(P_1))
%\crate(M,V,\sigma_{P^i_l}(P')) \\
%& \qquad \teq{ii)} \sum_{P' \in ds(P^i_1)} p_\alpha(P', P_1) \crate(M,V^\sigma,P') \\
%& \qquad = p_\alpha(P_0, P_1) \crate(M,V^\sigma,P_0) .
%\end{align*}
%Hence, as the equality $\crate(M,V^\sigma,P_0) = \crate(M,\widehat{V},P_0) > 0$ implies $p_\alpha(P_0, P_1) = $ $p_\alpha(\sigma_{P^i_j}(P_0), \sigma_{P^i_j}(P_1))$, $iii)$ yields the desired equality
%\[ \sum_{P_0 \xrightarrow{(\alpha, r)} P_1} r = \sum_{\sigma_{P^i_j}(P_0) \xrightarrow{(\alpha, r)} \sigma_{P^i_j}(P_1)} r . \]
%If, instead, $\arate(\sigma_{P^i_j}(P_0)) = 0$, the above equality follows from $iii)$.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%The following lemma will be needed in the proof of Theorem \ref{thm_peturbation_0}.
%
%\begin{lemma}\label{lem_gronwall}
%Let $c_1,c_2,c_3$ be positive constants and $u$ a continuous function on $0\leq t < \infinity$ such that
%\begin{align}\label{GL}
%u(t)\leq c_3+ c_2 t + c_1\int_0^t u(s) ds
%\end{align}
%Then,
%\[
%u(t)\leq \left( \frac{c_2}{c_1}+c_3\right)
%\xe^{c_1 t}-\frac{c_2}{c_1}\;.
%\]
%\end{lemma}
%\begin{proof}
%The inequality condition \eqref{GL} can be written as
%\[
%u(t)+\frac{c_2}{c_1}\leq \left(\frac{c_2}{c_1}+c_3\right)
%+c_1\int_0^t \left(u(s)+\frac{c_2}{c_1} \right) ds .
%\]
%As this yields together with $\tilde{u}(s) := u(s)+\frac{c_2}{c_1}, \tilde{\alpha} := \frac{c_2}{c_1}+c_3$ and $\tilde{v}(s) := c_1$
%\[
%\tilde{u}(t) \leq \tilde{\alpha} + \int_0^t \tilde{u}(s) \tilde{v}(s) ds ,
%\]
%Gronwall's inequality ensures $\dsl \tilde{u}(t) \leq \tilde{\alpha} \cdot \xe^{\int_0^t \tilde{v}(s) ds}$ and the proof is complete.
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\begin{proof}[Theorem \ref{thm_peturbation_0}]
%We observe first
%\begin{align*}
%\norm{x_\zeta(t)-x_\xi(t)}&\leq \norm{\underline{x}_\zeta - \underline{x}_\xi}+\norm{\int_0^t f(\zeta,x_\zeta(s))-f(\xi,x_\xi(s))ds}\\
%&\leq \delta + \norm{\int_0^t f(\zeta,x_\zeta(s))-f(\zeta,x_\xi(s))ds}\\
%&\qquad
%+\norm{\int_0^t f(\zeta,x_\xi(s))-f(\xi,x_\xi(s))ds}\\
%&\leq \delta + L_\zeta \int_0^t \norm{x_\zeta(s)-x_\xi(s)}ds + \varepsilon K t .
%\end{align*}
%The claim follows by applying Lemma \ref{lem_gronwall} for $u(s) := \norm{x_\zeta(s)-x_\xi(s)}$, $c_3 := \delta$, $c_1 := L_\zeta$ and $c_2 := \varepsilon K$.
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%The statements below will be needed in the proof of Theorem \ref{thm_perturbation}.
%
%\begin{lemma}\label{lem_bdd}
%For a FEPA model $M$ and an initial population function $V(0)$ of $M$, it holds that $\|V(t)\|_\infinity \leq c$ for $c := \max_{P \in \mathcal{G}(M)} \sum_{P' \in ds(P)} V_{P'}(0)$.
%\end{lemma}
%
%\begin{proof}
%Using structural induction one shows:
%\begin{enumerate}[$i)$]
%    \item For all $P \in \mathcal{B}(M)$ there exists a nonnegative function $\zeta$ such that $\dot{V}_{P} \geq - V_{P}\, \zeta(V)$.
%    \item For all $P \in \mathcal{G}(M)$ it holds that $\sum_{P' \in ds(P)} \dot{V}_{P'} = 0$.
%\end{enumerate}
%Since $i)$ ensures that $V_{P} \geq 0$ for all $P \in \mathcal{B}(M)$ and $ii)$ yields the conservation of mass, the proof is complete.
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\begin{proposition}\label{prop_lip_min}
%Let us assume that $\rho = \min$. For a FEPA model $M \equiv M(\xi)$ and
%\[ F_P(\xi,V) := \sum_{\alpha \in \mathcal{A}} \bigg( \sum_{P' \in \mathcal{B}(M)} p_\alpha(P', P) \crate(M, V, P') - \crate(M, V, P) \bigg) , \]
%$F(\xi,\cdot)$ and $F(\cdot,V)$ are globally Lipschitz. Moreover, given that $K_V$ denotes the Lipschitz constant of $F(\cdot,V)$ for some fixed $V$, it holds that
%$ \sup_{\| V \| \leq c} K_V < \infinity$ for all $c > 0$.
%\end{proposition}
%
%\begin{proof}
%The first claim follows from Lemma B.1 in \cite{HaydenLipschitz}. We show the second claim by exchanging the roles of parameters and variables in the proof of Lemma B.1. Specifically, one shows first that for each $Q \in \mathcal{B}(M)$ and $\alpha \in \mathcal{A}$ the set $\mathbb{R}_{>0}^{|\nu(M)|}$ can be covered by closed convex subsets $A_1, \ldots, A_I$ such that for a given $1 \leq \iota \leq I$, it holds that
%\[ \crate(M,V,Q) = V_Q \arate(Q)(\xi) \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)} \]
%for all $\xi \in A_\iota$, where $D \geq 1$ and $a_1,b_1, \ldots, a_n,b_n$ are apparent rates which satisfy
%\[ a_n(\xi,V) \geq b_n(\xi,V) \quad \land \quad a_n(\xi,V) \geq V_Q \arate(Q)(\xi) \prod_{m=1}^{n-1} \frac{b_m(\xi,V)}{a_m(\xi,V)} \]
%for all $1 \leq n \leq D$. As $\arate(Q)(\xi) = \sum_{k \in S} \xi_k$ for some set $S \subseteq \{1, \ldots, |\nu(M)|\}$, the first equation rewrites to
%\[ \crate(M,V,Q) = V_Q \sum_{k \in S} \xi_k \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)} . \]
%From the definition of $F_P(\cdot,V)$ it becomes evident that we have to consider also $p_\alpha(Q,P)(\xi) \crate(M,V,Q)$, where $Q \in \mathcal{B}(M)$. Noting that
%\[ \arate(Q)(\xi) \geq p_\alpha(Q,P)(\xi) \arate(Q)(\xi) = \sum_{Q \act{(\alpha,\xi_i)} P} \xi_i = \sum_{k \in S'} \xi_k \]
%for some set $S' \subseteq \{1, \ldots, |\nu(M)|\}$, we infer
%\[ p_\alpha(Q,P)(\xi) \crate(M,V,Q) = V_Q \sum_{k \in S'} \xi_k \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)} \]
%and
%\[ a_n(\xi,V) \geq V_Q \arate(Q)(\xi) \prod_{m=1}^{n-1} \frac{b_m(\xi,V)}{a_m(\xi,V)} \geq V_Q \sum_{k \in S'} \xi_k \prod_{m = 1}^{n-1} \frac{b_m(\xi,V)}{a_m(\xi,V)} \]
%for all $1 \leq n \leq D$, meaning that both $p_\alpha(Q,P)(\xi) \crate(M,V,Q)$ and $\crate(M,V,Q)$ belong to the same class of functions.
%
%Clearly, the function $f(\xi,V) := V_Q \sum_{k \in S} \xi_k \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)}$ is continuous in all $\xi \in A_\iota$. Moreover, each $\text{int}(A_\iota)$ may be assumed to be nonempty: if $\xi \in A_i$ and $\text{int}(A_i) = \emptyset$, there exists a sequence $(\xi^n)_n$ in $\mathbb{R}_{>0}^{|\nu(M)|} \setminus A_i \subseteq \cup_{\iota \neq i} A_\iota$ which converges to $\xi$. As $\cup_{\iota \neq i} A_\iota$ is closed, it follows that $\xi \in \cup_{\iota \neq i} A_\iota$, meaning that $\cup_{\iota \neq i} A_\iota$ covers already $\mathbb{R}_{>0}^{|\nu(M)|}$. Thus, $A_\iota$ can be disregarded.
%
%Now, for each partial derivative $\partial_{\xi_j} f(\cdot,V)$ we give a bound on $\text{int}(A_\iota)$ which depends only on $M$ and $V$. For this, let us fix some $\xi \in \text{int}(A_\iota)$. Then it holds that
%\[ \partial_{\xi_j} f(\xi,V) =
%\begin{cases}
%\dsl V_Q c_j \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)} + \Big( V_Q \sum_{k \in S} \xi_k \Big) \partial_{\xi_j} \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)} & , \ j \in S \\
%\dsl \Big( V_Q \sum_{k \in S} \xi_k \Big) \partial_{\xi_j} \prod_{n = 1}^D \frac{b_n(\xi,V)}{a_n(\xi,V)} & , \ j \notin S
%\end{cases}\]
%Using $F[n](\xi,V) := \prod_{m=1}^n \frac{b_m(\xi,V)}{a_m(\xi,V)}$ for $0 \leq n \leq D$ we infer
%\[ \partial_{\xi_j} F[n] = \frac{F[n-1]}{a_n} \partial_{\xi_j} b_n - \frac{F[n]}{a_n} \partial_{\xi_j} a_n + \frac{b_n}{a_n} \partial_{\xi_j} F[n-1] . \]
%Thus, the above inequalities yield
%\[
%\Big| \partial_{\xi_j} F[n](\xi,V) \Big( V_Q \sum_{k \in S} \xi_k \Big) \Big| \leq  | \partial_{\xi_j} b_n(\xi,V) | + | \partial_{\xi_j} a_n(\xi,V) | \\
%+ \Big| \partial_{\xi_j} F[n-1](\xi,V) \Big( V_Q \sum_{k \in S} \xi_k \Big) \Big|
%\]
%and we conclude
%\[ | \partial_{\xi_j} f(\xi,V) | \leq \sum_{n = 1}^D | \partial_{\xi_j} b_n(\xi,V) | + \sum_{n = 1}^D | \partial_{\xi_j} a_n(\xi,V) | \leq C \| V \|_\infinity \]
%for some $C > 0$ which depends on $j$ and $Q \in \mathcal{B}(M)$. One proceeds then similarly to the proof of Lemma B.1 from \cite{HaydenLipschitz}.
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\begin{proposition}\label{prop_lip_prod}
%Let us assume that $\rho = \cdot$. Then, for a FEPA model $M$ and
%\[
%F_P(\xi,V) := \sum_{\alpha \in \mathcal{A}} \bigg( \sum_{P' \in \mathcal{B}(M)} p_\alpha(P', P) \crate(M, V, P') - \crate(M, V, P) \bigg) ,
%\]
%it holds that $F(\cdot,\cdot)$ is locally Lipschitz.
%\end{proposition}
%
%\begin{proof}
%As in Proposition \ref{prop_lip_min}, one shows using structural induction that there exists a $\mathcal{D}(M) \geq 1$ such that for each $Q \in \mathcal{B}(M)$ and $\alpha \in \mathcal{A}$ it holds that
%\[ \crate(M(\xi),V,Q) = V_Q \arate(Q)(\xi) \prod_{n = 1}^D a_n(\xi,V) \]
%for some $1 \leq D \leq \mathcal{D}(M)$ and apparent rate functions $a_1, \ldots, a_n$. Since $\arate(Q)(\xi)$ $ = \sum_{k \in S} \xi_k$ for some set $S \subseteq \{1, \ldots, |\nu(M)|\}$ and
%\[ p_\alpha(Q,P)(\xi) \arate(Q)(\xi) = \sum_{Q \act{(\alpha,\xi_i)} P} \xi_i = \sum_{k \in S'} \xi_k \]
%for some set $S' \subseteq \{1, \ldots, |\nu(M)|\}$, this yields
%\[ \crate(M(\xi),V,Q) = V_Q \sum_{k \in S} \xi_k \prod_{n = 1}^D a_n(\xi,V) \]
%and
%\[ p_\alpha(Q,P)(\xi) \crate(M(\xi),V,Q) = V_Q \sum_{k \in S'} \xi_k \prod_{n = 1}^D a_n(\xi,V) . \]
%Obviously, $F(\cdot,\cdot)$ is partially continuously differentiable and the proof is complete.
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\begin{proof}[Theorem \ref{thm_perturbation}]
%The claim follows with Theorem \ref{thm_peturbation_0}, Lemma \ref{lem_bdd} and Propositions \ref{prop_lip_min} and \ref{prop_lip_prod}.
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_eps_cong}]
%Theorem \ref{thm_proj_label_eq} shows the case of exact fluid lumpability, whereas Theorem \ref{thm_cong} yields the case of ordinary fluid lumpability.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\begin{proof}[Theorem \ref{thm_eps_char}]
%As $\{\mathcal{P}^1, \ldots, \mathcal{P}^n\}$ is an OFL/EFL partition of $\mathcal{G}(M(\zeta))$, Theorem \ref{thm_char_efl} and \ref{thm_char_ofl} assure that the fluid atoms $P^i_j, P^i_{j'}$ in $M(\zeta)$ are semi-isomorphic. Moreover, there exists a $C' > 0$ such that $\norm{\cdot}_\infinity \leq C'
%\norm{\cdot}$. For any $\alpha \in \mathcal{A}$ and $P,P' \in \mathcal{B}(M)$, let $N_{(P,\alpha,P')}$ denote the number of $\alpha$-transitions from $P$ to $P'$ and $N := \max \{ N_{(P,\alpha,P')} \mid P,P' \in \mathcal{B}(M) \land \alpha \in \mathcal{A} \}$. The constant $C := N C'$ yields then the claim.
%\end{proof}



\end{document}


qapl14.bbl
\begin{thebibliography}{10}
\providecommand{\bibitemdeclare}[2]{}
\providecommand{\surnamestart}{}
\providecommand{\surnameend}{}
\providecommand{\urlprefix}{Available at }
\providecommand{\url}[1]{\texttt{#1}}
\providecommand{\href}[2]{\texttt{#2}}
\providecommand{\urlalt}[2]{\href{#1}{#2}}
\providecommand{\doi}[1]{doi:\urlalt{http://dx.doi.org/#1}{#1}}
\providecommand{\bibinfo}[2]{#2}

\bibitemdeclare{incollection}{worrell01:approximate}
\bibitem{worrell01:approximate}
\bibinfo{author}{Franck \surnamestart van Breugel\surnameend} \&
  \bibinfo{author}{James \surnamestart Worrell\surnameend}
  (\bibinfo{year}{2001}): \emph{\bibinfo{title}{Towards Quantitative
  Verification of Probabilistic Transition Systems}}.
\newblock In: {\sl \bibinfo{booktitle}{ICALP}}, {\sl \bibinfo{series}{LNCS}}
  \bibinfo{volume}{2076}, \bibinfo{publisher}{Springer}, pp.
  \bibinfo{pages}{421--432}, \doi{10.1007/3-540-48224-5\_35}.

\bibitemdeclare{article}{BuchholzOrdinaryExact}
\bibitem{BuchholzOrdinaryExact}
\bibinfo{author}{Peter \surnamestart Buchholz\surnameend}
  (\bibinfo{year}{1994}): \emph{\bibinfo{title}{{Exact and Ordinary Lumpability
  in Finite Markov Chains}}}.
\newblock {\sl \bibinfo{journal}{Journal of Applied Probability}}
  \bibinfo{volume}{31}(\bibinfo{number}{1}), pp. \bibinfo{pages}{59--75},
  \doi{10.2307/3215235}.

\bibitemdeclare{inproceedings}{bucholz94papm}
\bibitem{bucholz94papm}
\bibinfo{author}{Peter \surnamestart Buchholz\surnameend}
  (\bibinfo{year}{1994}): \emph{\bibinfo{title}{{Markovian Process Algebra:
  Composition and Equivalence}}}.
\newblock In: {\sl \bibinfo{booktitle}{Proc. 2nd {PAPM} Workshop}},
  \bibinfo{address}{Erlangen, Germany}.

\bibitemdeclare{article}{Cardelli_TCS_08}
\bibitem{Cardelli_TCS_08}
\bibinfo{author}{Luca \surnamestart Cardelli\surnameend}
  (\bibinfo{year}{2008}): \emph{\bibinfo{title}{On process rate semantics}}.
\newblock {\sl \bibinfo{journal}{Theor. Comput. Sci.}} \bibinfo{volume}{391},
  pp. \bibinfo{pages}{190--215}, \doi{10.1016/j.tcs.2007.11.012}.

\bibitemdeclare{article}{Ciocchetta_Hillston_09}
\bibitem{Ciocchetta_Hillston_09}
\bibinfo{author}{Federica \surnamestart Ciocchetta\surnameend} \&
  \bibinfo{author}{Jane \surnamestart Hillston\surnameend}
  (\bibinfo{year}{2009}): \emph{\bibinfo{title}{{Bio-PEPA}: A framework for the
  modelling and analysis of biological systems}}.
\newblock {\sl \bibinfo{journal}{Theor. Comput. Sci.}}
  \bibinfo{volume}{410}(\bibinfo{number}{33--34}), pp.
  \bibinfo{pages}{3065--3084}, \doi{10.1016/j.tcs.2009.02.037}.

\bibitemdeclare{inproceedings}{DBLP:conf/lics/DanosFFHK10}
\bibitem{DBLP:conf/lics/DanosFFHK10}
\bibinfo{author}{Vincent \surnamestart Danos\surnameend},
  \bibinfo{author}{Jerome \surnamestart Feret\surnameend},
  \bibinfo{author}{Walter \surnamestart Fontana\surnameend},
  \bibinfo{author}{Russell \surnamestart Harmer\surnameend} \&
  \bibinfo{author}{Jean \surnamestart Krivine\surnameend}
  (\bibinfo{year}{2010}): \emph{\bibinfo{title}{Abstracting the Differential
  Semantics of Rule-Based Models: Exact and Automated Model Reduction}}.
\newblock In: {\sl \bibinfo{booktitle}{LICS}}, pp. \bibinfo{pages}{362--381}.
\newblock
  \urlprefix\url{http://doi.ieeecomputersociety.org/10.1109/LICS.2010.44}.

\bibitemdeclare{article}{Danos200469}
\bibitem{Danos200469}
\bibinfo{author}{Vincent \surnamestart Danos\surnameend} \&
  \bibinfo{author}{Cosimo \surnamestart Laneve\surnameend}
  (\bibinfo{year}{2004}): \emph{\bibinfo{title}{Formal molecular biology}}.
\newblock {\sl \bibinfo{journal}{Theoretical Computer Science}}
  \bibinfo{volume}{325}(\bibinfo{number}{1}), pp. \bibinfo{pages}{69--110},
  \doi{10.1016/j.tcs.2004.03.065}.

\bibitemdeclare{inproceedings}{1029849}
\bibitem{1029849}
\bibinfo{author}{Josee \surnamestart Desharnais\surnameend},
  \bibinfo{author}{Radha \surnamestart Jagadeesan\surnameend},
  \bibinfo{author}{Vineet \surnamestart Gupta\surnameend} \&
  \bibinfo{author}{Prakash \surnamestart Panangaden\surnameend}
  (\bibinfo{year}{2002}): \emph{\bibinfo{title}{The metric analogue of weak
  bisimulation for probabilistic processes}}.
\newblock In: {\sl \bibinfo{booktitle}{LICS}}, pp. \bibinfo{pages}{413--422},
  \doi{10.1109/LICS.2002.1029849}.

\bibitemdeclare{inproceedings}{DBLP:conf/concur/PierroHW03}
\bibitem{DBLP:conf/concur/PierroHW03}
\bibinfo{author}{Alessandra \surnamestart {Di Pierro}\surnameend},
  \bibinfo{author}{Chris \surnamestart Hankin\surnameend} \&
  \bibinfo{author}{Herbert \surnamestart Wiklicky\surnameend}
  (\bibinfo{year}{2003}): \emph{\bibinfo{title}{Quantitative Relations and
  Approximate Process Equivalences}}.
\newblock In: {\sl \bibinfo{booktitle}{CONCUR}}, pp. \bibinfo{pages}{498--512}.
\newblock \urlprefix\url{http://dx.doi.org/10.1007/978-3-540-45187-7_33}.

\bibitemdeclare{incollection}{Faeder:2009aa}
\bibitem{Faeder:2009aa}
\bibinfo{author}{James~R. \surnamestart Faeder\surnameend},
  \bibinfo{author}{Michael~L. \surnamestart Blinov\surnameend} \&
  \bibinfo{author}{William~S. \surnamestart Hlavacek\surnameend}
  (\bibinfo{year}{2009}): \emph{\bibinfo{title}{Rule-Based Modeling of
  Biochemical Systems with {BioNetGen}}}.
\newblock {\sl \bibinfo{series}{Methods Mol. Biol.}} \bibinfo{volume}{500}, pp.
  \bibinfo{pages}{113--167}, \doi{10.1007/978-1-59745-525-1\_5}.

\bibitemdeclare{article}{canonical}
\bibitem{canonical}
\bibinfo{author}{S.~\surnamestart Gilmore\surnameend},
  \bibinfo{author}{J.~\surnamestart Hillston\surnameend} \&
  \bibinfo{author}{M.~\surnamestart Ribaudo\surnameend} (\bibinfo{year}{2001}):
  \emph{\bibinfo{title}{An efficient algorithm for aggregating {PEPA} models}}.
\newblock {\sl \bibinfo{journal}{IEEE Transactions on Software Engineering}}
  \bibinfo{volume}{27}(\bibinfo{number}{5}), pp. \bibinfo{pages}{449--464},
  \doi{10.1109/32.922715}.

\bibitemdeclare{article}{HaydenLipschitz}
\bibitem{HaydenLipschitz}
\bibinfo{author}{Richard~A. \surnamestart Hayden\surnameend},
  \bibinfo{author}{Anton \surnamestart Stefanek\surnameend} \&
  \bibinfo{author}{Jeremy~T. \surnamestart Bradley\surnameend}
  (\bibinfo{year}{2012}): \emph{\bibinfo{title}{Fluid computation of
  passage-time distributions in large {Markov} models}}.
\newblock {\sl \bibinfo{journal}{Theor. Comput. Sci.}}
  \bibinfo{volume}{413}(\bibinfo{number}{1}), pp. \bibinfo{pages}{106--141},
  \doi{10.1016/j.tcs.2011.07.017}.

\bibitemdeclare{inproceedings}{DBLP:conf/esm/HermannsR98}
\bibitem{DBLP:conf/esm/HermannsR98}
\bibinfo{author}{Holger \surnamestart Hermanns\surnameend} \&
  \bibinfo{author}{Marina \surnamestart Ribaudo\surnameend}
  (\bibinfo{year}{1998}): \emph{\bibinfo{title}{Exploiting Symmetries in
  Stochastic Process Algebras}}.
\newblock In: {\sl \bibinfo{booktitle}{European Simulation Multiconference}},
  \bibinfo{address}{Manchester, UK}, pp. \bibinfo{pages}{763--770}.

\bibitemdeclare{inproceedings}{Hillston_QEST05}
\bibitem{Hillston_QEST05}
\bibinfo{author}{J.~\surnamestart Hillston\surnameend} (\bibinfo{year}{2005}):
  \emph{\bibinfo{title}{{Fluid flow approximation of PEPA models}}}.
\newblock In: {\sl \bibinfo{booktitle}{Proceedings of Quantitative Evaluation
  of Systems}}, \bibinfo{publisher}{IEEE Computer Society Press}, pp.
  \bibinfo{pages}{33--43}, \doi{10.1109/QEST.2005.12}.

\bibitemdeclare{book}{pepa}
\bibitem{pepa}
\bibinfo{author}{Jane \surnamestart Hillston\surnameend}
  (\bibinfo{year}{1996}): \emph{\bibinfo{title}{A compositional approach to
  performance modelling}}.
\newblock \bibinfo{publisher}{Cambridge University Press},
  \bibinfo{address}{New York, NY, USA}, \doi{10.1017/CBO9780511569951}.

\bibitemdeclare{inproceedings}{dsn2013}
\bibitem{dsn2013}
\bibinfo{author}{Giulio \surnamestart Iacobelli\surnameend} \&
  \bibinfo{author}{Mirco \surnamestart Tribastone\surnameend}
  (\bibinfo{year}{2013}): \emph{\bibinfo{title}{Lumpability of fluid models
  with heterogeneous agent types}}.
\newblock In: {\sl \bibinfo{booktitle}{DSN}}, pp. \bibinfo{pages}{1--11},
  \doi{10.1109/DSN.2013.6575346}.

\bibitemdeclare{incollection}{springerlink:10.1007/BFb0039071}
\bibitem{springerlink:10.1007/BFb0039071}
\bibinfo{author}{Chi-Chang \surnamestart Jou\surnameend} \&
  \bibinfo{author}{Scott \surnamestart Smolka\surnameend}
  (\bibinfo{year}{1990}): \emph{\bibinfo{title}{Equivalences, congruences, and
  complete axiomatizations for probabilistic processes}}.
\newblock In: {\sl \bibinfo{booktitle}{CONCUR}}, {\sl \bibinfo{series}{LNCS}}
  \bibinfo{volume}{458}, pp. \bibinfo{pages}{367--383},
  \doi{10.1007/BFb0039071}.

\bibitemdeclare{article}{10.3389/fgene.2012.00131}
\bibitem{10.3389/fgene.2012.00131}
\bibinfo{author}{Ovidiu \surnamestart Radulescu\surnameend},
  \bibinfo{author}{Alexander~N. \surnamestart Gorban\surnameend},
  \bibinfo{author}{Andrei \surnamestart Zinovyev\surnameend} \&
  \bibinfo{author}{Vincent \surnamestart Noel\surnameend}
  (\bibinfo{year}{2012}): \emph{\bibinfo{title}{Reduction of dynamical
  biochemical reactions networks in computational biology}}.
\newblock {\sl \bibinfo{journal}{Frontiers in Genetics}}
  \bibinfo{volume}{3}(\bibinfo{number}{131}), \doi{10.3389/fgene.2012.00131}.

\bibitemdeclare{article}{DBLP:journals/tse/TribastoneGH12}
\bibitem{DBLP:journals/tse/TribastoneGH12}
\bibinfo{author}{Mirco \surnamestart Tribastone\surnameend},
  \bibinfo{author}{Stephen \surnamestart Gilmore\surnameend} \&
  \bibinfo{author}{Jane \surnamestart Hillston\surnameend}
  (\bibinfo{year}{2012}): \emph{\bibinfo{title}{Scalable Differential Analysis
  of Process Algebra Models}}.
\newblock {\sl \bibinfo{journal}{IEEE Transactions on Software Engineering}}
  \bibinfo{volume}{38}(\bibinfo{number}{1}), pp. \bibinfo{pages}{205--219}.
\newblock
  \urlprefix\url{http://doi.ieeecomputersociety.org/10.1109/TSE.2010.82}.

\bibitemdeclare{inproceedings}{concur12}
\bibitem{concur12}
\bibinfo{author}{Max \surnamestart Tschaikowski\surnameend} \&
  \bibinfo{author}{Mirco \surnamestart Tribastone\surnameend}
  (\bibinfo{year}{2012}): \emph{\bibinfo{title}{{Exact fluid lumpability for
  Markovian process algebra}}}.
\newblock In: {\sl \bibinfo{booktitle}{CONCUR}}, {\sl \bibinfo{series}{LNCS}}
  \bibinfo{volume}{7545}, pp. \bibinfo{pages}{380--394},
  \doi{10.1007/978-3-642-32940-1\_27}.
\newblock
  \urlprefix\url{http://www.pst.ifi.lmu.de/Personen/team/tschaikowski/concur12%
_techreport.pdf}.

\bibitemdeclare{article}{efl_tcs}
\bibitem{efl_tcs}
\bibinfo{author}{Max \surnamestart Tschaikowski\surnameend} \&
  \bibinfo{author}{Mirco \surnamestart Tribastone\surnameend}
  (\bibinfo{year}{2013}): \emph{\bibinfo{title}{Exact fluid lumpability in
  Markovian process algebra}}.
\newblock {\sl \bibinfo{journal}{Theoretical Computer Science}},
  \doi{10.1016/j.tcs.2013.07.029}.
\newblock
  \urlprefix\url{http://www.sciencedirect.com/science/article/pii/S03043975130%
05598}.

\bibitemdeclare{article}{Tschaikowski2013a}
\bibitem{Tschaikowski2013a}
\bibinfo{author}{Max \surnamestart Tschaikowski\surnameend} \&
  \bibinfo{author}{Mirco \surnamestart Tribastone\surnameend}
  (\bibinfo{year}{2013}): \emph{\bibinfo{title}{Tackling continuous state-space
  explosion in a {Markovian} process algebra}}.
\newblock {\sl \bibinfo{journal}{Theoretical Computer Science}},
  \doi{10.1016/j.tcs.2013.08.016}.
\newblock
  \urlprefix\url{http://www.sciencedirect.com/science/article/pii/S03043975130%
06403}.

\bibitemdeclare{article}{Wang22052009}
\bibitem{Wang22052009}
\bibinfo{author}{Pu~\surnamestart Wang\surnameend}, \bibinfo{author}{Marta~C.
  \surnamestart Gonz{\'a}lez\surnameend}, \bibinfo{author}{C{\'e}sar~A.
  \surnamestart Hidalgo\surnameend} \& \bibinfo{author}{Albert-L{\'a}szl{\'o}
  \surnamestart Barab{\'a}si\surnameend} (\bibinfo{year}{2009}):
  \emph{\bibinfo{title}{Understanding the Spreading Patterns of Mobile Phone
  Viruses}}.
\newblock {\sl \bibinfo{journal}{Science}}
  \bibinfo{volume}{324}(\bibinfo{number}{5930}), pp.
  \bibinfo{pages}{1071--1076}, \doi{10.1126/science.1167053}.

\bibitemdeclare{article}{Watson1972}
\bibitem{Watson1972}
\bibinfo{author}{Raymond~Keith \surnamestart Watson\surnameend}
  (\bibinfo{year}{1972}): \emph{\bibinfo{title}{On an Epidemic in a Stratified
  Population}}.
\newblock {\sl \bibinfo{journal}{Journal of Applied Probability}}
  \bibinfo{volume}{9}(\bibinfo{number}{3}), pp. \bibinfo{pages}{659--666},
  \doi{10.2307/3212334}.

\end{thebibliography}
