  Statistical Relational Models and, more recently, Probabilistic Programming,
have been making strides towards an integration of logic and probabilistic
reasoning. A natural expectation for this project is that a probabilistic logic
reasoning algorithm reduces to a logic reasoning algorithm when provided a
model that only involves 0-1 probabilities, exhibiting all the advantages of
logic reasoning such as short-circuiting, intelligibility, and the ability to
provide proof trees for a query answer. In fact, we can take this further and
require that these characteristics be present even for probabilistic models
with probabilities \emph{near} 0 and 1, with graceful degradation as the model
becomes more uncertain. We also seek inference that has amortized constant time
complexity on a model's size (even if still exponential in the induced width of
a more directly relevant portion of it) so that it can be applied to huge
knowledge bases of which only a relatively small portion is relevant to typical
queries. We believe that, among the probabilistic reasoning algorithms, Belief
Propagation is the most similar to logic reasoning: messages are propagated
among neighboring variables, and the paths of message-passing are similar to
proof trees. However, Belief Propagation is either only applicable to tree
models, or approximate (and without guarantees) for precision and convergence.
In this paper we present work in progress on an Anytime Exact Belief
Propagation algorithm that is very similar to Belief Propagation but is exact
even for graphical models with cycles, while exhibiting soft short-circuiting,
amortized constant time complexity in the model size, and which can provide
probabilistic proof trees.
