15 research outputs found
SDDs are Exponentially More Succinct than OBDDs
Introduced by Darwiche (2011), sentential decision diagrams (SDDs) are
essentially as tractable as ordered binary decision diagrams (OBDDs), but tend
to be more succinct in practice. This makes SDDs a prominent representation
language, with many applications in artificial intelligence and knowledge
compilation. We prove that SDDs are more succinct than OBDDs also in theory, by
constructing a family of boolean functions where each member has polynomial SDD
size but exponential OBDD size. This exponential separation improves a
quasipolynomial separation recently established by Razgon (2013), and settles
an open problem in knowledge compilation
On the Role of Canonicity in Bottom-up Knowledge Compilation
We consider the problem of bottom-up compilation of knowledge bases, which is
usually predicated on the existence of a polytime function for combining
compilations using Boolean operators (usually called an Apply function). While
such a polytime Apply function is known to exist for certain languages (e.g.,
OBDDs) and not exist for others (e.g., DNNF), its existence for certain
languages remains unknown. Among the latter is the recently introduced language
of Sentential Decision Diagrams (SDDs), for which a polytime Apply function
exists for unreduced SDDs, but remains unknown for reduced ones (i.e. canonical
SDDs). We resolve this open question in this paper and consider some of its
theoretical and practical implications. Some of the findings we report question
the common wisdom on the relationship between bottom-up compilation, language
canonicity and the complexity of the Apply function
“What if?” in Probabilistic Logic Programming
A ProbLog program is a logic program with facts that only hold with a specified probability. In this contribution, we extend this ProbLog language by the ability to answer “What if” queries. Intuitively, a ProbLog program defines a distribution by solving a system of equations in terms of mutually independent predefined Boolean random variables. In the theory of causality, Judea Pearl proposes a counterfactual reasoning for such systems of equations. Based on Pearl’s calculus, we provide a procedure for processing these counterfactual queries on ProbLog programs, together with a proof of correctness and a full implementation. Using the latter, we provide insights into the influence of different parameters on the scalability of inference. Finally, we also show that our approach is consistent with CP-logic, that is with the causal semantics for logic programs with annotated with disjunctions
Distribution-Aware Sampling and Weighted Model Counting for SAT
Given a CNF formula and a weight for each assignment of values to variables,
two natural problems are weighted model counting and distribution-aware
sampling of satisfying assignments. Both problems have a wide variety of
important applications. Due to the inherent complexity of the exact versions of
the problems, interest has focused on solving them approximately. Prior work in
this area scaled only to small problems in practice, or failed to provide
strong theoretical guarantees, or employed a computationally-expensive maximum
a posteriori probability (MAP) oracle that assumes prior knowledge of a
factored representation of the weight distribution. We present a novel approach
that works with a black-box oracle for weights of assignments and requires only
an {\NP}-oracle (in practice, a SAT-solver) to solve both the counting and
sampling problems. Our approach works under mild assumptions on the
distribution of weights of satisfying assignments, provides strong theoretical
guarantees, and scales to problems involving several thousand variables. We
also show that the assumptions can be significantly relaxed while improving
computational efficiency if a factored representation of the weights is known.Comment: This is a full version of AAAI 2014 pape
Efficient Computation of Shap Explanation Scores for Neural Network Classifiers via Knowledge Compilation
The use of Shap scores has become widespread in Explainable AI. However,
their computation is in general intractable, in particular when done with a
black-box classifier, such as neural network. Recent research has unveiled
classes of open-box Boolean Circuit classifiers for which Shap can be computed
efficiently. We show how to transform binary neural networks into those
circuits for efficient Shap computation. We use logic-based knowledge
compilation techniques. The performance gain is huge, as we show in the light
of our experiments.Comment: Conference submission. It replaces the previously uploaded paper
"Opening Up the Neural Network Classifier for Shap Score Computation", by the
same authors. This version considerably revised the previous on
"What if?" in Probabilistic Logic Programming
A ProbLog program is a logic program with facts that only hold with a
specified probability. In this contribution we extend this ProbLog language by
the ability to answer "What if" queries. Intuitively, a ProbLog program defines
a distribution by solving a system of equations in terms of mutually
independent predefined Boolean random variables. In the theory of causality,
Judea Pearl proposes a counterfactual reasoning for such systems of equations.
Based on Pearl's calculus, we provide a procedure for processing these
counterfactual queries on ProbLog programs, together with a proof of
correctness and a full implementation. Using the latter, we provide insights
into the influence of different parameters on the scalability of inference.
Finally, we also show that our approach is consistent with CP-logic, i.e. with
the causal semantics for logic programs with annotated with disjunctions