Solving #SAT and Bayesian Inference with Backtracking Search

Abstract

Inference in Bayes Nets (BAYES) is an important problem with numerous applications in probabilistic reasoning. Counting the number of satisfying assignments of a propositional formula (#SAT) is a closely related problem of fundamental theoretical importance. Both these problems, and others, are members of the class of sum-of-products (SUMPROD) problems. In this paper we show that standard backtracking search when augmented with a simple memoization scheme (caching) can solve any sum-of-products problem with time complexity that is at least as good any other state-of-the-art exact algorithm, and that it can also achieve the best known time-space tradeoff. Furthermore, backtracking's ability to utilize more flexible variable orderings allows us to prove that it can achieve an exponential speedup over other standard algorithms for SUMPROD on some instances. The ideas presented here have been utilized in a number of solvers that have been applied to various types of sum-of-product problems. These system's have exploited the fact that backtracking can naturally exploit more of the problem's structure to achieve improved performance on a range of probleminstances. Empirical evidence of this performance gain has appeared in published works describing these solvers, and we provide references to these works

Anytime AND/OR Depth-First Search for Combinatorial Optimization

“…As such our results are also relevant for related schemes like recursive conditioning (Darwiche 2001) and value elimination (Bacchus, Dalmao, and Pitassi 2003) in the area of probabilistic reasoning, or BTD (Backtracking Tree Decomposition (Jégou and Terrioux 2004)) in constraint optimization. We point out, however, that the presented concepts carry over to combinatorial AND/OR search spaces in general.…”

Algorithms and complexity results for #SAT and Bayesian inference

“…Besides these theoretical results, there are also good reasons to believe that our DPLL based algorithms have the potential to perform much better than their worst case guarantees on problems that arise from real domains. In other work, [2], we have investigated in more depth the practical application of the ideas presented here to the problem of BAYES, with very promising results.…”

“…Variable Elimination: The most fundamental algorithm for BAYES is variable or bucket elimination (VE) [11]. 2 Given an instance (V, F ) of SUMPROD, we define its underlying hypergraph H. The vertices of H are the variables V, and its hyperedges are the domain sets E i of the functions f i . Variable elimination begins by choosing an elimination ordering, π for the variables V = {X 1 , .…”

A model counter for constraints over unbounded strings

“…Therefore, the relation between the old and new password can be expressed as a set of constraints, and the attacker can use it to compute the size of solutions for new p. For example, consider the code snippet in Figure 1. Under the input old p = ab@123 and a new valid password, Lines 22,24,13,4,6,8,14,15,4,6,8,and 28 are executed. This path imposes the following constraints on new p:…”

“…This problem arises in many fields of computer science including artificial intelligence, program optimizations and information flow analysis [30,39]. For example, probabilistic inference problems in Bayesian networks can be solved by first representing the network as a set of propositional clauses, and then model counting the clause set to compute all the marginal probabilities [14,17,39]. Similarly, model counting has applications to various program transformation and optimization problems such as memory size minimization [47], worst case execution time estimation [33], increasing parallelism [47], and improving cache effectiveness [19].…”