Hashing-based model counting has emerged as a promising approach for large-scale probabilistic inference on graphical models. A key component of these techniques is the use of xor-based 2-universal hash functions that operate over Boolean domains. Many counting problems arising in probabilistic inference are, however, naturally encoded over finite discrete domains. Techniques based on bit-level (or Boolean) hash functions require these problems to be propositionalized, making it impossible to leverage the remarkable progress made in SMT (Satisfiability Modulo Theory) solvers that can reason directly over words (or bit-vectors). In this work, we present the first approximate model counter that uses word-level hashing functions, and can directly leverage the power of sophisticated SMT solvers. Empirical evaluation over an extensive suite of benchmarks demonstrates the promise of the approach.
Abstract Bayesian belief networks provide a natural, efficient method for representing probabilistic dependencies among a set of variables. For these reasons, numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence. Algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks. More general classes of belief networks, however, have eluded efforts to develop efficient inference algorithms. We show that probabilistic inference using belief networks is NP-hard. Therefore, it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks. This result suggests that research should be directed away from the search for a general, efficient probabilistic inference algorithm, and toward the design of efficient special-case, average-case, and approximation algorithms. The class of @math -complete problems is a class of computationally eqivalent counting problems (defined by the author in a previous paper) that are at least as difficult as the @math -complete problems. Here we show, for a large number of natural counting problems for which there was no previous indication of intractability, that they belong to this class. The technique used is that of polynomial time reduction with oracles via translations that are of algebraic or arithmetic nature. A recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting (WMC) on a propositional knowledge base. Specifically, the approach calls for encoding the probabilistic model, typically a Bayesian network, as a propositional knowledge base in conjunctive normal form (CNF) with weights associated to each model according to the network parameters. Given this CNF, computing the probability of some evidence becomes a matter of summing the weights of all CNF models consistent with the evidence. A number of variations on this approach have appeared in the literature recently, that vary across three orthogonal dimensions. The first dimension concerns the specific encoding used to convert a Bayesian network into a CNF. The second dimensions relates to whether weighted model counting is performed using a search algorithm on the CNF, or by compiling the CNF into a structure that renders WMC a polytime operation in the size of the compiled structure. The third dimension deals with the specific properties of network parameters (local structure) which are captured in the CNF encoding. In this paper, we discuss recent work in this area across the above three dimensions, and demonstrate empirically its practical importance in significantly expanding the reach of exact probabilistic inference. We restrict our discussion to exact inference and model counting, even though other proposals have been extended for approximate inference and approximate model counting. A uniform generation procedure for NP is an algorithm that, given any input in a fixed NP-language, outputs a uniformly distributed NP-witness for membership of the input in the language. We present a uniform generation procedure for NP that runs in probabilistic polynomial time with an NP-oracle. This improves upon results of M. (1986, Theoret. Comput. Sci.43, 169?188), which either require a ?P2 oracle or obtain only almost uniform generation. Our procedure utilizes ideas originating in the works of M. Sipser, and L. Stockmeyer (respectively, 1983, in Proceedings of the 15th Annual Symposium on the Theory of Computing, ACM, New York), and (1986). Abstract The class of problems involving the random generation of combinatorial structures from a uniform distribution is considered. Uniform generation problems are, in computational difficulty, intermediate between classical existence and counting problems. It is shown that exactly uniform generation of ‘efficiently verifiable’ combinatorial structures is reducible to approximate counting (and hence, is within the third level of the polynomial hierarchy). Natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation, and between their generation and counting versions. It is further shown that for self-reducible problems, almost uniform generation and randomized approximate counting are inter-reducible, and hence, of similar complexity. Abstract Many AI problems, when formalized, reduce to evaluating the probability that a propositional expression is true. In this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability. We consider various methods used in approximate reasoning such as computing degree of belief and Bayesian belief networks, as well as reasoning techniques such as constraint satisfaction and knowledge compilation, that use approximation to avoid computational difficulties, and reduce them to model-counting problems over a propositional domain. We prove that counting satisfying assignments of propositional languages is intractable even for Horn and monotone formulae, and even when the size of clauses and number of occurrences of the variables are extremely limited. This should be contrasted with the case of deductive reasoning, where Horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms. What is even more surprising is that, as we show, even approximating the number of satisfying assignments (i.e., “approximating” approximate reasoning), is intractable for most of these restricted theories. We also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given. 