N ov 2 01 5 Approximate Probabilistic Inference via Word-Level Counting ( Extended Abstract ) ∗ †

Probabilistic inference on large and uncertain data sets is increasingly being used in a wide range of applications. It is well-known that probabilistic inference is polynomially inter-reducible to model counting (Roth 1996). In a recent line of work, it has been shown (Chakraborty, Meel, and Vardi 2013; Chakraborty et al. 2014; Ermon et al. 2014; Ermon et al. 2013) that one can strike a fine balance between performance and approximation guarantees for propositional model counting, using2-universal hashing functions (Carter and Wegman 1977) on Boolean domains. This has propelled the model-counting formulation to emerge as a promising “assembly language” (Belle, Passerini, and Van den Broeck 2015) for inferencing in probabilistic graphical models. In a large class of probabilistic inference problems, an important case being lifted inference on first order represent ations (Kersting 2012), the values of variables come from finite but large (exponential in the size of the representatio n) domains. Data values coming from such domains are naturally encoded as fixed-width words, where the width is logarithmic in the size of the domain. Conditions on observed values are, in turn, encoded as word-level constraints, and the corresponding model-counting problem asks to count the number of solutions of a word-level constraint. It is theref o natural to ask if the success of approximate propositional model counters can be replicated for word-level constraint s. The balance between efficiency and strong guarantees of hashing-based algorithms for approximate propositiona l model counting crucially depends on two factors: (i) use of XOR-based2-universal bit-level hash functions, and (ii) use of state-of-the-art propositional satisfiability solvers , viz. CryptoMiniSAT (Cry ), that can efficiently reason about formulas that combined disjunctive clauses with XOR clauses. In recent years, the performance of SMT (SatisfiabilThe author list has been sorted alphabetically by last name; this should not be used to determine the extent of authors’ co ntributions. The full version along with the tool and benchmarks is available at http://www.cs.rice.edu/CS/Verification/Projects/CUSP/ Copyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. ity Modulo Theories) solvers has witnessed spectacular improvements (Barrett et al. 2012). Indeed, several highly optimized SMTsolvers for fixed-width words are now available in the public domain (Brummayer and Biere 2009; Jha, Limaye, and Seshia 2009; Hadarean et al. 2014; De Moura and Bjørner 2008). Nevertheless, 2-universal hashing functions for fixed-width words that are also amenable to efficient reasoning by SMT solvers have hitherto not been studied. The reasoning power of SMTsolvers for fixed-width words has therefore remained untapped for word-level model counting. Thus, it is not surprising that a ll existing work on probabilistic inference using model count ing (viz. (Chistikov, Dimitrova, and Majumdar 2015; Belle, Passerini, and Van den Broeck 2015; Ermon et al. 2013)) effectively reduce the problem to propositional model counting. Such an approach is often referred to as “bit blasting” (Barrett et al. 2012). The primary contribution of this paper is an efficient word-level approximate model counting algorithm SMTApproxMCthat can be employed to answer inference queries over high-dimensional discrete domains. Our algorithm uses a new class of word-level hashing functions that are2-universal and can be solved by word-level SMTsolvers capable of reasoning about linear equalities on words. Therefore, unlike previous works, SMTApproxMC is able to leverage the power of sophisticated SMT solvers. To illustrate the practical utility ofSMTApproxMC, we implemented a prototype and evaluated it on a suite of benchmarks. Our experiments demonstrate that SMTApproxMC can significantly outperform the prevalent approach of bit-blasting a word-level constraint and using an approximate propositional model counter that employs XOR-based hashing functions. Our proposed word-level hash functions embed the domain of all variables in a large enough finite domain. Thus, one would not expect our approach to work well for constraints that exhibit a heterogeneous mix of word widths. or for problems that are difficult for word-level SMT solvers. Indeed, our experiments suggest that the use of word-level hashing functions provides significant benefits when the original word-level constraint is such that (i) the words ap pearing in it have long and similar widths, and (ii) the SMTsolver can reason about the constraint at the word-level, without extensive bit-blasting.