Model Counting

Propositional model counting or #SAT is the problem of computing the number of models for a given propositional formula, i.e., the number of distinct truth assignments to variables for which the formula evaluates to true. For a proposi-tional formula F , we will use #F to denote the model count of F. This problem is also referred to as the solution counting problem for SAT. It generalizes SAT and is the canonical #P-complete problem. There has been significant theoretical work trying to characterize the worst-case complexity of counting problems, with some surprising results such as model counting being hard even for some polynomial-time solvable problems like 2-SAT. The model counting problem presents fascinating challenges for practitioners and poses several new research questions. Efficient algorithms for this problem will have a significant impact on many application areas that are inherently beyond SAT ('beyond' under standard complexity theoretic assumptions), such as bounded-length adversarial and contingency planning, and probabilistic reasoning. For example, various probabilistic inference problems, such as Bayesian net reasoning, can be effectively translated into model counting problems [cf. Another application is in the study of hard combinatorial problems, such as combinatorial designs, where the number of solutions provides further insights into the problem. Even finding a single solution can be a challenge for such problems; counting the number of solutions is much harder. Not surprisingly, the largest formulas we can solve for the model counting problem with state-of-the-art model counters are orders of magnitude smaller than the formulas we can solve with the best SAT solvers. Generally speaking, current exact counting methods can tackle problems with a couple of hundred variables, while approximate counting methods push this to around 1,000 variables. #SAT can be solved, in principle and to an extent in practice, by extending the two most successful frameworks for SAT algorithms, namely, DPLL and local search. However, there are some interesting issues and choices that arise when extending SAT-based techniques to this harder problem. In general, solving #SAT requires the solver to, in a sense, be cognizant of all solutions in the search space, thereby reducing the effectiveness and relevance of commonly used SAT heuristics designed to quickly narrow down the search to a single solution. The resulting