An Approximate Skolem Function Counter (2312.12026v2)
Abstract: One approach to probabilistic inference involves counting the number of models of a given Boolean formula. Here, we are interested in inferences involving higher-order objects, i.e., functions. We study the following task: Given a Boolean specification between a set of inputs and outputs, count the number of functions of inputs such that the specification is met. Such functions are called Skolem functions. We are motivated by the recent development of scalable approaches to Boolean function synthesis. This stands in relation to our problem analogously to the relationship between Boolean satisfiability and the model counting problem. Yet, counting Skolem functions poses considerable new challenges. From the complexity-theoretic standpoint, counting Skolem functions is not only #P-hard; it is quite unlikely to have an FPRAS (Fully Polynomial Randomized Approximation Scheme) as the problem of even synthesizing one Skolem function remains challenging, even given access to an NP oracle. The primary contribution of this work is the first algorithm, SkolemFC, that computes an estimate of the number of Skolem functions. SkolemFC relies on technical connections between counting functions and propositional model counting: our algorithm makes a linear number of calls to an approximate model counter and computes an estimate of the number of Skolem functions with theoretical guarantees. Moreover, we show that Skolem function count can be approximated through a polynomial number of calls to a SAT oracle. Our prototype displays impressive scalability, handling benchmarks comparably to state-of-the-art Skolem function synthesis engines, even though counting all such functions ostensibly poses a greater challenge than synthesizing a single function.
- Knowledge compilation for Boolean functional synthesis. In Proc. of FMCAD, 2019.
- SAT-based MaxSAT algorithms. Artificial Intelligence, 2013.
- What’s hard about Boolean functional synthesis? In Proc. of CAV, 2018.
- Towards parallel boolean functional synthesis. In Proc. of TACAS. Springer, 2017.
- Maximal specification synthesis. ACM SIGPLAN Notices, 2016.
- Allqbf solving by computational learning. In Proc. of ATVA, 2012.
- Resolution proofs and skolem functions in QBF evaluation and applications. In Proc. of CAV, 2011.
- The multiple facets of software diversity: Recent developments in year 2000 and beyond. ACM Computing Surveys (CSUR), 2015.
- Approximate Model Counting: Is SAT Oracle More Powerful than NP Oracle? In Proc. of ICALP, 2023.
- Approximate counting in SMT and value estimation for probabilistic programs. In Proc. of TACAS, 2015.
- Functional synthesis via input-output separation. In Proc. of FMCAD, 2018.
- Approximate probabilistic inference via word-level counting. In Proc. of AAAI, volume 30, 2016.
- A scalable approximate model counter. In Proc. of CP, 2013.
- Balancing scalability and uniformity in sat-witness generator. In Proc. of DAC, 6 2014.
- Algorithmic improvements in approximate counting for probabilistic inference: From linear to logarithmic sat calls. In Proc. of IJCAI, 2016.
- Approximate model counting. In Handbook of Satisfiability. 2021.
- Descriptive complexity of #P functions: A new perspective. Journal of Computer and System Sciences, 2021.
- An optimal algorithm for monte carlo estimation. In Proc. of FOCS, 1995.
- An optimal approximation algorithm for bayesian inference. Artificial Intelligence, 1997.
- On almost-uniform generation of sat solutions: The power of 3-wise independent hashing. In Proc. of LICS, 2022.
- ADDMC: Weighted model counting with algebraic decision diagrams. In Proc. of AAAI, 2020.
- Sat competition 2020. Artificial Intelligence, 2021.
- Manthan: A data-driven approach for boolean function synthesis. In Proc. of CAV, 2020.
- Program synthesis as dependency quantified formula modulo theory. In Proc. of IJCAI, 2021.
- Engineering an efficient boolean functional synthesis engine. In Proc. of ICCAD, 2021.
- Model counting. In Handbook of satisfiability. IOS press, 2021.
- A model-theoretic characterization of constant-depth arithmetic circuits. Annals of Pure and Applied Logic, 2019.
- Jie-Hong R Jiang. Quantifier elimination via functional composition. In Proc. of CAV, 2009.
- Interpolating functions from large boolean relations. In Proc. of ICCAD, 2009.
- Skolem functions for factored formulas. In Proc. of FMCAD, 2015.
- Decision procedures. Springer, 2016.
- Maxsat, hard and soft constraints. In Handbook of satisfiability. 2021.
- Specification synthesis with constrained horn clauses. In Proc. of PLDI, 2021.
- Enumerative level-2 solution counting for quantified boolean formulas (short paper). In Proc. of CP, 2023.
- Markus N Rabe. Incremental determinization for quantifier elimination and functional synthesis. In Proc. of CAV, 2019.
- Incremental determinization. In Proc. of SAT, 2016.
- Understanding and extending incremental determinization for 2QBF. In Proc. of CAV, 2018.
- BIRD: engineering an efficient CNF-XOR SAT solver and its applications to approximate model counting. In Proc. of AAAI, 2019.
- Outercount: A first-level solution-counter for quantified boolean formulas. In Proc. of CICM, 2022.
- Extending sat solvers to cryptographic problems. In Proc. of SAT, 2009.
- GANAK: A Scalable Probabilistic Exact Model Counter. In Proc. of IJCAI, 2019.
- Larry Stockmeyer. The complexity of approximate counting. In Proc. of ACM symposium on Theory of computing, 1983.
- Marc Thurley. sharpSAT–counting models with advanced component caching and implicit BCP. In Proc. of SAT, 2006.
- Factored boolean functional synthesis. In Proc. of FMCAD. IEEE, 2017.
- Rounding Meets Approximate Model Counting. In Proc. of CAV, 2023.