Abstract.Propositional model counting (#SAT), i.e., counting the number of satisfying assignments of a propositional formula, is a problem of significant theoretical and practical interest. Due to the inherent complexity of the problem, approximate model counting, which counts the number of satisfying assignments to within given tolerance and confidence level, was proposed as a practical alternative to exact model counting. Yet, approximate model counting has been studied essentially only theoretically. The only reported implementation of approximate model counting, due to Karp and Luby, worked only for DNF formulas. A few existing tools for CNF formulas are bounding model counters; they can handle realistic problem sizes, but fall short of providing counts within given tolerance and confidence, and, thus, are not approximate model counters.We present here a novel algorithm, as well as a reference implementation, that is the first scalable approximate model counter for CNF formulas. The algorithm works by issuing a polynomial number of calls to a SAT solver. Our tool, ApproxMC, scales to formulas with tens of thousands of variables. Careful experimental comparisons show that ApproxMC reports, with high confidence, bounds that are close to the exact count, and also succeeds in reporting bounds with small tolerance and high confidence in cases that are too large for computing exact model counts.

Abstract. Constrained-random verification (CRV) is widely used in industry for validating hardware designs. The effectiveness of CRV depends on the uniformity of test stimuli generated from a given set of constraints. Most existing techniques sacrifice either uniformity or scalability when generating stimuli. While recent work based on random hash functions has shown that it is possible to generate almost uniform stimuli from constraints with 100,000+ variables, the performance still falls short of today's industrial requirements. In this paper, we focus on pushing the performance frontier of uniform stimulus generation further. We present a random hashing-based, easily parallelizable algorithm, UniGen2, for sampling solutions of propositional constraints. UniGen2 provides strong and relevant theoretical guarantees in the context of CRV, while also offering significantly improved performance compared to existing almostuniform generators. Experiments on a diverse set of benchmarks show that UniGen2 achieves an average speedup of about 20× over a state-ofthe-art sampling algorithm, even when running on a single core. Moreover, experiments with multiple cores show that UniGen2 achieves a nearlinear speedup in the number of cores, thereby boosting performance even further.

Functional verification constitutes one of the most challenging tasks in the development of modern hardware systems, and simulationbased verification techniques dominate the functional verification landscape. A dominant paradigm in simulation-based verification is directed random testing, where a model of the system is simulated with a set of random test stimuli that are uniformly or near-uniformly distributed over the space of all stimuli satisfying a given set of constraints. Uniform or near-uniform generation of solutions for large constraint sets is therefore a problem of theoretical and practical interest. For boolean constraints, prior work offered heuristic approaches with no guarantee of performance, and theoretical approaches with proven guarantees, but poor performance in practice. We offer here a new approach with theoretical performance guarantees and demonstrate its practical utility on large constraint sets.

Given a Boolean formula φ, the problem of model counting, also referred to as #SAT is to compute the number of solutions of φ. Model counting is a fundamental problem in artificial intelligence with a wide range of applications including probabilistic reasoning, decision making under uncertainty, quantified information flow, and the like. Motivated by the success of SAT solvers, there has been surge of interest in the design of hashing-based techniques for approximate model counting for the past decade. We profiled the state of the art approximate model counter ApproxMC2 and observed that over 99.99% of time is consumed by the underlying SAT solver, CryptoMiniSat. This observation motivated us to ask: Can we design an efficient underlying CNF-XOR SAT solver that can take advantage of the structure of hashing-based algorithms and would this lead to an efficient approximate model counter? The primary contribution of this paper is an affirmative answer to the above question. We present a novel architecture, called BIRD, to handle CNF-XOR formulas arising from hashingbased techniques. The resulting hashing-based approximate model counter, called ApproxMC3, employs the BIRD framework in its underlying SAT solver, CryptoMiniSat. To the best of our knowledge, we conducted the most comprehensive study of evaluation performance of counting algorithms involving 1896 benchmarks with computational effort totaling 86400 computational hours. Our experimental evaluation demonstrates significant runtime performance improvement for ApproxMC3 over ApproxMC2. In particular, we solve 648 benchmarks more than ApproxMC2, the state of the art approximate model counter and for all the formulas where both ApproxMC2 and ApproxMC3 did not timeout and took more than 1 seconds, the mean speedup is 284.40 – more than two orders of magnitude.

Constrained-random simulation is the predominant approach used in the industry for functional verification of complex digital designs. The effectiveness of this approach depends on two key factors: the quality of constraints used to generate test vectors, and the randomness of solutions generated from a given set of constraints. In this paper, we focus on the second problem, and present an algorithm that significantly improves the state-of-the-art of (almost-)uniform generation of solutions of large Boolean constraints. Our algorithm provides strong theoretical guarantees on the uniformity of generated solutions and scales to problems involving hundreds of thousands of variables.