Short XORs for Model Counting: From Theory to Practice

Gomes, Carla P.; Hoffmann, Joerg; Sabharwal, Ashish; Selman, Bart

doi:10.1007/978-3-540-72788-0_13

Cited by 26 publications

(31 citation statements)

References 2 publications

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“…It was empirically demonstrated that short XORs, surprisingly, perform quite well for wide variety of benchmarks. The earlier works [16,12] failed, however, to obtain provable bounds on adequate size of XOR constraints that are close to empirical observations. In contrast, our results imply that we can substitute n by |I| in Theorem 3 of [12] to obtain new provable bounds on the size of XORs required for approximate model counting.…”

Section: Applications To Sampling and Countingmentioning

confidence: 83%

“…Since counting techniques require weaker guarantees of universality, recent research efforts have focused on obtaining precise bounds on the size of XORs required. These efforts are motivated by investigations into shorter XORs by Gomes et al [16]. It was empirically demonstrated that short XORs, surprisingly, perform quite well for wide variety of benchmarks.…”

Section: Applications To Sampling and Countingmentioning

confidence: 99%

“…Since approximation techniques for model counting only requires weaker guarantees of universality [8], several techniques have been proposed on employing shorter XORs for model counting [16,12]. The investigations into shorter XORs [16,12] empirically demonstrated that short XORs, surprisingly, perform quite well for wide variety of benchmarks, even without a theoretical guarantee, but have failed to obtain provable bounds on adequate size of XOR constraints that are close to empirical observations.…”

Section: Impact On Xor Size Bounds For Model Counting Techniquesmentioning

confidence: 99%

“…The investigations into shorter XORs [16,12] empirically demonstrated that short XORs, surprisingly, perform quite well for wide variety of benchmarks, even without a theoretical guarantee, but have failed to obtain provable bounds on adequate size of XOR constraints that are close to empirical observations. By computing the size of XOR constraints based on the size of minimal independent support and then applying Theorem 3 of [12], we provide the first theoretically proven bounds on adequate size of XOR constraints that are very close to empirically observed bounds.…”

Section: Impact On Xor Size Bounds For Model Counting Techniquesmentioning

confidence: 99%

“…Since combinatorial reasoning tools are invoked to search for solutions of the conjunction of input constraints and XOR-based universal hash functions, recent investigations have focused on determining the relationship between runtime performance of combinatorial search and features of XOR-constraints. It has been observed that lower density XORs are easy to reason in practice and runtime performance of solvers greatly enhances with the decrease in the density of XOR-constraints [16]. This has led to recent work focused on designing hash functions with low density XOR-constraints [12] but such hash functions provide very weak guarantees of universality that are not adequate for hashing-based techniques for sampling.…”

Section: Introductionmentioning

confidence: 99%

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On computing minimal independent support and its applications to sampling and counting

et al. 2015

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Abstract. Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in state-of-the-art constrained sampling and counting tools, while still retaining theoretical guarantees.

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Section: Applications To Sampling and Countingmentioning

confidence: 83%

Section: Applications To Sampling and Countingmentioning

confidence: 99%

Section: Impact On Xor Size Bounds For Model Counting Techniquesmentioning

confidence: 99%

Section: Impact On Xor Size Bounds For Model Counting Techniquesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On computing minimal independent support and its applications to sampling and counting

et al. 2015

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The Return of xorro

Everardo

Janhunen

Kaminski

et al. 2019

Logic Programming and Nonmonotonic Reasoning

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Although parity constraints are at the heart of many relevant reasoning modes like sampling or model counting, little attention has so far been paid to their integration into ASP systems. We address this shortcoming and investigate a variety of alternative approaches to implementing parity constraints, ranging from rather basic ASP encodings to more sophisticated theory propagators (featuring Gauss-Jordan elimination). All of them are implemented in the xorro system by building on the theory reasoning capabilities of the ASP system clingo. Our comparative empirical study investigates the impact of the number and size of parity constraints on performance and indicates the merits of the respective implementation techniques. Finally, we benefit from parity constraints to equip xorro with means to sample answer sets, paving the way for new applications of ASP.

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A Study of Symmetry Breaking Predicates and Model Counting

Wang

Usman

Almaawi

et al. 2020

Tools and Algorithms for the Construction and Analysis of Systems

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Propositional model counting is a classic problem that has recently witnessed many technical advances and novel applications. While the basic model counting problem requires computing the number of all solutions to the given formula, in some important application scenarios, the desired count is not of all solutions, but instead, of all unique solutions up to isomorphism. In such a scenario, the user herself must try to either use the full count that the model counter returns to compute the count up to isomorphism, or ensure that the input formula to the model counter adequately captures the symmetry breaking predicates so it can directly report the count she desires. We study the use of CNF-level and domain-level symmetry breaking predicates in the context of the state-of-the-art in model counting, specifically the leading approximate model counter ApproxMC and the recently introduced exact model counter ProjMC. As benchmarks, we use a range of problems, including structurally complex specifications of software systems and constraint satisfaction problems. The results show that while it is sometimes feasible to compute the model counts up to isomorphism using the full counts that are computed by the model counters, doing so suffers from poor scalability. The addition of symmetry breaking predicates substantially assists model counters. Domain-specific predicates are particularly useful, and in many cases can provide full symmetry breaking to enable highly efficient model counting up to isomorphism. We hope our study motivates new research on designing model counters that directly account for symmetries to facilitate further applications of model counting.

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Short XORs for Model Counting: From Theory to Practice

Cited by 26 publications

References 2 publications

On computing minimal independent support and its applications to sampling and counting

On computing minimal independent support and its applications to sampling and counting

The Return of xorro

A Study of Symmetry Breaking Predicates and Model Counting

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