Generating SAT instances with community structure

Giráldez-Cru, Jesús; Levy, Jordi

doi:10.1016/j.artint.2016.06.001

Cited by 32 publications

(29 citation statements)

References 21 publications

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“…Sonobe et al [ 9 ] improved the performance of a parallel SAT solver by using community structured partitioning. Giráldez-Cru and Levy [ 10 , 11 ] used the formula community to generate some highly modular pseudo-random industrial instances.…”

Section: Introductionmentioning

confidence: 99%

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

Zhang

Zhou

2021

Entropy

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The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.

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Section: Introductionmentioning

confidence: 99%

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

Zhang

Zhou

2021

Entropy

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“…The main aim of these studies is to better understand the success of CDCL techniques and help to improve them. They can also help develop new benchmarks generation models with more realistic properties [34][35][36][37][38][39][40]. This problem has been identified as an important challenge in the SAT community [41][42][43].…”

Section: Related Workmentioning

confidence: 99%

The Self-Similarity of Industrial SAT Instances

Ansótegui¹,

Bonet²,

Giráldez-Cru³

et al. 2020

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In the last years, we have witnessed a remarkable progress of algorithms solving Boolean satisfiability (SAT). The success of these algorithms has been especially relevant in a large number of industrial or real-world applications, for which these SAT solvers are nowadays an essential core part of their solving processes. Interestingly enough, these applications include a very diverse and heterogeneous range of domains, such as hardware verification, planning, and cryptography, among others. Unfortunately, the reasons of the good performance of these solvers on this variety of industrial benchmarks are not completely understood yet. Since SAT solvers’ efficiency is fundamental in various domains, obtaining a better understanding of these algorithms and the reasons of their good performance is crucial. In order to shed light on this question, SAT solvers are often viewed as complex systems with many interconnected components (e.g., conflict analysis and learning mechanism, database management, search restarts) interacting in many unpredictable ways. There is the common belief that the resulting emergent behavior of these complex systems takes advantage of a certain underlying structure of the SAT formula, which is shared by the majority of these industrial problems regardless the domain they come from. Recently, there have been some attempts of characterizing this structure under the lens of complex networks, with the purpose of better understanding the success of the solvers, and potentially improving them. In this paper, we analyze the structure of industrial SAT instances under the lens of self-similarity, and study how the execution of SAT solvers affect that structure. Many real-world graphs exhibit self-similar structure (with small fractal dimension), which means that after rescaling (replacing groups of nodes by a single node), the same kind of structure can be observed. In our analysis, in which we represent SAT instances as graphs, we observe that many industrial SAT formulas exhibit the same kind of structure. Moreover, we analyze how this structure evolves by effects of learning new clauses during the search. In particular, we observe that learned clauses usually contain variables that are close in the graph representation of the formula. This is, the learning mechanism tends to work locally. On the contrary, this learning mechanism on random SAT formulas –which do not exhibit any structure at all– is unable to generate these local clauses. This difference contributes to explain the success of modern SAT solvers on industrial problems.

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“…Markström in [ 26 ] proposed a constructor method of SAT instance based on Eulerian graphs, and discussed how a solver can try to avoid at least some of the pitfalls presented by these instances. Giraldez-Cru and Levy in [ 27 ] proposed a new model of generation of random SAT instances with community structure, and showed that modern solvers do actually exploit this community structure. In [ 28 ], they presented a random SAT instances generator based on the notion of locality, and showed that CDCL SAT solvers take advantage of both popularity and similarity.…”

Section: Related Workmentioning

confidence: 99%

Uniquely Satisfiable d-Regular (k,s)-SAT Instances

2020

Entropy

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Unique k-SAT is the promised version of k-SAT where the given formula has 0 or 1 solution and is proved to be as difficult as the general k-SAT. For any k ≥ 3 , s ≥ f ( k , d ) and ( s + d ) / 2 > k − 1 , a parsimonious reduction from k-CNF to d-regular (k,s)-CNF is given. Here regular (k,s)-CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular (k,s)-CNF formula is a regular (k,s)-CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all k ≥ 3 , f ( k , d ) ≤ u ( k , d ) + 1 and f ( k , d + 1 ) ≤ u ( k , d ) . The critical function f ( k , d ) is the maximal value of s, such that every d-regular (k,s)-CNF formula is satisfiable. In this study, u ( k , d ) denotes the minimal value of s such that there exists a uniquely satisfiable d-regular (k,s)-CNF formula. We further show that for s ≥ f ( k , d ) + 1 and ( s + d ) / 2 > k − 1 , there exists a uniquely satisfiable d-regular ( k , s + 1 ) -CNF formula. Moreover, for k ≥ 7 , we have that u ( k , d ) ≤ f ( k , d ) + 1 .

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Generating SAT instances with community structure

Cited by 32 publications

References 21 publications

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

The Self-Similarity of Industrial SAT Instances

Uniquely Satisfiable d-Regular (k,s)-SAT Instances

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