The Self-Similarity of Industrial SAT Instances

Abstract

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.

The impact of heterogeneity and geometry on the proof complexity of random satisfiability

“…With this article, we provide a theoretical foundation that matches the observations in practice by studying the proof complexity of k-SAT instances (for constant k) drawn from the power-law SAT model, and from a very general model with underlying geometry. The former was introduced by Ansótegui et al [4], the latter is a generalization of the geometric model by Giráldez-Cru and Levy [40] in the same way as geometric inhomogeneous random graphs [18] are a generalization of hyperbolic random graphs [44]. Our findings are that heterogeneous instances are hard asymptotically almost surely 1 in that modern SAT solvers require superpolynomial or even exponential running time to refute unsatisfiable instances.…”

“…The 𝑑-dimensional torus T 𝑑 is defined as the 𝑑-dimensional hypercube [0, 1] 𝑑 in which opposite borders are identified, that is, a coordinate of 0 is identical to a coordinate of 1. 4 It is equipped with the 𝔭-norm as metric, for arbitrary but fixed 𝔭 ∈ N + ∪ {∞}. To define it formally for the torus, let p = (p 1 , … , p 𝑑 ) and q = (q 1 , … , q 𝑑 ) be two points in T 𝑑 .…”

“…Besides serving as somewhat realistic benchmarks for SAT competitions [39], these SAT models can be used to study solver behavior depending on heterogeneity and locality. One can experimentally observe that a high level of heterogeneity improves the performance of SAT solvers that also perform well on industrial instances [4,13]. Moreover, locality seems very beneficial as solvers appear to implicitly use the locality of a given instance [40].…”

“…Inspired by network science, researchers have studied models that resemble industrial instances with respect to these properties. Particularly, Ansótegui et al [4] introduced a power-law SAT model for heterogeneous instances, which has been theoretically studied in terms of satisfiability thresholds [33][34][35]. A different model with heterogeneous degree distributions has been studied by Cooper et al [22], Ansótegui et al [3], and Omelchenko and Bulatov [52].…”

“…Moreover, Giráldez-Cru and Levy [40] introduced a model in which variable weights lead to heterogeneity while an underlying geometry facilitates locality. Comparing this to network models, the former model [4] is the SAT-variant of Chung-Lu graphs [19,20]. The latter [40] is based on the popularity-similarity model [53], which is closely related to hyperbolic random graphs [44] and geometric inhomogeneous random graphs [18].…”