Community Structure in Industrial SAT Instances

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there are few works trying to exactly characterize the main features of this structure. The research community on complex networks has developed techniques of analysis and algorithms to study real-world graphs that can be used by the SAT community. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them. In this paper, inspired by the results on complex networks, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. In our analysis, we represent SAT instances as graphs, and we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver. In particular, we use the community structure to detect that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. This is, learned clauses tend to contain variables of distinct communities

Community structure in industrial SAT instances

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them. In this paper, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.Peer ReviewedPostprint (published version

Explaining SAT Solving Using Causal Reasoning

The past three decades have witnessed notable success in designing efficient SAT solvers, with modern solvers capable of solving industrial benchmarks containing millions of variables in just a few seconds. The success of modern SAT solvers owes to the widely-used CDCL algorithm, which lacks comprehensive theoretical investigation. Furthermore, it has been observed that CDCL solvers still struggle to deal with specific classes of benchmarks comprising only hundreds of variables, which contrasts with their widespread use in real-world applications. Consequently, there is an urgent need to uncover the inner workings of these seemingly weak yet powerful black boxes. In this paper, we present a first step towards this goal by introducing an approach called {CausalSAT}, which employs causal reasoning to gain insights into the functioning of modern SAT solvers. {CausalSAT} initially generates observational data from the execution of SAT solvers and learns a structured graph representing the causal relationships between the components of a SAT solver. Subsequently, given a query such as whether a clause with low literals blocks distance (LBD) has a higher clause utility, {CausalSAT} calculates the causal effect of LBD on clause utility and provides an answer to the question. We use {CausalSAT} to quantitatively verify hypotheses previously regarded as "rules of thumb" or empirical findings, such as the query above or the notion that clauses with high LBD experience a rapid drop in utility over time. Moreover, {CausalSAT} can address previously unexplored questions, like which branching heuristic leads to greater clause utility in order to study the relationship between branching and clause management. Experimental evaluations using practical benchmarks demonstrate that {CausalSAT} effectively fits the data, verifies four "rules of thumb", and provides answers to three questions closely related to implementing modern solvers

SAT and CP: Parallelisation and Applications

This thesis is considered with the parallelisation of solvers which search for either an arbitrary, or an optimum, solution to a problem stated in some formal way. We discuss the parallelisation of two solvers, and their application in three chapters.In the first chapter, we consider SAT, the decision problem of propositional logic, and algorithms for showing the satisfiability or unsatisfiability of propositional formulas. We sketch some proof-theoretic foundations which are related to the strength of different algorithmic approaches. Furthermore, we discuss details of the implementations of SAT solvers, and show how to improve upon existing sequential solvers. Lastly, we discuss the parallelisation of these solvers with a focus on clause exchange, the communication of intermediate results within a parallel solver. The second chapter is concerned with Contraint Programing (CP) with learning. Contrary to classical Constraint Programming techniques, this incorporates learning mechanisms as they are used in the field of SAT solving. We present results from parallelising CHUFFED, a learning CP solver. As this is both a kind of CP and SAT solver, it is not clear which parallelisation approaches work best here. In the final chapter, we will discuss Sorting networks, which are data oblivious sorting algorithms, i. e., the comparisons they perform do not depend on the input data. Their independence of the input data lends them to parallel implementation. We consider the question how many parallel sorting steps are needed to sort some inputs, and present both lower and upper bounds for several cases

Understanding and Enhancing CDCL-based SAT Solvers

Modern conflict-driven clause-learning (CDCL) Boolean satisfiability (SAT) solvers routinely solve formulas from industrial domains with millions of variables and clauses, despite the Boolean satisfiability problem being NP-complete and widely regarded as intractable in general. At the same time, very small crafted or randomly generated formulas are often infeasible for CDCL solvers. A commonly proposed explanation is that these solvers somehow exploit the underlying structure inherent in industrial instances. A better understanding of the structure of Boolean formulas not only enables improvements to modern SAT solvers, but also lends insight as to why solvers perform well or poorly on certain types of instances. Even further, examining solvers through the lens of these underlying structures can help to distinguish the behavior of different solving heuristics, both in theory and practice. The first issue we address relates to the representation of SAT formulas. A given Boolean satisfiability problem can be represented in arbitrarily many ways, and the type of encoding can have significant effects on SAT solver performance. Further, in some cases, a direct encoding to SAT may not be the best choice. We introduce a new system that integrates SAT solving with computer algebra systems (CAS) to address representation issues for several graph-theoretic problems. We use this system to improve the bounds on several finitely-verified conjectures related to graph-theoretic problems. We demonstrate how our approach is more appropriate for these problems than other off-the-shelf SAT-based tools. For more typical SAT formulas, a better understanding of their underlying structural properties, and how they relate to SAT solving, can deepen our understanding of SAT. We perform a largescale evaluation of many of the popular structural measures of formulas, such as community structure, treewidth, and backdoors. We investigate how these parameters correlate with CDCL solving time, and whether they can effectively be used to distinguish formulas from different domains. We demonstrate how these measures can be used as a means to understand the behavior of solvers during search. A common theme is that the solver exhibits locality during search through the lens of these underlying structures, and that the choice of solving heuristic can greatly influence this locality. We posit that this local behavior of modern SAT solvers is crucial to their performance. The remaining contributions dive deeper into two new measures of SAT formulas. We first consider a simple measure, denoted “mergeability,” which characterizes the proportion of input clauses pairs that can resolve and merge. We develop a formula generator that takes as input a seed formula, and creates a sequence of increasingly more mergeable formulas, while maintaining many of the properties of the original formula. Experiments over randomly-generated industrial-like instances suggest that mergeability strongly negatively correlates with CDCL solving time, i.e., as the mergeability of formulas increases, the solving time decreases, particularly for unsatisfiable instances. Our final contribution considers whether one of the aforementioned measures, namely backdoor size, is influenced by solver heuristics in theory. Starting from the notion of learning-sensitive (LS) backdoors, we consider various extensions of LS backdoors by incorporating different branching heuristics and restart policies. We introduce learning-sensitive with restarts (LSR) backdoors and show that, when backjumping is disallowed, LSR backdoors may be exponentially smaller than LS backdoors. We further demonstrate that the size of LSR backdoors are dependent on the learning scheme used during search. Finally, we present new algorithms to compute upper-bounds on LSR backdoors that intrinsically rely upon restarts, and can be computed with a single run of a SAT solver. We empirically demonstrate that this can often produce smaller backdoors than previous approaches to computing LS backdoors

JFPC 2019 - Actes des 15es Journées Francophones de Programmation par Contraintes

National audienceLes JFPC (Journées Francophones de Programmation par Contraintes) sont le principal congrès de la communauté francophone travaillant sur les problèmes de satisfaction de contraintes (CSP), le problème de la satisfiabilité d'une formule logique propositionnelle (SAT) et/ou la programmation logique avec contraintes (CLP). La communauté de programmation par contraintes entretient également des liens avec la recherche opérationnelle (RO), l'analyse par intervalles et différents domaines de l'intelligence artificielle.L'efficacité des méthodes de résolution et l'extension des modèles permettent à la programmation par contraintes de s'attaquer à des applications nombreuses et variées comme la logistique, l'ordonnancement de tâches, la conception d'emplois du temps, la conception en robotique, l'étude du génôme en bio-informatique, l'optimisation de pratiques agricoles, etc.Les JFPC se veulent un lieu convivial de rencontres, de discussions et d'échanges pour la communauté francophone, en particulier entre doctorants, chercheurs confirmés et industriels. L'importance des JFPC est reflétée par la part considérable (environ un tiers) de la communauté francophone dans la recherche mondiale dans ce domaine.Patronnées par l'AFPC (Association Française pour la Programmation par Contraintes), les JFPC 2019 ont lieu du 12 au 14 Juin 2019 à l'IMT Mines Albi et sont organisées par Xavier Lorca (président du comité scientifique) et par Élise Vareilles (présidente du comité d'organisation)