35,233 research outputs found

    Community structure in industrial SAT instances

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    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

    Characterizing the Temperature of SAT Formulas

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    The remarkable advances in SAT solving achieved in the last years have allowed to use this technology to solve many real-world applications, such as planning, formal verification and cryptography, among others. Interestingly, these industrial SAT problems are commonly believed to be easier than classical random SAT formulas, but estimating their actual hardness is still a very challenging question, which in some cases even requires to solve them. In this context, realistic pseudo-industrial random SAT generators have emerged with the aim of reproducing the main features of these application problems to better understand the success of those SAT solving techniques on them. In this work, we present a model to estimate the temperature of real-world SAT instances. This temperature represents the degree of distortion into the expected structure of the formula, from highly structured benchmarks (more similar to real-world SAT instances) to the complete absence of structure (observed in the classical random SAT model). Our solution is based on the popularity–similarity random model for SAT, which has been recently presented to reproduce two crucial features of application SAT benchmarks: scale-free and community structures. This model is able to control the hardness of the generated formula by introducing some randomizations in the expected structure. Using our regression model, we observe that the estimated temperature of the applications benchmarks used in the last SAT Competitions correlates to their hardness in most of the cases.Juan de la Cierva program, fellowship IJC2019-040489-I, funded by MCIN and AE

    Beyond the structure of SAT formulas

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    Hoy en día, muchos problemas del mundo real son codificados en instancias SAT y resueltos eficientemente por modernos SAT solvers. Estos solvers, usualmente conocidos como Conflict-Driven Clause Learning (CDCL: Aprendizaje de cláusulas guiado por conflictos) SAT solvers, incluyen una variedad de sofisticadas técnicas, como el aprendizaje de cláusulas, estructuras de datos perezosas, heurísticas de ramificación adaptativas basadas en los conflictos, o reinicios aleatorios, entre otros. Sin embargo, las razones de su eficiencia resolviendo problemas SAT del mundo real, o industriales, son todavía desconocidas. La creencia común en la comunidad SAT es que estas técnicas explotan alguna estructura oculta de los problemas del mundo real. En esta tesis, primeramente se caracteriza algunas importantes características de la estructura subyacente de las instancias SAT industriales. Específicamente, estas son la estructura de comunidades y la estructura auto-similar. Se observa que la mayoría de las fórmulas SAT industriales, vistas como grafos, tienen estas dos propiedades. Esto significa que (i) en un grafo con una estructura de comunidades clara; es decir, alta modularidad, se puede encontrar una partición de sus nodos en comunidades de tal forma que la mayoría de las aristas conectan nodos de la misma comunidad; y (ii) en un grafo con el patrón de auto-similitud; es decir, siendo fractal, su forma se mantiene después de re-escalados (agrupando conjuntos de nodos en uno). Se analiza también cómo estas estructuras están afectadas por los efectos de las técnicas CDCL durante la búsqueda. Usando los estudios estructurales previos, se proponen tres aplicaciones. Primero, se aborda el problema de la generación aleatoria de instancias SAT pseudo-industriales usando la noción de modularidad. Nuestro modelo genera instancias similares a las (clásicas) fórmulas SAT aleatorias cuando la modularidad es baja, pero cuando este valor es alto, nuestro modelo también es adecuado para modelar problemas pseudo-industriales realísticamente. Segundo, se propone un método basado en la estructura en comunidades de la instancia para detectar cláusulas aprendidas relevantes. Nuestra técnica aumenta la instancia original con un conjunto de cláusulas relevantes, y esto resulta en una mejora general de la eficiencia de varios CDCL SAT solvers. Finalmente, se analiza la clasificación de instancias SAT industrial en familias usando las características estructurales previamente analizadas, y se comparan con otros clasificadores comúnmente usados en aproximaciones SAT portfolio. En resumen, esta disertación extiende nuestro conocimiento sobre la estructura de las instancias SAT, con el objetivo de explicar mejor el éxito de las técnicas CDCL, con la posibilidad de mejorarlas; y propone una serie de aplicaciones basadas en este análisis de la estructura subyacente de las fórmulas SAT.Nowadays, many real-world problems are encoded into SAT instances and efficiently solved by modern SAT solvers. These solvers, usually known as Conflict-Driven Clause Learning (CDCL) SAT solvers, include a variety of sophisticated techniques, such as clause learning, lazy data structures, conflict-based adaptive branching heuristics, or random restarts, among others. However, the reasons of their efficiency in solving real-world, or industrial, SAT instances are still unknown. The common wisdom in the SAT community is that these technique exploit some hidden structure of real-world problems. In this thesis, we characterize some important features of the underlying structure of industrial SAT instances. Namely, they are the community structure and the self-similar structure. We observe that most industrial SAT formulas, viewed as graphs, have these two properties. This means that~(i) in a graph with a clear community structure, i.e. having high modularity, we can find a partition of its nodes into communities such that most edges connect nodes of the same community; and~(ii) in a graph with a self-similar pattern, i.e. being fractal, its shape is kept after re-scalings, i.e., grouping sets of nodes into a single node. We also analyze how these structures are affected by the effects of CDCL techniques during the search. Using the previous structural studies, we propose three applications. First, we face the problem of generating pseudo-industrial random SAT instances using the notion of modularity. Our model generates instances similar to (classical) random SAT formulas when the modularity is low, but when this value is high, our model is also adequate to model realistic pseudo-industrial problems. Second, we propose a method based on the community structure of the instance to detect relevant learnt clauses. Our technique augments the original instance with this set of relevant clauses, and this results into an overall improvement of the efficiency of several state-of-the-art CDCL SAT solvers. Finally, we analyze the classification of industrial SAT instances into families using the previously analyzed structure features, and we compare them to other classifiers commonly used in portfolio SAT approaches. In summary, this \dissertation extends the understandings of the structure of SAT instances, with the aim of better explaining the success of CDCL techniques and possibly improve them, and propose a number of applications based on this analysis of the underlying structure of SAT formulas

    Understanding and Enhancing CDCL-based SAT Solvers

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    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

    A New Look at Survey Propagation and Its Generalizations

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    This article provides a new conceptual perspective on survey propagation, which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random k-SAT problems even with densities close to the satisfiability threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number ρ ∈ [0, 1]. We then show that applying belief propagation---a well-known “message-passing” technique for estimating marginal probabilities---to this family of MRFs recovers a known family of algorithms, ranging from pure survey propagation at one extreme (ρ = 1) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme (ρ = 0). Configurations in these MRFs have a natural interpretation as partial satisfiability assignments, on which a partial order can be defined. We isolate cores as minimal elements in this partial ordering, which are also fixed points of survey propagation and the only assignments with positive probability in the MRF for ρ = 1. Our experimental results for k = 3 suggest that solutions of random formulas typically do not possess non-trivial cores. This makes it necessary to study the structure of the space of partial assignments for ρ \u3c 1 and investigate the role of assignments that are very close to being cores. To that end, we investigate the associated lattice structure, and prove a weight-preserving identity that shows how any MRF with ρ \u3e 0 can be viewed as a “smoothed” version of the uniform distribution over satisfying assignments (ρ = 0). Finally, we isolate properties of Gibbs sampling and message-passing algorithms that are typical for an ensemble of k-SAT problems

    Community Structure in Industrial SAT Instances

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    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
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