Conflict-driven Clause Learning SAT Solvers Research Articles

On CDCL-Based Proof Systems with the Ordered Decision Strategy

We prove that conflict-driven clause learning SAT-solvers with the ordered decision strategy and the DECISION learning scheme are equivalent to ordered resolution. We also prove that, by replacing this learning scheme with its opposite that stops after the first new clause when backtracking, they become equivalent to general resolution. To the best of our knowledge, this is the first theoretical study of the interplay between specific decision strategies and clause learning. For both results, we allow nondeterminism in the solver's ability to perform unit propagation, conflict analysis, and restarts, in a way that is similar to previous works in the literature. To aid the presentation of our results, and possibly future research, we define a model and language for discussing CDCL-based proof systems that allows for succinct and precise theorem statements.

Popularity-similarity random SAT formulas

In the last decades, we have witnessed a remarkable success of algorithms solving the Boolean Satisfiability problem (SAT) on instances encoding application or real-world problems arising from a very diverse number of domains, such as hardware and software verification, planning or cryptography. These algorithms are the so known Conflict-Driven Clause Learning (CDCL) SAT solvers. Interestingly enough, the reasons for the success of these solvers on this diverse range of problems are not completely understood yet.A common issue when facing this open challenge is the heterogeneity of this set of benchmarks. Another problem is the limited number of existing instances. In this context, random models of SAT formulas capturing features shared by the majority of these application benchmarks become crucial, for both theoretical and practical purposes. On the one hand, it is undoubtedly necessary to have random models where theoretical properties, like hardness, can be studied. Therefore, realistic random SAT models may contribute to explain the success of these solvers on these industrial problems. On the other hand, the limited number of benchmarks and their hardness in practice makes the evaluation of new solving techniques a costly task. Therefore, these realistic random SAT generators can provide an unlimited number of pseudo-industrial random SAT instances with some desired properties.In this work, we present a random SAT instances generator based on the notion of locality. This notion is complementary to the popularity of variables, which is present in the scale-free structure, observable in actual application problems and achievable by previous generators. Our random SAT model combines both locality and popularity, and we show that they are two decisive dimensions of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of them. Locality is closely related to the community structure, another important feature of application SAT benchmarks, which is indirectly achieved by this model. To the best of our knowledge, this is the first random SAT model that generates both scale-free structure and community structure at once.

A branching heuristic for SAT solvers based on complete implication graphs

The performance of modern conflict-driven clause learning (CDCL) SAT solvers strongly depends on branching heuristics. State-of-the-art branching heuristics, such as variable state independent decaying sum (VSIDS) and learning rate branching (LRB), are computed and maintained by aggregating the occurrences of the variables in conflicts. However, these heuristics are not sufficiently accurate at the beginning of the search because they are based on very few conflicts. We propose the distance branching heuristic, which, given a conflict, constructs a complete implication graph and increments the score of a variable considering the longest distance between the variable and the conflict rather than the simple presence of the variable in the graph. We implemented the proposed distance branching heuristic in Maple_LCM and Glucose-3.0, two of the best CDCL SAT solvers, and used the resulting solvers to solve instances from the application and crafted tracks of the 2014 and 2016 SAT competitions and the main track of the 2017 SAT competition. The empirical results demonstrate that using the proposed distance branching heuristic in the initialization phase of Maple_LCM and Glucose-3.0 solvers improves performance. The Maple_LCM solver with the proposed distance branching heuristic in the initialization phase won the main track of the 2017 SAT competition.

Beyond the structure of SAT formulas

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.