On Continuous Local BDD-Based Search for Hybrid SAT Solving

We explore the potential of continuous local search (CLS) in SAT solving by proposing a novel approach for finding a solution of a hybrid system of Boolean constraints. The algorithm is based on CLS combined with belief propagation on binary decision diagrams (BDDs). Our framework accepts all Boolean constraints that admit compact BDDs, including symmetric Boolean constraints and small-coefficient pseudo-Boolean constraints as interesting families. We propose a novel algorithm for efficiently computing the gradient needed by CLS. We study the capabilities and limitations of our versatile CLS solver, GradSAT, by applying it on many benchmark instances. The experimental results indicate that GradSAT can be a useful addition to the portfolio of existing SAT and MaxSAT solvers for solving Boolean satisfiability and optimization problems.Comment: AAAI 2

Pseudo-Boolean Constraint Encodings for Conjunctive Normal Form and their Applications

In contrast to a single clause a pseudo-Boolean (PB) constraint is much more expressive and hence it is easier to define problems with the help of PB constraints. But while PB constraints provide us with a high-level problem description, it has been shown that solving PB constraints can be done faster with the help of a SAT solver. To apply such a solver to a PB constraint we have to encode it with clauses into conjunctive normal form (CNF). While we can find a basic encoding into CNF which is equivalent to a given PB constraint, the solving time of a SAT solver significantly depends on different properties of an encoding, e.g. the number of clauses or if generalized arc consistency (GAC) is maintained during the search for a solution. There are various PB encodings that try to optimize or balance these properties. This thesis is about such encodings. For a better understanding of the research field an overview about the state-of-the art encodings is given. The focus of the overview is a simple but complete description of each encoding, such that any reader could use, implement and extent them in his own work. In addition two novel encodings are presented: The Sequential Weight Counter (SWC) encoding and the Binary Merger Encoding. While the SWC encoding provides a very simple structure – it is listed in four lines – empirical evaluation showed its practical usefulness in various applications. The Binary Merger encoding reduces the number of clauses a PB encoding needs while having the important GAC property. To the best of our knowledge currently no other encoding has a lower upper bound for the number of clauses produced by a PB encoding with this property. This is an important improvement of the state-of-the art, since both GAC and a low number of clauses are vital for an improved solving time of the SAT solver. The thesis also contributes to the development of new applications for PB constraint encodings. The programming library PBLib provides researchers with an open source implementation of almost all PB encodings – including the encodings for the special cases at-most-one and cardinality constraints. The PBLib is also the foundation of the presented weighted MaxSAT solver optimax, the PBO solver pbsolver and the WBO, PBO and weighted MaxSAT solver npSolver

Arithmetic and Modularity in Declarative Languages for Knowledge Representation

The past decade has witnessed the development of many important declarative languages for knowledge representation and reasoning such as answer set programming (ASP) languages and languages that extend first-order logic. Also, since these languages depend on background solvers, the recent advancements in the efficiency of solvers has positively affected the usability of such languages. This thesis studies extensions of knowledge representation (KR) languages with arithmetical operators and methods to combine different KR languages. With respect to arithmetic in declarative KR languages, we show that existing KR languages suffer from a huge disparity between their expressiveness and their computational power. Therefore, we develop an ideal KR language that captures the complexity class NP for arithmetical search problems and guarantees universality and efficiency for solving such problems. Moreover, we introduce a framework to language-independently combine modules from different KR languages. We study complexity and expressiveness of our framework and develop algorithms to solve modular systems. We define two semantics for modular systems based on (1) a model-theoretical view and (2) an operational view on modular systems. We prove that our two semantics coincide and also develop mechanisms to approximate answers to modular systems using the operational view. We augment our algorithm these approximation mechanisms to speed up the process of solving modular system. We further generalize our modular framework with supported model semantics that disallows self-justifying models. We show that supported model semantics generalizes our two previous model-theoretical and operational semantics. We compare and contrast the expressiveness of our framework under supported model semantics with another framework for interlinking knowledge bases, i.e., multi-context systems, and prove that supported model semantics generalizes and unifies different semantics of multi-context systems. Motivated by the wide expressiveness of supported models, we also define a new supported equilibrium semantics for multi-context systems and show that supported equilibrium semantics generalizes previous semantics for multi-context systems. Furthermore, we also define supported semantics for propositional programs and show that supported model semnatics generalizes the acclaimed stable model semantics and extends the two celebrated properties of rationality and minimality of intended models beyond the scope of logic programs