This paper introduces a new hybrid method for efficiently integrating Pseudo-Boolean (PB) constraints into generic SAT solvers in order to solve PB satisfiability and optimization problems. To achieve this, we adopt the cutting-plane technique to draw inferences among PB constraints and combine it with generic implication graph analysis for conflictinduced learning. Novel features of our approach include a light-weight and efficient hybrid learning and backjumping strategy for analyzing PB constraints and CNF clauses in order to simultaneously learn both a CNF clause and a PB constraint with minimum overhead and use both to determine the backtrack level. Several techniques for handling the original and learned PB constraints are introduced. Overall, our method benefits significantly from the pruning power of the learned PB constraints, while keeping the overhead of adding them into the problem low. In this paper, we also address two other methods for solving PB problems, namely Integer Linear Programming (ILP) and pre-processing to CNF SAT, and present a thorough comparison between them and our hybrid method. Experimental comparison of our method against other hybrid approaches is also demonstrated. Additionally, we provide details of the MiniSAT-based implementation of our solver Pueblo to enable the reader to construct a similar one.

Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents

The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,

This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying zap, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the Davis-Putnam-Logemann-Loveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method. The third paper in this series discusses zap’s implementation and presents experimental performance results. 1.

We present a novel low-overhead framework for encoding and utilizing structural symmetry in propositional satisfiability algorithms (SAT solvers). We use the notion of complete multi-class symmetry and demonstrate the efficacy of our technique through a solver SymChaff that achieves exponential speedup by using simple tags in the specification of problems from both theory and practice. Efficient implementations of DPLL-based SAT solvers are routinely used in areas as diverse as planning, scheduling, design automation, model checking, verification, testing, and algebra. A natural feature of many application domains is the presence of symmetry, such as that amongst all trucks at a certain location in logistics planning and all wires connecting two switch boxes in an FPGA circuit. Many of these problems turn out to have a concise description in many-sorted first order logic. This description can be easily specified by the problem designer and almost as easily inferred automatically. SymChaff, an extension of the popular SAT solver zChaff, uses information obtained from the “sorts” in the first order logic constraints to create symmetry sets that are used to partition variables into classes and to maintain and utilize symmetry information dynamically. Current approaches designed to handle symmetry include: (A) symmetry breaking predicates (SBPs), (B) pseudo-Boolean solvers with implicit representation for counting, (C) modifications of DPLL that handle symmetry dynamically, and (D) techniques based on ZBDDs. SBPs are prohibitively many, often large, and expensive to compute for problems such as the ones we report experimental results for. Pseudo-Boolean solvers are provably exponentially slow in certain symmetric situations and their implicit counting representation is not always appropriate. Suggested modifications of DPLL either work on limited global symmetry and are difficult to extend, or involve expensive algebraic group computations. Finally, techniques based on ZBDDs often do not compare well even with ordinary DPLL-based solvers. Sym-Chaff addresses and overcomes most of these limitations.

This is the third of three papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern highperformance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal has been to define a representation in which this structure is apparent and can be exploited to improve computational performance. The first paper surveyed existing work that (knowingly or not) exploited problem structure to improve the performance of satisfiability engines, and the second paper showed that this structure could be understood in terms of groups of permutations acting on individual clauses in any particular Boolean theory. We conclude the series by discussing the techniques needed to implement our ideas, and by reporting on their performance on a variety of problem instances. 1.

ir. M.J.H. Heule dr. H. van Maaren prof.dr.ir. H.J. Sips

Tools for modeling and solving search

Representing and reasoning about time dependent information is a key research issue in many areas of computer science and artificial intelligence. One of the best known and widely used formalisms for representing interval-based qualitative temporal information is Allen’s interval algebra (IA). The fundamental reasoning task in IA is to find a scenario that is consistent with the given information. This problem is in general NP-complete. In this paper, we investigate how an interval-based representation, or IA network, can be encoded into a propositional formula of Boolean variables and/or predicates in decidable theories. Our task is to discover whether satisfying such a formula can be more efficient than finding a consistent scenario for the original problem. There are two basic approaches to modelling an IA network: one represents the relations between intervals as variables and the other represents the end-points of each interval as variables. By combining these two approaches with three different Boolean satisfiability (SAT) encoding schemes, we produced six encoding schemes for converting IA to SAT. In addition, we also showed how IA networks can be formulated into satisfiability modulo theories (SMT) formulae based on the quantifier-free integer difference logic (QF-IDL). These encodings were empirically studied using randomly generated IA problems of sizes

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