Effective SAT solving

A growing number of problem domains are successfully being tackled by SAT solvers. This thesis contributes to that trend by pushing the state-of-the-art of core SAT algorithms and their implementation, but also in several important application areas. It consists of five papers: the first details the implementation of the SAT solver MiniSat and the other four papers discuss specific issues related to different application domains. In the first paper, catering to the trend of extending and adapting SAT solvers, we present a detailed description of MiniSat, a SAT solver designed for that particular purpose. The description additionally bridges a gap between theory and practice, serving as a tutorial on modern SAT solving algorithms. Among other things, we describe how to solve a series of related SAT problems efficiently, called incremental SAT solving. For finding finite first order models the MACE-style method that is based on SAT solving is well-known. In the second paper we improve the basic method with several techniques that can be loosely classified as either transformations that make the reduction to SAT result in fewer clauses or techniques that are designed to speed up the search of the SAT solver. The resulting tool, called Paradox, won the SAT/Models division of the CASC competition in 2003 and has not been beaten since by a single general purpose model finding tool. In the last decade the interest in methods for safety property verification that are based on SAT solving has been steadily growing. One example of such a method is temporal induction. The method requires a sequence of increasingly stronger induction proofs to be performed. In the third paper we show how this sequence of proofs can be solved efficiently using incremental SAT solving. The last two papers consider two frequently occurring types of encodings: (1) the problem of encoding circuits into CNF, and (2) encoding 0-1 integer linear programming into CNF and how to use incremental SAT to solve the intended ptimization problem. There are several encoding patterns that occur over and over again in this thesis but also elsewhere. The most noteworthy are: incremental SAT, lazy encoding of constraints, and bit-wise encoding of arithmetic influenced by hardware designs for adders and multipliers. The general conclusion is: deploying SAT solvers effectively requires implementations that are efficient, yet easily adaptable to specific application needs. Moreover, to get the best results, it is worth spending effort to make sure that one uses the best codings possible for an application. However, it is important to note that this is not absolutely necessary. For some applications naive problem codings work just fine which is indeed part of the appeal of using SAT solving

SAT-Solving in Practice, with a Tutorial Example from Supervisory Control

Satisfiability solving, the problem of deciding whether the variables of a propositional formula can be assigned in such a way that the formula evaluates to true, is one of the classic problems in computer science. It is of theoretical interest because it is the canonical NP-complete problem. It is of practical interest because modern SAT-solvers can be used to solve many important and practical problems. In this tutorial paper, we show briefly how such SAT-solvers are implemented, and point to some typical applications of them. Our aim is to provide sufficient information (much of it through the reference list) to kick-start researchers from new fields wishing to apply SAT-solvers to their problems. Supervisory control theory originated within the control community and is a framework for reasoning about a plant to be controlled and a specification that the closed-loop system must fulfil. This paper aims to bridge the gap between the computer science community and the control community by illustrating how SAT-based techniques can be used to solve some supervisory control related problems

Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems.Comment: 9 pages; update shortcut constraint discussio

Applications of SAT solving

In the area of formal verification it is well known that there can be no single logic that suits all needs. This insight motivates the diversity of this dissertation: it contains contributions to SAT solving, First Order theorem proving and Model Finding, and Symbolic Model Checking. A growing number of problem domains are successfully being tackled by SAT solvers. Following the current trend of extending and adapting SAT solvers we present a detailed description of a SAT solver designed for that particular purpose. The description bridges a gap between theory and practice, serving as a tutorial on modern SAT solving algorithms. Among other things we describe how to solve a series of related SAT problems efficiently, called incremental SAT solving. For finding finite first order models, the MACE-style method that is based on SAT solving, is well-known. We improve the basic method by several techniques, that can be loosely classified as either transformations that make the reduction to SAT result in fewer clauses, or techniques that are designed to speed up the search of the SAT solver. The resulting tool, called PARADOX, performed well in the SAT division of the CASC-19 competition. Recently, there has been large interest in methods for safety property verification that are based on SAT solving. One example is temporal induction, also called k-induction. The method requires a sequence of increasingly stronger induction proofs to be performed. We show how this sequence of proofs can be solved more efficiently using incremental SAT solving

Translating pseudo-boolean constraints into SAT

In this paper, we describe and evaluate three different techniques for translating pseudoboolean constraints (linear constraints over boolean variables) into clauses that can be handled by a standard SAT-solver. We show that by applying a proper mix of translation techniques, a SAT-solver can perform on a par with the best existing native pseudo-boolean solvers. This is particularly valuable in those cases where the constraint problem of interest is naturally expressed as a SAT problem, except for a handful of constraints. Translating those constraints to get a pure clausal problem will take full advantage of the latest improvements in SAT research. A particularly interesting result of this work is the efficiency of sorting networks to express pseudo-boolean constraints. Although tangential to this presentation, the result gives a suggestion as to how synthesis tools may be modified to produce arithmetic circuits more suitable for SAT based reasoning. Keywords: pseudo-Boolean, SAT-solver, SAT translation, integer linear programmin

Temporal Induction by Incremental SAT Solving

We show how a very modest modi cation to a typical modern SAT-solver enables it to solve a series of related SAT-instances eciently

MINISAT v2.0 (Beta)

This document describes the current state of the open source SAT solver MINISAT. The latest version brings new features that was proven to be successful during the SAT competition 2005, namely the variable elimination based simplification techniques of SATELITEGTI (winner of the industrial category). The resulting solver is mainly intended to be easier to use, understand, and extend — therefore performance is estimated to be roughly similar to that o

Applying Logic Synthesis for Speeding Up SAT

Abstract. SAT solvers are often challenged with very hard problems that remain unsolved after hours of CPU time. The research community meets the challenge in two ways: (1) by improving the SAT solver technology, for example, perfecting heuristics for variable ordering, and (2) by inventing new ways of constructing simpler SAT problems, either using domain specific information during the translation from the original problem to CNF, or by applying a more universal CNF simplification procedure after the translation. This paper explores preprocessing of circuitbased SAT problems using recent advances in logic synthesis. Two fast logic synthesis techniques are considered: DAG-aware logic minimization and a novel type of structural technology mapping, which reduces the size of the CNF derived from the circuit. These techniques are experimentally compared to CNF-based preprocessing. The conclusion is that the proposed techniques are complementary to CNF-based preprocessing and speedup SAT solving substantially on industrial examples.

New Techniques that Improve MACE-style Finite Model Finding

We describe a new method for nding nite models of unsorted rst-order logic clause sets. The method is a MACE-style method, i.e. it "attens" the rst-order clauses, and for increasing model sizes, instantiates the resulting clauses into propositional clauses which are consecutively solved by a SAT-solver. We enhance the standard method by using 4 novel techniques: term de nitions, which reduce the number of variables in attened clauses, incremental SAT, which enables reuse of search information between consecutive model sizes, static symmetry reduction, which reduces the number of isomorphic models by adding extra constraints to the SAT problem, and sort inference, which allows the symmetry reduction to be applied at a ner grain. All techniques have been implemented in a new model nder, called Paradox, with very promising results