Obtaining Memory-Efficient Solutions to Boolean Equation Systems

AbstractThis paper is concerned with memory-efficient solution techniques for Boolean fixed-point equations. We show how certain structures of fixed-point equation systems, often encountered in solving verification problems, can be exploited in order to substantially improve the performance of fixed-point computations. Also, we investigate the space complexity of the problem of solving Boolean equation systems, showing a NL-hardness result. A prototype of the proposed technique has been implemented and experimental results on a series of protocol verification benchmarks are reported

Techniques for solving Boolean equation systems

Boolean equation systems are ordered sequences of Boolean equations decorated with least and greatest fixpoint operators. Boolean equation systems provide a useful framework for formal verification because various specification and verification problems, for instance, μ-calculus model checking can be represented as the problem of solving Boolean equation systems. The general problem of solving a Boolean equation system is a computationally hard task, and no polynomial time solution technique for the problem has been discovered so far. In this thesis, techniques for finding solutions to Boolean equation systems are studied and new methods for solving such systems are devised. The thesis presents a general framework that allows for dividing Boolean equation systems into individual blocks and solving these blocks in isolation with special techniques. Three special techniques are presented, namely: (i) new specialized algorithms for disjunctive and conjunctive form Boolean equation systems, (ii) a new encoding of a general form Boolean equation system into answer set programming, and (iii) new encodings of a general form Boolean equation systems into satisfiability problems. The approaches (ii) and (iii) are motivated by the recent success of answer set programming solvers and satisfiability solvers in formal verification. First, the thesis presents especially fast solution algorithms for disjunctive and conjunctive classes of Boolean equation systems. These special algorithms are useful because many practically relevant model checking problems can be represented as Boolean equation systems that are disjunctive or conjunctive. The new algorithms have been implemented and the performance of the algorithms has been compared experimentally on communication protocol verification examples. Second, the thesis gives a translation of the problem of solving a general form Boolean equation system into the problem of finding a stable model of a logic program. The translation allows to use implementations of answer set programming solvers to solve Boolean equation systems. Experimental tests have been performed using the presented approach and these experiments indicate the usefulness of answer set programming in this problem domain. Third, the thesis presents reductions from the problem of solving general form Boolean equation systems to the satisfiability problems of difference logic and propositional logic. The reductions allow to use implementations of satisfiability solvers to solve Boolean equation systems. The presented reductions have been implemented and it is shown via experiments that the new approach leads to practically efficient methods to solve general Boolean equation systems.Boolen yhtälöryhmät ovat kiintopisteoperaattoreilla varustettuja Boolen yhtälöitä. Boolen yhtälöryhmät luovat hyödyllisen viitekehyksen tietokoneavusteiselle verifioinnille, sillä monet määrittely- ja verifiointiongelmat voidaan kuvata tällaisten kiintopisteyhtälöiden avulla. Työssä kehitetään uusia menetelmiä Boolen yhtälöryhmien ratkaisemiseen. Työssä esitetään yleinen viitekehys Boolen yhtälöryhmien ratkaisemiseen, joka yksinkertaistaa ratkaisun laskemista jakamalla yhtälöryhmät yksinkertaisempiin aliongelmiin. Työssä esitetään kolme uutta mentelmää Boolen yhtälöryhmien ratkaisemiseen. Konjunktiivisten ja disjunktiivisten Boolen yhtälöryhmien ratkaisemiseen kehitetään uusia algoritmeja, sekä esitetään näiden toteutukset ja suorituskykyjä koskevia koetuloksia. Työssä kehitetään käännös Boolen yhtälöryhmän ratkaisemisesta logiikkaohjelman stabiilin mallin löytämiseen sekä menetelmän toimivuutta koskevia koetuloksia. Käännös mahdollistaa logiikkaohjelmointiympäristöjen toteutusten käytön Boolen yhtälöryhmien ratkaisemiseen. Koetulokset osoittavat rajoitepohjaisen logiikkaohjelmointiympäristön tehokkuuden Boolen yhtälöryhmien ratkaisemisessa. Työssä kehitetään myös käännökset Boolen yhtälöryhmän ratkaisemisesta differenssilogiikan sekä lauselogiikan toteutuvuusongelmiin. Käännökset mahdollistavat toteutuvuustarkastimien käytön Boolen yhtälöryhmien ratkaisemiseen. Koetulokset osoittavat esitettyjen menetelmien tehokkuuden Boolen yhtälöryhmien ratkaisemisessa.reviewe

SOLVING BOOLEAN EQUATION SYSTEMS

ABSTRACT: Boolean equation systems are ordered sequences of Boolean equations decorated with fixpoint operators. Boolean equation systems provide a useful framework for computer aided verification because various specification and verification problems can be encoded as the problem of solving such fixpoint equation systems. In this work, techniques for finding solutions to Boolean equation systems are studied, and new methods for solving such systems are devised. An approach to solve a general form Boolean equation system, which simplifies the process of finding the solution by dividing the system into more simple subsystems and solving these by optimized procedures, is introduced and analyzed. New solution algorithms for disjunctive and conjunctive classes of Boolean equation systems are presented, together with an implementation and experimental evaluation of a solver for these classes. A novel translation of the problem of solving a general form Boolean equation system into the problem of finding a stable model of a logic program is given. The translation allows to use an implementation of an answe

TECHNIQUES FOR SOLVING BOOLEAN EQUATION SYSTEMS

ABSTRACT: Boolean equation systems are ordered sequences of Boolean equations decorated with least and greatest fixpoint operators. Boolean equation systems provide a useful framework for formal verification because various specification and verification problems, for instance, µ-calculus model checking can be represented as the problem of solving Boolean equation systems. The general problem of solving a Boolean equation system is a computationally hard task, and no polynomial time solution technique for the problem has been discovered so far. In this thesis, techniques for finding solutions to Boolean equation systems are studied and new methods for solving such systems are devised. The thesis presents a general framework that allows for dividing Boolean equation systems into individual blocks and solving these blocks in isolation with special techniques. Three special techniques are presented, namely

Solving Alternating Boolean Equation Systems in Answer Set Programming

Abstract. In this paper we apply answer set programming to solve alternating Boolean equation systems. We develop a novel characterization of solutions for variables in disjunctive and conjunctive Boolean equation systems. Based on this we devise a mapping from Boolean equation systems with alternating fixed points to normal logic programs such that the solution of a given variable of an equation system can be determined by the existence of a stable model of the corresponding logic program. The technique can be used to model check alternating formulas of modal µ-calculus.

Solving Alternating Boolean Equation Systems in Answer Set Programming

In this paper we apply answer set programming to solve alternating Boolean equation systems. We develop a novel characterization of solutions for variables in disjunctive and conjunctive Boolean equation systems. Based on this we devise a mapping from Boolean equation systems with alternating fixed points to normal logic programs such that the solution of a given variable of an equation system can be determined by the existence of a stable model of the corresponding logic program