Strict General Setting for Building Decision Procedures into Theorem Provers

The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of well-known or new schemes within this framework a very easy task even for a non-expert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas

A Comparison of Decision Procedures in Presburger Arithmetic

It is part of the tradition and folklore of automated reasoning that the intractability of Cooper's decision procedure for Presburger integer arithmetic makes is too expensive for practical use. More than 25 years of work has resulted in numerous approximate procedures via rational arithmetic, all of which are incomplete and restricted to the quantifier-free fragment. In this paper we report on an experiment which strongly questions this tradition. We measured the performance of procedures due to Hodes, Cooper (and heuristic variants thereof which detect counterexamples), across a corpus of 10 000 randomly generated quantifierfree Presburger formulae. The results are startling: a variant of Cooper's procedure outperforms Hodes' procedure on both valid and invalid formulae, and is fast enough for practical use. These results contradict much perceived wisdom that decision procedures for integer arithmetic are too expensive to use in practice. 1 Introduction A decis..

URSA: A System for Uniform Reduction to SAT

There are a huge number of problems, from various areas, being solved by reducing them to SAT. However, for many applications, translation into SAT is performed by specialized, problem-specific tools. In this paper we describe a new system for uniform solving of a wide class of problems by reducing them to SAT. The system uses a new specification language URSA that combines imperative and declarative programming paradigms. The reduction to SAT is defined precisely by the semantics of the specification language. The domain of the approach is wide (e.g., many NP-complete problems can be simply specified and then solved by the system) and there are problems easily solvable by the proposed system, while they can be hardly solved by using other programming languages or constraint programming systems. So, the system can be seen not only as a tool for solving problems by reducing them to SAT, but also as a general-purpose constraint solving system (for finite domains). In this paper, we also describe an open-source implementation of the described approach. The performed experiments suggest that the system is competitive to state-of-the-art related modelling systems

URSA: A System for Uniform Reduction to SAT

There are a huge number of problems, from various areas, being solved byreducing them to SAT. However, for many applications, translation into SAT isperformed by specialized, problem-specific tools. In this paper we describe anew system for uniform solving of a wide class of problems by reducing them toSAT. The system uses a new specification language URSA that combines imperativeand declarative programming paradigms. The reduction to SAT is definedprecisely by the semantics of the specification language. The domain of theapproach is wide (e.g., many NP-complete problems can be simply specified andthen solved by the system) and there are problems easily solvable by theproposed system, while they can be hardly solved by using other programminglanguages or constraint programming systems. So, the system can be seen notonly as a tool for solving problems by reducing them to SAT, but also as ageneral-purpose constraint solving system (for finite domains). In this paper,we also describe an open-source implementation of the described approach. Theperformed experiments suggest that the system is competitive tostate-of-the-art related modelling systems.Comment: 39 pages, uses tikz.st

Formalization of Abstract State Transition Systems for SAT

We present a formalization of modern SAT solvers and their properties in a form of abstract state transition systems. SAT solving procedures are described as transition relations over states that represent the values of the solver's global variables. Several different SAT solvers are formalized, including both the classical DPLL procedure and its state-of-the-art successors. The formalization is made within the Isabelle/HOL system and the total correctness (soundness, termination, completeness) is shown for each presented system (with respect to a simple notion of satisfiability that can be manually checked). The systems are defined in a general way and cover procedures used in a wide range of modern SAT solvers. Our formalization builds up on the previous work on state transition systems for SAT, but it gives machine-verifiable proofs, somewhat more general specifications, and weaker assumptions that ensure the key correctness properties. The presented proofs of formal correctness of the transition systems can be used as a key building block in proving correctness of SAT solvers by using other verification approaches

Formalization of Abstract State Transition Systems for SAT

We present a formalization of modern SAT solvers and their properties in aform of abstract state transition systems. SAT solving procedures are describedas transition relations over states that represent the values of the solver'sglobal variables. Several different SAT solvers are formalized, including boththe classical DPLL procedure and its state-of-the-art successors. Theformalization is made within the Isabelle/HOL system and the total correctness(soundness, termination, completeness) is shown for each presented system (withrespect to a simple notion of satisfiability that can be manually checked). Thesystems are defined in a general way and cover procedures used in a wide rangeof modern SAT solvers. Our formalization builds up on the previous work onstate transition systems for SAT, but it gives machine-verifiable proofs,somewhat more general specifications, and weaker assumptions that ensure thekey correctness properties. The presented proofs of formal correctness of thetransition systems can be used as a key building block in proving correctnessof SAT solvers by using other verification approaches