Termination orderings for associative-commutative rewriting systems

In this paper we describe a new class of orderings—associative path orderings—for proving termination of associative-commutative term rewriting systems .These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to E - congruence classes, where E is an equational theory consisting of associativity and commutativity axioms. Associative path orderings are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition,the associative path condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings,boolean algebra,etc.) and, in addition, point out ways to handle situations where the associative path condition is too restrictive

Efficient Encodings of First-Order Horn Formulas in Equational Logic

We present several translations from first-order Horn formulas to equational logic. The goal of these translations is to allow equational theorem provers to efficiently reason about non-equational problems. Using these translations we were able to solve 37 problems of rating 1.0 (i.e. which had not previously been automatically solved) from the TPTP

Formalizing Bachmair and Ganzinger’s Ordered Resolution Prover

We present a formalization of the first half of Bachmair and Ganzinger’s chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapter’s text, emphasizing the value of formal proofs in the field of automated reasoning

Superposition as a logical glue

The typical mathematical language systematically exploits notational and logical abuses whose resolution requires not just the knowledge of domain specific notation and conventions, but not trivial skills in the given mathematical discipline. A large part of this background knowledge is expressed in form of equalities and isomorphisms, allowing mathematicians to freely move between different incarnations of the same entity without even mentioning the transformation. Providing ITP-systems with similar capabilities seems to be a major way to improve their intelligence, and to ease the communication between the user and the machine. The present paper discusses our experience of integration of a superposition calculus within the Matita interactive prover, providing in particular a very flexible, "smart" application tactic, and a simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311

Basic completion strategies as another application of the Maude strategy language

The two levels of data and actions on those data provided by the separation between equations and rules in rewriting logic are completed by a third level of strategies to control the application of those actions. This level is implemented on top of Maude as a strategy language, which has been successfully used in a wide range of applications. First we summarize the Maude strategy language design and review some of its applications; then, we describe a new case study, namely the description of completion procedures as transition rules + control, as proposed by Lescanne.Comment: In Proceedings WRS 2011, arXiv:1204.531

Revisiting Enumerative Instantiation

International audienceFormal methods applications often rely on SMT solvers to automatically discharge proof obligations. SMT solvers handle quantified formulas using incomplete heuristic techniques like E-matching, and often resort to model-based quantifier instantiation (MBQI) when these techniques fail. This paper revisits enumerative instantiation, a technique that considers instantiations based on exhaustive enumeration of ground terms. Although simple, we argue that enumer-ative instantiation can supplement other instantiation techniques and be a viable alternative to MBQI for valid proof obligations. We first present a stronger Her-brand Theorem, better suited as a basis for the instantiation loop used in SMT solvers; it furthermore requires considering less instances than classical Herbrand instantiation. Based on this result, we present different strategies for combining enumerative instantiation with other instantiation techniques in an effective way. The experimental evaluation shows that the implementation of these new techniques in the SMT solver CVC4 leads to significant improvements in several benchmark libraries, including many stemming from verification efforts

Strict Basic Superposition and Chaining

The most efficient techniques that have been developed to date for equality handling in first-order theorem proving are based on superposition calculi. Superposition is a refinement of paramodulation in that various ordering constraints are imposed on inferences. For practical purposes, a key aspect of superposition is its compatibility with powerful simplification techniques. In this paper we solve a long-standing open problem by showing that strict superposition---that is, superposition without equality factoring---is refutationally complete. The difficulty of the problem arises from the fact that the strict calculus, in contrast to the standard calculus with equality factoring, is not compatible with arbitrary removal of tautologies, so that the usual techniques for proving the (refutational) completeness of paramodulation calculi are not directly applicable. We deal with the problem by introducing a suitable notion of direct rewrite proof and modifying proof techniques based on can..

Authors ’ Addresses

The most efficient techniques that have been developed to date for equality handling in first-order theorem proving are based on superposition calculi. Superposition is a refinement of paramodulation in that various ordering constraints are imposed on inferences. For practical purposes, a key aspect of superposition is its compatibility with powerful simplification techniques. In this paper we solve a long-standing open problem by showing that strict superposition—that is, superposition without equality factoring—is refutationally complete. The difficulty of the problem arises from the fact that the strict calculus, in contrast to the standard calculus with equality factoring, is not compatible with arbitrary removal of tautologies, so that the usual techniques for proving the (refutational) completeness of paramodulation calculi are not directly applicable. We deal with the problem by introducing a suitable notion of direct rewrite proof and modifying proof techniques based on candidate models and counterexamples in that we define these concepts, no

An LPO-based Termination Ordering for Higher-Order Terms without lambda-abstraction

. We present a new precedence-based termination ordering for (polymorphic) higher-order terms without -abstraction. The ordering has been designed to strictly generalize the lexicographic path ordering (on first-order terms). It is relatively simple, but can be used to prove termination of many higher-order rewrite systems, especially those corresponding to typical functional programs. We establish the relevant properties of the ordering, include a number of examples, and also discuss certain limitations of the ordering and possible extensions. 1 Introduction Specification and interactive reasoning systems, such as HOL [10], Isabelle [12], or PVS [11], typically employ some expressive, higher-order logic as their basis specification formalism. Rewrite techniques, in this context, provide a computational mechanism for the simplification or evaluation of expressions and form an essential part of the (equational) reasoning component of such systems. Termination is one of the key properti..