Leo Bachmair

Completion Without Failure

Publisher Summary nThis chapter discusses completion without failure. The design of efficient methods for dealing with the equality predicate is one of the major goals in automated theorem proving. Just adding equality axioms almost invariably leads to unacceptable inefficiencies. A number of special methods have been devised for reasoning about equality. Within resolution-based provers, demodulation, that is, using equations in only one direction to rewrite terms to a simpler form, is frequently employed. A complete method for handling equations is paramodulation in which equational consequences are generated by using all equations in both directions. Paramodulation is difficult to control and may produce hosts of irrelevant or redundant formulas. The chapter discusses the purely equational case in which a theory is presented as a set of equations and one is interested in proving a given equation to be valid in that equational theory. In important special cases, validity can be decided using canonical rewrite systems that have the property that all equal terms simplify to an identical form. Deciding validity in theories for which canonical systems are known is thus easy and reasonably efficient.Publisher Summary This chapter discusses completion without failure. The design of efficient methods for dealing with the equality predicate is one of the major goals in automated theorem proving. Just adding equality axioms almost invariably leads to unacceptable inefficiencies. A number of special methods have been devised for reasoning about equality. Within resolution-based provers, demodulation, that is, using equations in only one direction to rewrite terms to a simpler form, is frequently employed. A complete method for handling equations is paramodulation in which equational consequences are generated by using all equations in both directions. Paramodulation is difficult to control and may produce hosts of irrelevant or redundant formulas. The chapter discusses the purely equational case in which a theory is presented as a set of equations and one is interested in proving a given equation to be valid in that equational theory. In important special cases, validity can be decided using canonical rewrite systems that have the property that all equal terms simplify to an identical form. Deciding validity in theories for which canonical systems are known is thus easy and reasonably efficient.

Commutation, transformation, and termination

In this paper we study the use of commutation properties for proving termination of rewrite systems. Commutation properties may be used to prove termination of a combined system R∪S by proving termination of R and S separately. We present termination methods for ordinary and for equational rewrite systems. Commutation is also important for transformation techniques. We outline the application of transforms—mappings from terms to terms—to termination in general, and describe various specific transforms, including transforms for associative-commutative rewrite systems.

Completion for rewriting modulo a congruence

We present completion methods for rewriting modulo a congruence, generalizing previous methods by Peterson and Stickel (1981) and Jouannaud and Kirchner (1986). We formalize our methods as equational inference systems and describe techniques for reasoning about such systems.

On restrictions of ordered paramodulation with simplification

We consider a restricted version of ordered paramodulation, called strict superposition. We show that strict superposition (together with equality resolution) is refutationally complete for Horn clauses, but not for general first-order clauses. Two moderate enrichments of the strict superposition calculus are, however, sufficient to establish refutation completeness. This strictly improves previous results. We also propose a simple semantic notion of redundancy for clauses which covers most simplification and elimination techniques used in practice yet preserves completeness of the proposed calculi. The paper introduces a new and comparatively simple technique for completeness proofs based on the use of canonical rewrite systems to represent equality interpretations.

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.

Equational inference, canonical proofs, and proof orderings

We describe the application of proof orderings—a technique for reasoning about inference systems-to various rewrite-based theorem-proving methods, including refinements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huets procedure for rewriting modulo a congruence; ordered completion (a refutationally complete extension of standard completion); and a proof by consistency procedure for proving inductive theorems.

Associative-Commutative Discrimination Nets

Use of discrimination nets for many-to-one pattern matching has been shown to dramatically improve the performance of the Knuth-Bendix completion procedure used in rewriting. Many important applications of rewriting require associative-commutative (AC) function symbols and it is therefore quite natural to expect performance gains by using similar techniques for AC-completion. In this paper we propose such a technique, called AC-discrimination net, that is a natural generalization of the standard discrimination net in the sense that if no AC-symbols are present in the pattern, it specializes to the standard discrimination net. Moreover we show how AC-discrimination nets can be augmented so as to further improve the performance of AC-matching on problems that are typically seen in practice.

Ordered chaining calculi for first-order theories of transitive relations

We propose inference systems for binary relations that satisfy composition laws such as transitivity. Our inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods as they are used in the context of resolution-type theorem proving. We establish the refutational completeness of these calculi and prove that our methods are compatible with the usual simplification techniques employed in refutational theorem provers, such as subsumption or tautology deletion. Various optimizations of the basic chaining calculus will be discussed for theories with equality and for total orderings. A key to the practicality of chaining methods is the extent to which so-called variable chaining can be avoided. We demonstrate that rewrite techniques considerably restrict variable chaining and that further restrictions are possible if the transitive relation under consideration satisfies additional properties, such as symmetry. But we also show that variable chaining cannot be completely avoided in general.

Critical pair criteria for completion

We formulate the Knuth-Bendix completion method at an abstract level, as an equational inference system, and formalize the notion of critical pair criterion using orderings on equational proofs. We prove the correctness of standard completion and verify all known criteria for completion, including those for which correctness had not been established previously. What distinguishes our approach from others is that our results apply to a large class of completion procedures, not just to a particular version. Proof ordering techniques therefore provide a basis for the design and verification of specific completion procedures (with or without criteria).

Elimination of Equality via Transformation with Ordering Constraints

We refine Brands method for eliminating equality axioms by (i) imposing ordering constraints on auxiliary variables introduced during the transformation process and (ii) avoiding certain transformations of positive equations with a variable on one side. The refinements are both of theoretical and practical interest. For instance, the second refinement is implemented in Setheo and appears to be critical for that provers performance on equational problems. The correctness of this variant of Brands method was an open problem that is solved by the more general results in the present paper. The experimental results we obtained from a prototype implementation of our proposed method also show some dramatic improvements of the proof search in model elimination theorem proving. We prove the correctness of our refinements of Brands method by establishing a suitable connection to basic paramodulation calculi and thereby shed new light on the connection between different approaches to equational theorem proving.