A complete proof of correctness of the Knuth-Bendix completion algorithm

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A Formalization of the Theorem of Existence of First-Order Most General Unifiers

This work presents a formalization of the theorem of existence of most general unifiers in first-order signatures in the higher-order proof assistant PVS. The distinguishing feature of this formalization is that it remains close to the textbook proofs that are based on proving the correctness of the well-known Robinson's first-order unification algorithm. The formalization was applied inside a PVS development for term rewriting systems that provides a complete formalization of the Knuth-Bendix Critical Pair theorem, among other relevant theorems of the theory of rewriting. In addition, the formalization methodology has been proved of practical use in order to verify the correctness of unification algorithms in the style of the original Robinson's unification algorithm.Comment: In Proceedings LSFA 2011, arXiv:1203.542

Knuth-Bendix Completion with Modern Termination Checking, Master\u27s Thesis, August 2006

Knuth-Bendix completion is a technique for equational automated theorem proving based on term rewriting. This classic procedure is parametrized by an equational theory and a (well-founded) reduction order used at runtime to ensure termination of intermediate rewriting systems. Any reduction order can be used in principle, but modern completion tools typically implement only a few classes of such orders (e.g., recursive path orders and polynomial orders). Consequently, the theories for which completion can possibly succeed are limited to those compatible with an instance of an implemented class of orders. Finding and specifying a compatible order, even among a small number of classes, is challenging in practice and crucial to the success of the method. In this thesis, a new variant on the Knuth-Bendix completion procedure is developed in which no order is provided by the user. Modern termination-checking methods are instead used to verify termination of rewriting systems. We prove the new method correct and also present an implementation called Slothrop which obtains solutions for theories that do not admit typical orders and that have not previously been solved by a fully automatic tool

AC-KBO Revisited

Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous. Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms. In this paper we consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.Comment: 31 pages, To appear in Theory and Practice of Logic Programming (TPLP) special issue for the 12th International Symposium on Functional and Logic Programming (FLOPS 2014

On ground word problem of term equation systems

We give semi-decision procedures for the ground word problem of variable preserving term equation systems and term equation systems. They are natural improvements of two well known trivial semi-decision procedures. We show the correctness of our procedures

Infinite Runs in Abstract Completion

Completion is one of the first and most studied techniques in term rewriting and fundamental to automated reasoning with equalities. In an earlier paper we presented a new and formalized correctness proof of abstract completion for finite runs. In this paper we extend our analysis and our formalization to infinite runs, resulting in a new proof that fair infinite runs produce complete presentations of the initial equations. We further consider ordered completion - an important extension of completion that aims to produce ground-complete presentations of the initial equations. Moreover, we revisit and extend results of Métivier concerning canonicity of rewrite systems. All proofs presented in the paper have been formalized in Isabelle/HOL