Knuth-Bendix Research Articles

Chain properties of rule closures

Abstract This article introduces a generalisation of the crossed rule approach to the detection of Knuth-Bendix completion procedure divergence. It introduces closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a suitable formalism for progress in this direction. Supporting substitution algebra is developed first, followed by considerations concerning rule closures in general, concluding with an investigation of closure chain properties. Issues concerning the narrowing process are not discussed here.

Fast Knuth-Bendix completion with a term rewriting system compiler

A term rewriting system compiler can greatly improve the execution speed of reductions by transforming rewriting rules into target code. In this report, we present a new application of the term rewriting system compiler: the Knuth-Bendix completion algorithm. The compiling technique proposed in this algorithm, is dynamic in the sense that rewriting rules are repeatedly compiled in the completion process. The execution time of the completion with dynamic compiling is ten or more times as fast as that obtained with a traditional term rewriting system interpreter.

On the Knuth-Bendix completion for concurrent processes

We consider critical pairs for replacement systems over free partially commutative monoids. This is done in order to apply the Knuth-Bendix completion procedure to concurrent processes. We will see that there are one-rule systems which have no finite set of critical pairs. Therefore we develop a sufficient (and computable) condition such that finite trace replacement systems have a finite set of critical pairs. This condition is always satisfied in the purely free case of semi-Thue systems or purely commutative case of vector replacement systems. In fact, it generalizes (and unifies) both cases. We will give examples of how one can use this generalization.

Automatic proofs by induction in theories without constructors

Inductionless induction consists of using pure equational reasoning for proving the validity of an equation in the initial algebra of a set of equational axioms, which would normally require some kind of induction. Under given hypotheses, the equation is valid iff adding it to the set of axioms does not result in an inconsistency. This inconsistency can be found by the Knuth-Bendix completion algorithm, provided that the signature of the algebra is split into free constructors and defined symbols, which must be completely defined in terms of constructors. This is the base of the so-called inductive completion algorithm of Huet and Hullot. Two key concepts, inductive reducibility and inductive co-reducibility , allow us to extend these techniques in various directions: incomplete specifications, nonfree constructors, no constructors specified, equational term rewriting systems. The method is adapted for proving the consistency property of an enrichment of a specification by new operators and new equations. In addition, we get also a simple algorithm to exhibit a set of constructors of a specification. Finally, inductive co-reducibility is reduced to inductive reducibility and an algorithm for deciding inductive reducibility is given for left linear term rewriting systems.

Existence, Uniqueness, and Construction of Rewrite Systems

The construction of term-rewriting systems, specifically by the Knuth–Bendix completion procedure, is considered. We look for conditions that might ensure the existence of a finite canonical rewriting system for a given equational theory and that might guarantee that the completion procedure will find it. We define several notions of equivalence between rewriting systems in the ordinary and modulo case, and examine uniqueness of systems and the need for backtracking in implementing completion.

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).

Only prime superpositions need be considered in the Knuth-Bendix completion procedure

The Knuth and Bendix test for local confluence of a term rewriting system involves generating superpositions of the left-hand sides, and for each superposition deriving a critical pair of terms and checking whether these terms reduce to the same term. We prove that certain superpositions, which are called composite because they can be split into other superpositions, do not have to be subjected to the critical-pair-joinability test; it suffices to consider only prime superpositions. As a corollary, this result settles a conjecture of Lankford that unblocked superpositions can be omitted. To prove the result, we introduce new concepts and proof techniques which appear useful for other proofs relating to the Church-Rosser property. This test has been implemented in the completion procedures for ordinary term rewriting systems as well as term rewriting systems with associative-commutative operators. Performance of the completion procedures with this test and without the test are compared on a number of examples in the Rewrite Rule Laboratory (RRL) being developed at General Electric Research and Development Center.

Knuth-Bendix Algorithm and It's Applications.

Knuth-Bendix Algorithm and It's Applications.

The Knuth-Bendix Completion Procedure and Thue Systems

The Knuth-Bendix completion procedure for term rewriting systems in many cases provides a decision procedure for equational theories and has been found to have many applications in various areas. We discuss the application of the Knuth-Bendix procedure to Thue systems. We use the notion of a reduced Thue system and show that for every Church-Rosser Thue system, there is a unique reduced Church-Rosser Thue system equivalent to it. Furthermore, the Knuth-Bendix completion procedure, when applied to a Thue system T, always produces the finite reduced Church-Rosser Thue system equivalent to T whenever such a system exists. Similar results can also be proved for almost-confluent Thue systems. Using properties of reduced Church-Rosser systems, we develop conditions under which a class of special Thue systems have equivalent finite Church-Rosser systems. In addition, we show that the completion procedure always terminates on finite parenthesized Thue systems, from which the termination of the completion procedur...

Rewrite systems on a lattice of types

Re-writing systems for partial algebras are developed by modifying the Knuth-Bendix completion algorithm to permit the use of latticestructured domains. Some problems with the original algorithm, such as the treatment of division rings, are overcome conveniently by this means. The use of a type lattice also gives a natural framework for specifying data types in Computer Science without over-specifying error situations. The soundness and meaning of the major concepts involved in re-writing systems are reviewed when applied to such structures.

Refutational theorem proving using term-rewriting systems

In this paper we propose a new approach to theorem proving in first-order logic based on the term-rewriting method. First for propositional calculus, we introduce a canonical term-rewriting system for Boolean algebra. This system enables us to transform the first-order predicate calculus into a form of equational logic, and to develop several complete strategies (both clausal and nonclausal) for first-order theories based on the Knuth-Bendix Completion Procedure. More importantly, our strategies can deal with predicate logic and built-in (equational) theories in a uniform and effective way. We also describe an implementation and comparisons with some other first-order theorem-proving methods.

A criterion for eliminating unnecessary reductions in the Knuth-Bendix algorithm

Given a finite set E of multivariate polynomials, the second author's 1965 algorithm constructs a set E' of polynomials such that E and E' generate the same ideal congruence, but E' has the additional property that the congruence can be decided by "reduction to normal forms w.r.t. E'." The two essential ideas of this algorithm are: the consideration of certain "critical pairs" associated with elements in E and the successive "completion" of E by adjoining the difference of normal forms of critical pairs.

About the rewriting systems produced by the Knuth-Bendix completion algorithm

We show that in a free F-algebra endowed with a strict order we can associate with a complete rewriting system in the sense of Knuth-Bendix an equivalent proper (minimal) complete system. Furthermore, if the complete system is compatible with a strict order, then the equivalent proper complete system is unique.

Some reduction strategies for algebraic term rewriting

A general algorithm for reduction with algebraic term rewriting systems is presented and several aspects of rewriting are discussed.It turns out that the well known strategies of innermost and outermost rewriting, which are special cases of the general algorithm, exhibit good performance under different aspects. The general rewriting algorithm suggests improvements for naive reduction algorithms with specific strategies, which in some cases give them better asymptotic behaviour. Empirical results with an algebraic axiomatization of natural numbers show considerably faster runtimes for the improved algorithms.The algorithms presented here do not impose any restrictions on the rewriting systems. They are currently used both within an implementation of the Knuth-Bendix completion algorithm and in a separate term reduction system, whenever sufficiently large terms are involved.

Proofs by induction in equational theories with constructors

We show how to prove (and disprove) theorems in the initial algebra of an equational variety by a simple extension of the Knuth-Bendix completion algorithm. This allows us to prove by purely equational reasoning theorems whose proof usually requires induction. We show applications of this method to proofs of programs computing over data structures, and to proofs of algebraic-summation identities. This work extends and simplifies recent results of Musser, “Proc., 7th Annual ACM Symposium on Principles of Programming Languages,” Las Vegas, January 1980, pp. 154–162, and Goguen, “Proc., 5th Conf. on Automated Deduction,” Les Arcs, July 1980, pp. 356–373.

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

We give in this paper a complete description of the Knuth-Bendix completion algorithm. We prove its correctness in full, isolating carefully the essential abstract notions, so that the proof may be extended to other versions and extensions of the basic algorithm. We show that it defines a semidecision algorithm for the validity problem in the equational theories for which it applies, yielding a decision procedure whenever the algorithm terminates.