We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's 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. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

Following Buchberger's approach to computing a Gröbner basis of a polynomial ideal in polynomial rings, a completion procedure for finitely generated right ideals in Z[H] is given, where H is an ordered monoid presented by a finite, convergent semi-Thue system (\Sigma; T). Taking a finite set F ` Z[H] we get a (possibly infinite) basis of the right ideal generated by F, such that using this basis we have unique normal forms for all p 2 Z[H] (especially the normal form is 0 in case p is an element of the right ideal generated by F). As the ordering and multiplication on H need not be compatible, reduction has to be defined carefully in order to make it Noetherian. Further we no longer have p \Delta x! p 0 for p 2 Z[H]; x 2 H. Similar to Buchberger's s-polynomials, confluence criteria are developed and a completion procedure is given. In case T = ; or (\Sigma; T) is a convergent, 2--monadic presentation of a group providing inverses of length 1 for the generators or (\Sigma; T) is a convergent presentation of a commutative monoid, termination can be shown. So in this cases finitely generated right ideals admit finite Gröbner bases. The connection to the subgroup problem is discussed.

The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...

A first explicit connection between finitely presented commutative monoids and ideals in polynomial rings was used 1958 by Emelichev yielding a solution to the word problem in commutative monoids by deciding the ideal membership problem. The aim of this paper is to show in a similar fashion how congruences on monoids and groups can be characterized by ideals in respective monoid and group rings. These characterizations enable to transfer well known results from the theory of string rewriting systems for presenting monoids and groups to the algebraic setting of subalgebras and ideals in monoid respectively group rings. Moreover, natural one-sided congruences defined by subgroups of a group are connected to one-sided ideals in the respective group ring and hence the subgroup problem and the ideal membership problem are directly related. For several classes of finitely presented groups we show explicitly how Gröbner basis methods are related to existing solutions of the subgroup problem by rewriting methods. For the case of general monoids and submonoids weaker results are presented. In fact it becomes clear that string rewriting methods for monoids and groups can be lifted in a natural fashion to define reduction relations in monoid and group rings.

Abstract. Abstract canonical systems and inference (ACSI) were introduced to formalize the intuitive notions of good proof and good inference appearing typically in first-order logic or in Knuth-Bendix like completion procedures. Since this abstract framework is intended to be generic, it is of fundamental interest to show its adequacy to represent the main systems of interest. This has been done for ground completion (where all equational axioms are ground) but was still an open question for the general completion process. By showing that the standard completion is an instance of the ACSI framework we close the question. For this purpose, two proof representations, proof terms and proofs by replacement, are compared to built canonical system framework. Classification: Logic in computer science, rewriting and deduction, completion, good proof, proof representation, canonicity.

der Technischen Fakult"at der Universit"at des Saarlandes Saarbr"ucken

We present an algorithm to decide whether a homogeneous linear partial difference equation with constant coefficients provides an unfalsified model for a finite set of observations, which consist in multiindexed signals, known on a finite subset of N n.To this aim we introduce the concept of “generalized term order ” and extend the theory of Gröbner bases accordingly. c ○ 1996 Academic Press Limited 1.

In this paper we present an automated proof for the confluence of a rewrite system for integro-differential operators (given in Table 1). We also outline a generic prototype implementation of the integro-differential polynomials—the key tool for this proof—realized using the Theorema system. With its generic functor mechanism—detailed in Section 2—we are able to provide a formalization of the theory of integrodifferential

We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.

We present novel Gröbner basis algorithms based on saturation loops used by modern superposition theorem provers. We illustrate the practical value of the algorithms through an experimental implementation within the Z3 SMT solver.