Results 1 
5 of
5
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1
Cancellative Abelian Monoids in Refutational Theorem Proving
 PHD THESIS, INSTITUT FÜR INFORMATIK, UNIVERSITÄT DES SAARLANDES
, 1997
"... ..."
(Show Context)
A Generalized Approach to Equational Unification
 MIT LABORATORY FOR COMPUTER SCIENCE
, 1985
"... ..."
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
Generic Completion (Version 1)
, 1998
"... Abstract rewrite systems define a reduction relation by a set of rules. An important aspect of such rewrite relations is their behavior in an arbitrary context associated with the underlying congruence relation. This behavior is described by "compatibility properties": a rewrite system is ..."
Abstract
 Add to MetaCart
Abstract rewrite systems define a reduction relation by a set of rules. An important aspect of such rewrite relations is their behavior in an arbitrary context associated with the underlying congruence relation. This behavior is described by "compatibility properties": a rewrite system is compatible if applying the same context to two objects preserves the rewrite relation among them. We present a generic completion procedure for abstract rewrite systems and characterize the minimal compatibility requirement for rewrite relations that are necessary to describe completion procedures as proof transformation procedures. This requirement (called sufficient compatibility) subsumes the class of semicompatible rewrite relations that play an important role in algebraic completion. In order to allow for noncompatible rewrite relations we must replace the concept of equations (critical pairs) by promises which leads to a new proof transformation relation and a new proof ordering for abstract r...