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38
Completion Without Failure
, 1989
"... We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational the ..."
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Cited by 121 (18 self)
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We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational theories. The method can also be applied to Horn clauses with equality, in which case it corresponds to positive unit resolution plus oriented paramodulation, with unrestricted simplification.
A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier)
"... Recently, Craig Squier introduced the notion of finite derivation type to show that some finitely presentable monoid has no presentation by means of a finite complete rewriting system. A similar result was already obtained by the same author using homology, but the new method is more direct and more ..."
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Cited by 19 (4 self)
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Recently, Craig Squier introduced the notion of finite derivation type to show that some finitely presentable monoid has no presentation by means of a finite complete rewriting system. A similar result was already obtained by the same author using homology, but the new method is more direct and more powerful. Here, we present Squier's argument with a bit of categorical machinery, making proofs shorter and easier. In addition we prove that, if a monoid has finite derivation type, then its third homology group is of finite type. An invariant for a structure is something which can be calculated in many ways, but only depends on the structure itself. Typical examples are the dimension of a vector space or the genus of a surface. Squier's finiteness condition for monoids is of this kind: It can be defined in terms of a finite presentation, but does not depend on the choice of this presentation. To begin with a simpler case, consider the following theorem, which is not hard to prove: If M is...
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
Open Problems in Rewriting
 Proceeding of the Fifth International Conference on Rewriting Techniques and Application (Montreal, Canada), LNCS 690
, 1991
"... Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27 ..."
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Cited by 19 (2 self)
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Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation
Integrating linear arithmetic into superposition calculus
 In Computer Science Logic (CSL’07
, 2007
"... Abstract. We present a method of integrating linear rational arithmetic into superposition calculus for firstorder logic. One of our main results is completeness of the resulting calculus under some finiteness assumptions. 1 ..."
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Cited by 17 (3 self)
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Abstract. We present a method of integrating linear rational arithmetic into superposition calculus for firstorder logic. One of our main results is completeness of the resulting calculus under some finiteness assumptions. 1
String rewriting and Gröbner bases  a general approach to monoid and group rings
 Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita
, 1995
"... 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 tech ..."
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Cited by 15 (5 self)
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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 semiThue system. For certain presentations, including free groups and contextfree groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...
Rewriting Modulo a Rewrite System
, 1995
"... . We introduce rewriting with two sets of rules, the first interpreted equationally and the second not. A semantic view considers equational rules as defining an equational theory and reduction rules as defining a rewrite relation modulo this theory. An operational view considers both sets of rules ..."
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Cited by 13 (3 self)
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. We introduce rewriting with two sets of rules, the first interpreted equationally and the second not. A semantic view considers equational rules as defining an equational theory and reduction rules as defining a rewrite relation modulo this theory. An operational view considers both sets of rules as similar. We introduce sufficient properties for these two views to be equivalent (up to different notions of equivalence). The paper ends with a collection of example showing the effectiveness of this approach. Rewriting can be viewed simultaneously as the most basic symbolmanipulating method, and as a very expressive specification framework, given the expressive power of rewriting modulo equations. It is a primary candidate to the role of a general logical framework [Mes92, MOM93]. Historically, rewriting has been given an equational semantics, saying that a rewrite rule u \Gamma! v is interpreted as u is equal to v. This is the case for instance when defining functions or solving the w...
Superposition Theorem Proving for Abelian Groups Represented as Integer Modules
 THEORETICAL COMPUTER SCIENCE
, 1996
"... We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equation ..."
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Cited by 13 (4 self)
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We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the lefthand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the ACunification problems which are generated. That is, ACunification is not necessary at the top of a term, only below some nonACsymbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible.
Tame combings, almost convexity, and rewriting systems for groups
 Math. Z
, 1997
"... Abstract: A finite complete rewriting system for a group is a finite presentation which gives an algorithmic solution to the word problem. Finite complete rewriting systems have proven to be useful objects in geometric group theory, yet little is known about the geometry of groups admitting such rew ..."
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Cited by 8 (5 self)
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Abstract: A finite complete rewriting system for a group is a finite presentation which gives an algorithmic solution to the word problem. Finite complete rewriting systems have proven to be useful objects in geometric group theory, yet little is known about the geometry of groups admitting such rewriting systems. We show that a group G with a finite complete rewriting system admits a tame 1combing; it follows (by work of Mihalik and Tschantz) that if G is an infinite fundamental group of a closed irreducible 3manifold M, then the universal cover of M is R 3. We also establish that a group admitting a geodesic rewriting system is almost convex in the sense of Cannon, and that almost convex groups are tame
Combining Algebra and Universal Algebra in FirstOrder Theorem Proving: The Case of Commutative Rings
 In Proc. 10th Workshop on Specification of Abstract Data Types
, 1995
"... . We present a general approach for integrating certain mathematical structures in firstorder equational theorem provers. More specifically, we consider theorem proving problems specified by sets of firstorder clauses that contain the axioms of a commutative ring with a unit element. Associativeco ..."
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Cited by 7 (4 self)
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. We present a general approach for integrating certain mathematical structures in firstorder equational theorem provers. More specifically, we consider theorem proving problems specified by sets of firstorder clauses that contain the axioms of a commutative ring with a unit element. Associativecommutative superposition forms the deductive core of our method, while a convergent rewrite system for commutative rings provides a starting point for more specialized inferences tailored to the given class of formulas. We adopt ideas from the Grobner basis method to show that many inferences of the superposition calculus are redundant. This result is obtained by the judicious application of the simplification techniques afforded by convergent rewriting and by a process called symmetrization that embeds inferences between single clauses and ring axioms. 1 Introduction 1.1 Motivation Specifications of programs include both symbols with their usual mathematical meaning as well as additional f...