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15
InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 821 (23 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
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 140 (21 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 Completion Procedure for Computing a Canonical Basis for a kSubalgebra
 IN COMPUTERS AND MATHEMATICS
, 1989
"... A completion procedure for computing a canonical basis for a ksubalgebra is proposed. Using this canonical basis, the membership problem for a ksubalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewritin ..."
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Cited by 38 (0 self)
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A completion procedure for computing a canonical basis for a ksubalgebra is proposed. Using this canonical basis, the membership problem for a ksubalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewriting concepts. A canonical basis produced by the completion procedure shares many properties of a Grobner basis such as reducing an element of a ksubalgebra to 0 and generating unique normal forms for the equivalence classes generated by a ksubalgebra. In contrast to Shannon and Sweedler's approach using tag variables, this approach is direct. One of the limitations of the approach however is that the procedure may not terminate for some term orderings thus giving an infinite canonical basis. The procedure is illustrated using examples.
Theorem proving using equational matings and rigid E–unification
 Journal of the ACM
"... In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (firstorder) languages with equality. A decidable version of Eunification called rigid Eunification is introduced, and it is shown that the method of equational matings remains complete when used in ..."
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Cited by 38 (2 self)
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In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (firstorder) languages with equality. A decidable version of Eunification called rigid Eunification is introduced, and it is shown that the method of equational matings remains complete when used in conjunction with rigid Eunification. Checking that a family of mated sets is an equational mating is equivalent to the following restricted kind of Eunification. Problem: Given →/E = {Ei  1 ≤ i ≤ n} a family of n finite sets of equations and S = {〈ui, vi 〉  1 ≤ i ≤ n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(Ei) as a set of ground equations (i.e. holding the variables in θ(Ei) "rigid"), θ(ui) and θ(vi) are provably equal from θ(Ei) for i = 1,...,n? Equivalently, is there a substitution θ such that θ(ui) and θ(vi) can be shown congruent from θ(Ei) by the congruence closure method for i 1,..., n? A substitution θ solving the above problem is called a rigid →/Eunifier of S, and a pair (→/E, S) such that S has some rigid →/Eunifier is called an equational premating. It is shown that deciding whether a pair 〈→/E, S 〉 is an
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
Abstract saturationbased inference
 IN PROCEEDINGS OF THE 18TH ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 2003
"... Solving goals—like deciding word problems or resolving constraints—is much easier in some theory presentations than in others. What have been called “completion processes”, in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a g ..."
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Cited by 12 (5 self)
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Solving goals—like deciding word problems or resolving constraints—is much easier in some theory presentations than in others. What have been called “completion processes”, in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a given class of problems. We provide a general prooftheoretic setting within which completionlike processes can be modelled and studied. This framework centers around wellfounded orderings of proofs. It allows for abstract definitions and very general characterizations of saturation processes and redundancy criteria.
Paramodulation with NonMonotonic Orderings
 In 14th IEEE Symposium on Logic in Computer Science (LICS
, 1999
"... All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique ..."
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Cited by 10 (8 self)
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All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS). By a careful further analysis of our technique, we obtain the first KnuthBendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly nontotalizable) reduction ordering Ø whenever it exists 2 . Note that being a reduction ordering is the minimal possible requirement on Ø, since a TRS terminates if, and only if, it is contained in a reduction ordering. Keywords: term rewriting, automated deduction. 1 Introduction Deduction with equality is fundamental in mathematics, logics and many applications of ...
Vademecum of Divergent Term Rewriting Systems
 CRIN, Research Report
, 1990
"... This paper presents two structural patterns to detect divergence of the completion procedure, followed by a detailed overview of dioeerent examples of divergent rewrite systems. Further it introduces five different empirical methods to avoid divergence, applicable during a session with a rewrite rul ..."
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Cited by 5 (1 self)
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This paper presents two structural patterns to detect divergence of the completion procedure, followed by a detailed overview of dioeerent examples of divergent rewrite systems. Further it introduces five different empirical methods to avoid divergence, applicable during a session with a rewrite rule laboratory.