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1987], `Termination of rewriting systems by polynomial interpretations and its implementation', Science of Computer Programming
"... Abstract. This paper describes the actual implementation in the rewrite rule laboratory REVE of an elementary procedure that checks inequalities between polynomials and is used for proving termination of rewriting systems, especially in the more difficult case of associativecommutative rewriting sy ..."
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Abstract. This paper describes the actual implementation in the rewrite rule laboratory REVE of an elementary procedure that checks inequalities between polynomials and is used for proving termination of rewriting systems, especially in the more difficult case of associativecommutative rewriting systems, for which a complete characterization is given. 1. The origin of the problem Termination is central in programming and in particular in termrewriting systems, the latter being both a theoretical and a practical basis for functional and logic languages. Indeed the problem is not only a key for ensuring that a program and its procedures eventually produce the expected result, it is also important in concur rent programming where liveness results rely on termination of the components. Termrewriting systems are also used for proving equational theorems and are a basic tool for checking specifications of abstract data types. Again, the termination problem is crucial in the implementation of the Kn~hBendix algorithm, which tests the local confluence and needs the termination to be able to infer the total confluence. Termination is also necessary to direct equations properly. Until now, methods based on recursive path ordering were satisfactory [8,19], but when we recently ran experiments on transformation of FP programs [2], we were faced with a problem that the recursive path ordering could not handle. The problem, motivated by a simple example of code optimization, is just Associativity + Endomorphism: (XI * x2) * X3 = XI * (x2 * x 3), f(x I * x2) = f(xJ * f(x 2). The variables are functions, * is the composition and f is a mapcarlike operator. In order to optimize the program, the user wants to decrease the number of uses * This work was supported by the Greco de Programmation.
Completion of Rewrite Systems with Membership Constraints Part II: Constraint Solving
 J. Symbolic Computation
, 1992
"... this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. Thi ..."
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Cited by 71 (2 self)
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this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed,
Equational Formulae with Membership Constraints
 Information and Computation
, 1994
"... We propose a set of transformation rules for first order formulae whose atoms are either equations between terms or "membership constraints" t 2 i. i can be interpreted as a regular tree language (i is called a sort in the algebraic specification community) or as a tree language in any cla ..."
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Cited by 38 (3 self)
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We propose a set of transformation rules for first order formulae whose atoms are either equations between terms or "membership constraints" t 2 i. i can be interpreted as a regular tree language (i is called a sort in the algebraic specification community) or as a tree language in any class of languages which satisfies some adequate closure and decidability properties. This set of rules is proved to be correct, terminating and complete. This provides a quantifier elimination procedure: for every regular tree language L, the first order theory of some structure defining L is decidable. This extends the results of Mal'cev (1971), Maher (1988), Comon and Lescanne (1989). We also show how to apply our results to automatic inductive proofs in equational theories. Introduction To unify two terms s and t means to turn the equation s = t into an equivalent solved form which is either ? (this means that s = t has no solution, or, in other words, that s and t are not unifiable) or else a form...
Termination of Term Rewriting
, 2000
"... Contents 1 Introduction 2 2 Semantical methods 3 2.1 Wellfounded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicograp ..."
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Cited by 32 (6 self)
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Contents 1 Introduction 2 2 Semantical methods 3 2.1 Wellfounded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicographic combinations . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 A hierarchy of termination 17 3.1 Simple termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Total termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 The hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Syntactical methods 25 4.1 Recursive path order . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Justi cation of recursive path order . . . . . . . . . . . . . . . . . . . 30 4.3 Extensions of recursive path order . . . . . . . . . . . . . . . . . . . 36 4.
A fully syntactic ACRPO
 Information and Computation
, 1999
"... . We present the first fully syntactic (i.e., noninterpretationbased) ACcompatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is ACtotal, and defined uniformly for both ground and nonground ter ..."
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Cited by 27 (4 self)
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. We present the first fully syntactic (i.e., noninterpretationbased) ACcompatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is ACtotal, and defined uniformly for both ground and nonground terms, as well as for partial precedences. More importantly, it is the first one that can deal incrementally with partial precedences, an aspect that is essential, together with its intuitive behaviour, for interactive applications like KnuthBendix completion. 1
Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations
, 1996
"... this paper is to make the similarity between KnuthBendix completion and the Buchberger algorithm explicit, by describing a general algorithm called Snormalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equatio ..."
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Cited by 27 (0 self)
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this paper is to make the similarity between KnuthBendix completion and the Buchberger algorithm explicit, by describing a general algorithm called Snormalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations 3 instances of this general algorithm for a particular choice of S. This has been achieved in two steps.
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 ...
Any Ground AssociativeCommutative Theory Has a Finite Canonical System
 Proceedings 4th Conference on Rewriting Techniques and Applications
, 1991
"... We show that theories presented by a set of ground equations with several associativecommutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering which is ACcompatible and total on the set of congruence classes generated ..."
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Cited by 18 (4 self)
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We show that theories presented by a set of ground equations with several associativecommutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering which is ACcompatible and total on the set of congruence classes generated by the associativity and commutativity axioms. As far as we know, this is the first ordering with such properties, when several AC function symbols and free function symbols are allowed. Such an ordering is also a fundamental tool for deriving complete theorem proving strategies with builtin associative commutative unification.