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31
Termination of Term Rewriting Using Dependency Pairs
 Comput. Sci
, 2000
"... We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left and righthand sides of rewrite rules, but introduce the notion of dependency pairs to compare lefthand sides with special subter ..."
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Cited by 266 (49 self)
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We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left and righthand sides of rewrite rules, but introduce the notion of dependency pairs to compare lefthand sides with special subterms of the righthand sides. This results in a technique which allows to apply existing methods for automated termination proofs to term rewriting systems where they failed up to now. In particular, there are numerous term rewriting systems where a direct termination proof with simplification orderings is not possible, but in combination with our technique, wellknown simplification orderings (such as the recursive path ordering, polynomial orderings, or the KnuthBendix ordering) can now be used to prove termination automatically. Unlike previous methods, our technique for proving innermost termination automatically can also be applied to prove innermost termination of term rewriting systems that are not terminating. Moreover, as innermost termination implies termination for certain classes of term rewriting systems, this technique can also be used for termination proofs of such systems.
The SizeChange Principle for Program Termination
, 2001
"... The \sizechange termination" principle for a rstorder functional language with wellfounded data is: a program terminates on all inputs if every innite call sequence (following program control ow) would cause an innite descent in some data values. Sizechange analysis is based only on local ..."
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Cited by 212 (15 self)
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The \sizechange termination" principle for a rstorder functional language with wellfounded data is: a program terminates on all inputs if every innite call sequence (following program control ow) would cause an innite descent in some data values. Sizechange analysis is based only on local approximations to parameter size changes derivable from program syntax. The set of innite call sequences that follow program ow and can be recognized as causing innite descent is an !regular set, representable by a Buchi automaton. Algorithms for such automata can be used to decide sizechange termination. We also give a direct algorithm operating on \sizechange graphs" (without the passage to automata). Compared to other results in the literature, termination analysis based on the sizechange principle is surprisingly simple and general: lexical orders (also called lexicographic orders), indirect function calls and permuted arguments (descent that is not insitu) are all handled auto...
ELAN from a rewriting logic point of view
 Theoretical Computer Science
, 2002
"... ELAN implements computational systems, a concept that combines two first class entities: rewrite rules and rewriting strategies. ELAN can be used either as a logical framework or to describe and execute deterministic as well as nondeterministic rule based processes. With the general goal to make pr ..."
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Cited by 60 (5 self)
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ELAN implements computational systems, a concept that combines two first class entities: rewrite rules and rewriting strategies. ELAN can be used either as a logical framework or to describe and execute deterministic as well as nondeterministic rule based processes. With the general goal to make precise a rewriting logic based semantics of ELAN, this paper has three contributions: a presentation of the concepts of rules and strategies available in ELAN, an expression of rewrite rules with matching conditions in conditional rewriting logic, and finally an enrichment mechanism of a rewrite theory into a strategy theory in conditional rewriting logic.
Argument Filtering Transformation
 In Proc. 1st PPDP, LNCS 1702
, 1999
"... To simplify the task of proving termination of term rewriting systems, several elimination methods, such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this paper, we show that the argument lter ..."
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Cited by 51 (2 self)
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To simplify the task of proving termination of term rewriting systems, several elimination methods, such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this paper, we show that the argument ltering method combining with the dependency pair technique is essential in all the above elimination methods. We present remarkable simple proofs for the soundness of these elimination methods based on this observation. Moreover, we propose a new elimination method, called the argument ltering transformation, which is not only more powerful than all the other elimination methods but also especially useful to make clear the essential relation hidden behind these methods.
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.
An extension of dependency pair method for proving termination of higherorder rewrite systems
 IEICE Trans. on Information and Systems
, 2001
"... Abstract. This paper explores how to extend the dependency pair technique for proving termination of higherorder rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the nonexistence of an infinite Rchain of the dependency pairs. However, the t ..."
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Cited by 28 (2 self)
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Abstract. This paper explores how to extend the dependency pair technique for proving termination of higherorder rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the nonexistence of an infinite Rchain of the dependency pairs. However, the termination and the nonexistence of an infinite Rchain do not coincide in the higherorder case. We introduce a new notion of dependency forest that characterize infinite reductions and infinite Rchains, and show that the termination property of higherorder rewrite systems R can be checked by showing the nonexistence of an infinite Rchain, if R is strongly linear or nonnested. 1
On Proving Left Termination of Constraint Logic Programs
 ACM Transaction on Computational Logic
, 2001
"... The Constraint Logic Programming (CLP) Scheme merges logic programming with constraint solving over predefined domains. In this paper, we study proof methods for universal left termination of constraint logic programs. We provide a sound and complete characterization of left termination for ideal CL ..."
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Cited by 21 (8 self)
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The Constraint Logic Programming (CLP) Scheme merges logic programming with constraint solving over predefined domains. In this paper, we study proof methods for universal left termination of constraint logic programs. We provide a sound and complete characterization of left termination for ideal CLP languages which generalizes acceptability of logic programs. The characterization is then refined to the notion of partial acceptability, which is wellsuited for automatic modular inference. We describe a theoretical framework for automation of the approach, which is implemented. For nonideal CLP languages and without any assumption on their incomplete constraint solvers, even the most basic sound termination criterion from logic programming does not lift. We focus on a specific system, namely CLP(R), by proposing some additional conditions that make (partial) acceptability sound
Modularity of Termination Using Dependency Pairs
 PROC. 9TH RTA
, 1997
"... The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before. In this paper we present a refinement of this framework in order to prove termination in a modular way. Our modularity results significantly incre ..."
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Cited by 18 (10 self)
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The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before. In this paper we present a refinement of this framework in order to prove termination in a modular way. Our modularity results significantly increase the class of term rewriting systems where termination resp. innermost termination can be proved automatically. Moreover, the modular approach to dependency pairs yields new modularity criteria which extend previous results in this area considerably. In particular, existing results for modularity of innermost termination can easily be obtained as direct consequences of our new criteria.
Automated Incremental Termination Proofs for Hierarchically Defined Term Rewriting Systems
 In Proc. IJCAR 2001, LNAI 2083
, 2001
"... We propose the notion of rewriting modules in order to provide a structural and hierarchical approach of TRS. We define then relative dependency pairs built upon these modules which allow us to perform termination proofs incrementally. Important results can be expressed in that new framework (reg ..."
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Cited by 15 (2 self)
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We propose the notion of rewriting modules in order to provide a structural and hierarchical approach of TRS. We define then relative dependency pairs built upon these modules which allow us to perform termination proofs incrementally. Important results can be expressed in that new framework (regarding CE termination for instance), and with help of extendable orderings, we give effective new incremental methods for proving termination particularly suited for automation. 1
Termination of AssociativeCommutative Rewriting by Dependency Pairs
 9th International Conference on Rewriting Techniques and Applications, volume 1379 of Lecture
, 1998
"... A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak ACcompatible reduction orderings which may be used in this criterion. ..."
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Cited by 14 (1 self)
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A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak ACcompatible reduction orderings which may be used in this criterion.