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Generalized Sufficient Conditions for Modular Termination of Rewriting
 IN ENGINEERING, COMMUNICATION AND COMPUTING
, 1992
"... Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential cou ..."
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Cited by 49 (7 self)
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Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union R1 \Phi R2 of two (finitely branching) terminating term rewriting systems R1 , R2 is nonterminating, then one of the systems, say R1 , enjoys an interesting (undecidable) property, namely it is not termination preserving under nondeterministic collapses, i.e. R1 \Phi fG(x; y) ! x; G(x; y) ! yg is nonterminating, and the other system R2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of pote...
Relating Innermost, Weak, Uniform and Modular Termination of Term Rewriting Systems
, 1993
"... We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of ..."
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Cited by 28 (5 self)
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We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of term rewriting systems. These criteria provide interesting possibilities to infer completeness, i.e. termination plus confluence, from restricted termination and confluence properties. Using these basic results we are also able to prove some new results about modular termination of rewriting. In particular, we show that termination is modular for some classes of innermost terminating and locally confluent term rewriting systems, namely for nonoverlapping and even for overlay systems. As an easy consequence this latter result also entails a simplified proof of the fact that completeness is a decomposable property of socalled constructor systems. Furthermore we show how to obtain similar re...
Modular Termination of Term Rewriting Systems Revisited
, 1995
"... This paper is concerned with the impact of stepwise development methodologies on prototyping. ..."
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Cited by 25 (12 self)
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This paper is concerned with the impact of stepwise development methodologies on prototyping.
Modular & Incremental Proofs of ACTermination
 Journal of Symbolic Computation
, 2002
"... Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of ACdependency pairs. ..."
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Cited by 15 (3 self)
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Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of ACdependency pairs.
Sufficient Conditions for Modular Termination of Conditional Term Rewriting Systems
 In Proceedings of the 3rd International Workshop on Conditional Term Rewriting Systems
, 1993
"... . Recently we have shown the following abstract result for unconditional term rewriting systems (TRSs). Whenever the disjoint union R1 \Phi R2 of two (finite) terminating TRSs R1 , R2 is nonterminating, then one of the systems, say R1 , enjoys an interesting property, namely it is not termination ..."
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Cited by 13 (4 self)
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. Recently we have shown the following abstract result for unconditional term rewriting systems (TRSs). Whenever the disjoint union R1 \Phi R2 of two (finite) terminating TRSs R1 , R2 is nonterminating, then one of the systems, say R1 , enjoys an interesting property, namely it is not termination preserving under nondeterministic collapses, i.e. R1 \Phi fG(x; y) ! x; G(x; y) ! yg is nonterminating, and the other system R2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient syntactical criteria for modular termination of rewriting. Here we extend this result and derived sufficient criteria for modularity of termination to the case of conditional term rewriting systems (CTRSs). Moreover we relate various definitions of notions related to termination of CTRSs to each other and discuss some subtleties and problems concerning extra variables in the rules. 1 Introduction From a theoretical point of view and also for efficiency ...
Simple Termination is Difficult
 Applicable Algebra in Engineering, Communication and Computing
, 1993
"... A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termi ..."
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Cited by 5 (1 self)
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A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termination. We show that simple termination is an undecidable property, even for onerule systems. This contradicts a result by Jouannaud and Kirchner. The proof is based on the ingenious construction of Dauchet who showed the undecidability of termination for onerule systems. Our results may be summarized as follows: being simply terminating, (non)selfembedding, and (non)looping are undecidable properties of orthogonal, variable preserving, onerule constructor systems. 1. Introduction It is wellknown that termination is an undecidable property of term rewriting systems. This result was obtained by Huet and Lankford [9] in 1978. They showed that every Turing machine can be coded as a strin...
Combinations of Simplifying Conditional Term Rewriting Systems
 In Proceedings of the 3rd International Workshop on Conditional Term Rewriting Systems
, 1992
"... . A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the lefthand side of any rewrite rule is greater than the righthand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of ..."
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Cited by 4 (2 self)
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. A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the lefthand side of any rewrite rule is greater than the righthand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of finitely many rules, it is terminating and the applicability of a rewrite rule is decidable by recursively reducing the terms in the conditions. Consider two finite CTRSs R1 and R2 which may share constructors (symbols that do not occur at the root position of the lefthand side of any rewrite rule) but no other function symbols. It will be shown that the combined CTRS R = R1[R2 is simplifying if and only if R1 and R2 are simplifying. Moreover, confluence is a modular property of finite simplifying join CTRSs. 1 Introduction During the past decade, term rewriting has gained an enormous importance in fields of computer science concerned with symbolic manipulation. Among others, it may be vie...