Results 1  10
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30
Modular Properties of Composable Term Rewriting Systems
 Journal of Symbolic Computation
, 1995
"... this paper we prove several new modularity results for unconditional and conditional term rewriting systems. Most of the known modularity results for the former systems hold for disjoint or constructorsharing combinations. Here we focus on a more general kind of combination: socalled composable sy ..."
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Cited by 50 (6 self)
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this paper we prove several new modularity results for unconditional and conditional term rewriting systems. Most of the known modularity results for the former systems hold for disjoint or constructorsharing combinations. Here we focus on a more general kind of combination: socalled composable systems. As far as conditional term rewriting systems are concerned, all known modularity result but one apply only to disjoint systems. Here we investigate conditional systems which may share constructors. Furthermore, we refute a conjecture of Middeldorp (1990, 1993). 1. Introduction
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...
Generating Polynomial Orderings for Termination Proofs
 In Proc. 6th RTA, LNCS 914
, 1995
"... Most systems for the automation of termination proofs using polynomial orderings are only semiautomatic, i.e. the "right" polynomial ordering has to be given by the user. We show that a variation of Lankford's partial derivative technique leads to an easier and slightly more powerful method than mo ..."
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Cited by 46 (22 self)
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Most systems for the automation of termination proofs using polynomial orderings are only semiautomatic, i.e. the "right" polynomial ordering has to be given by the user. We show that a variation of Lankford's partial derivative technique leads to an easier and slightly more powerful method than most other semiautomatic approaches. Based on this technique we develop a method for the automated synthesis of a suited polynomial ordering.
Completeness of Combinations of Constructor Systems
 Journal of Symbolic Computation
, 1993
"... this paper we show that it is sufficient to impose the constructor discipline for obtaining the modularity of completeness. This result is a simple consequence of a quite powerful divide and conquer technique for establishing completeness of such constructor systems. Our approach is not limited to s ..."
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Cited by 31 (2 self)
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this paper we show that it is sufficient to impose the constructor discipline for obtaining the modularity of completeness. This result is a simple consequence of a quite powerful divide and conquer technique for establishing completeness of such constructor systems. Our approach is not limited to systems which are composed of disjoint parts. The importance of our method is that we may decompose a given constructor system into parts which possibly share function symbols and rewrite rules in order to infer completeness. We obtain a similar technique for semicompleteness, i.e. the combination of confluence and weak normalisation. 1. Introduction
On the Modularity of Termination of Term Rewriting Systems
 Theoretical Computer Science
, 1993
"... It is wellknown that termination is not a modular property of term rewriting systems, i.e., it is not preserved under disjoint union. The objective of this paper is to provide a "uniform framework" for sufficient conditions which ensure the modularity of termination. We will prove the following res ..."
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Cited by 29 (3 self)
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It is wellknown that termination is not a modular property of term rewriting systems, i.e., it is not preserved under disjoint union. The objective of this paper is to provide a "uniform framework" for sufficient conditions which ensure the modularity of termination. We will prove the following result. Whenever the disjoint union of two terminating term rewriting systems is nonterminating, then one of the systems is not C E terminating (i.e., it looses its termination property when extended with the rules Cons(x; y) ! x and Cons(x; y) ! y) and the other is collapsing. This result has already been achieved by Gramlich [7] for finitely branching term rewriting systems. A more sophisticated approach is necessary, however, to prove it in full generality. Most of the known sufficient criteria for the preservation of termination [24, 15, 13, 7] follow as corollaries from our result, and new criteria are derived. This paper particularly settles the open question whether simple termination ...
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 27 (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...
Unravelings and Ultraproperties
 In Proceedings of the Fifth International Conference on Algebraic and Logic Programming (ALP'96), volume 1139 of LNCS
, 1996
"... Conditional rewriting is universally recognized as being much more complicated than unconditional rewriting. In this paper we study how much of conditional rewriting can be automatically inferred from the simpler theory of unconditional rewriting. We introduce a new tool, called unraveling, to autom ..."
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Cited by 26 (3 self)
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Conditional rewriting is universally recognized as being much more complicated than unconditional rewriting. In this paper we study how much of conditional rewriting can be automatically inferred from the simpler theory of unconditional rewriting. We introduce a new tool, called unraveling, to automatically translate a conditional term rewriting system (CTRS) into a term rewriting system (TRS). An unraveling enables to infer properties of a CTRS by studying the corresponding ultraproperty on the corresponding TRS. We show how to rediscover properties like decreasingness, and to give easy proofs of some existing results on CTRSs. Moreover, we show how unravelings provide a valuable tool to study modularity of CTRSs, automatically giving a multitude of new results.
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.
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 inc ..."
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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. 1 Introduction Termination is one of the most important properties of a term rewriting system (TRS). While in general this problem is undecidable [HL78], several methods for proving termination have been developed (for surveys see e.g. [Der87, Ste95b, DH95]). However, most methods that are amenable to a...
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 ...