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25
Complete Monotonic Semantic Path Orderings
 In Proc. 17th CADE, LNAI 1831
, 2000
"... Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination ..."
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Cited by 31 (8 self)
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Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and generalpurpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several nontrivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO. 1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest po...
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 & Incremental Automated Termination Proofs
 Journal of Automated Reasoning
, 2004
"... We propose a modular approach of term rewriting systems, making the best of their hierarchical structure. We define rewriting modules and then provide a new method to prove termination incrementally. We obtain new and powerful termination criteria for standard rewriting. Our policy of restraining ..."
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Cited by 24 (5 self)
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We propose a modular approach of term rewriting systems, making the best of their hierarchical structure. We define rewriting modules and then provide a new method to prove termination incrementally. We obtain new and powerful termination criteria for standard rewriting. Our policy of restraining termination itself (thus relaxing constraints over hierarchies components) together with the generality of the module approach are sufficient to express previous results and methods the premisses of which either include restrictions over unions or make a particular reduction strategy compulsory.
On Proving Termination by Innermost Termination
 In Proc. 7th RTA, LNCS 1103
, 1996
"... We present a new approach for proving termination of rewrite systems by innermost termination. From the resulting abstract criterion we derive concrete conditions, based on critical peak properties, under which innermost termination implies termination (and confluence). Finally, we show how to apply ..."
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Cited by 20 (0 self)
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We present a new approach for proving termination of rewrite systems by innermost termination. From the resulting abstract criterion we derive concrete conditions, based on critical peak properties, under which innermost termination implies termination (and confluence). Finally, we show how to apply the main results for providing new sufficient conditions for the modularity of termination.
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...
Solution to the Problem of Zantema on a Persistent Property of Term Rewriting Systems
 Proc. 7th International Conf. on Algebraic and Logic Programming, LNCS, 1490
, 1998
"... . A property P of term rewriting systems is persistent if for any manysorted term rewriting system R, R has the property P iff its underlying term rewriting system 2(R), which results from R by omitting its sort information, has the property P . It is shown that termination is a persistent property ..."
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Cited by 16 (0 self)
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. A property P of term rewriting systems is persistent if for any manysorted term rewriting system R, R has the property P iff its underlying term rewriting system 2(R), which results from R by omitting its sort information, has the property P . It is shown that termination is a persistent property of manysorted term rewriting systems that contain only variables of the same sort. This is the positive solution to a problem of Zantema, which has been appeared as Rewriting Open Problem 60 in literature. 1 Introduction Sort introduction technique in term rewriting has caught attention recently [2][8]. We prove in this paper a conjecture which opens up a possibility of new applications of this technique. The conjecture reads: for any terminating manysorted term rewriting system R, if R contains only variables of the same sort then 2(R) is also terminating. Here, 2(R)called the underlying term rewriting system of Ris the term rewriting system obtained from R by omitting its sort i...
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 13 (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
Completeness of Combinations of Conditional Constructor Systems
, 1994
"... this paper we extend the divide and conquer technique of Middeldorp and Toyama for establishing (semi)completeness of constructor systems to conditional constructor systems. We show that both completeness (i.e. the combination of confluence and strong normalisation) and semicompleteness (confluenc ..."
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Cited by 12 (2 self)
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this paper we extend the divide and conquer technique of Middeldorp and Toyama for establishing (semi)completeness of constructor systems to conditional constructor systems. We show that both completeness (i.e. the combination of confluence and strong normalisation) and semicompleteness (confluence plus weak normalisation) are decomposable properties of conditional constructor systems without extra variables in the conditions of the rewrite rules. 1. Introduction
Pushing the Frontiers of Combining Rewrite Systems Farther Outwards
 In Proceedings of the Second International Workshop on Frontiers of Combining Systems, FroCos '98, Applied Logic Series
, 1998
"... It is well known that simple termination is modular for certain kinds of combinations of term rewriting systems (TRSs). This result is of practical relevance because most techniques for (automated) termination proofs use simplification orderings, so they show in fact simple termination. On the other ..."
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Cited by 11 (3 self)
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It is well known that simple termination is modular for certain kinds of combinations of term rewriting systems (TRSs). This result is of practical relevance because most techniques for (automated) termination proofs use simplification orderings, so they show in fact simple termination. On the other hand, in practice many systems are nonsimply terminating. In order to cope with such systems, Arts and Giesl developed the dependency pair approach. By using (quasi)simplification orderings in combination with dependency pairs, it is possible to prove termination of nonsimply terminating systems automatically. It is natural to ask whether modularity of simple termination can be extended to the class of those systems which can be handled by this technique. In this paper we show that this is indeed the case. In this way, the class of TRSs for which termination can be proved in a modular way is extended significantly. 1 Introduction Modularity is a wellknown paradigm in computer science. ...