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InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 751 (21 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
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 214 (46 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.
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 51 (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
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 ..."
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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.
Decidable Approximations of Sets of Descendants and Sets of Normal Forms
, 1997
"... We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations ..."
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Cited by 42 (13 self)
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We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of programs: we show how to compute a superset of results, to prove the sufficient completeness property, or to find a criterion for proving termination under a specific strategy, the sequential reduction strategy.
Transformation Techniques for ContextSensitive Rewrite Systems
, 2004
"... Contextsensitive rewriting is a computational restriction of term rewriting used to model nonstrict (lazy) evaluation in functional programming. The goal of this paper is the study and development of techniques to analyze the termination behavior of contextsensitive rewrite systems. For that purp ..."
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Cited by 36 (4 self)
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Contextsensitive rewriting is a computational restriction of term rewriting used to model nonstrict (lazy) evaluation in functional programming. The goal of this paper is the study and development of techniques to analyze the termination behavior of contextsensitive rewrite systems. For that purpose, several methods have been proposed in the literature which transform contextsensitive rewrite systems into ordinary rewrite systems such that termination of the transformed ordinary system implies termination of the original contextsensitive system. In this way, the huge variety of existing techniques for termination analysis of ordinary rewriting can be used for contextsensitive rewriting, too. We analyze the existing transformation techniques for proving termination of contextsensitive rewriting and we suggest two new transformations. Our first method is simple, sound, and more powerful than the previously proposed transformations. However, it is not complete, i.e., there are terminating contextsensitive rewrite systems that are transformed into nonterminating term rewrite systems. The second method that we present in this paper is both sound and complete. All these observations also hold for rewriting modulo associativity and commutativity.
Automatically Proving Termination Where Simplification Orderings Fail
, 1997
"... To prove termination of term rewriting systems (TRSs), several methods have been developed to synthesize suitable wellfounded orderings automatically. However, virtually all orderings that are amenable to automation are socalled simplification orderings. Unfortunately, there exist numerous interes ..."
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Cited by 32 (9 self)
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To prove termination of term rewriting systems (TRSs), several methods have been developed to synthesize suitable wellfounded orderings automatically. However, virtually all orderings that are amenable to automation are socalled simplification orderings. Unfortunately, there exist numerous interesting and relevant TRSs that cannot be oriented by orderings of this restricted class and therefore their termination cannot be proved automatically with the existing techniques. In this paper we present a new automatic approach which allows to apply the standard techniques for automated termination proofs to those TRSs where these techniques failed up to now. For that purpose we have developed a procedure which, given a TRS, generates a set of inequalities (constraints) automatically. If there exists a wellfounded ordering satisfying these constraints, then the TRS is terminating. It turns out that for many TRSs where a direct application of standard techniques fails, these standard techniq...
Termination analysis for functional programs using term orderings
 IN PROCEEDINGS OF THE SECOND INTERNATIONAL STATIC ANALYSIS SYMPOSIUM, LNCS 983
, 1995
"... To prove the termination of a functional program there has to be a wellfounded ordering such that the arguments in each recursive call are smaller than the corresponding inputs. In this paper we present a procedure for automated termination proofs of functional programs. In contrast to previously p ..."
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Cited by 31 (12 self)
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To prove the termination of a functional program there has to be a wellfounded ordering such that the arguments in each recursive call are smaller than the corresponding inputs. In this paper we present a procedure for automated termination proofs of functional programs. In contrast to previously presented methods a suited wellfounded ordering does not have to be fixed in advance by the user, but can be synthesized automatically. For that purpose we use approaches developed in the area of term rewriting systems for the automated generation of suited wellfounded term orderings. But unfortunately term orderings cannot be directly used for termination proofs of functional programs which call other algorithms in the arguments of their recursive calls. The reason is that while for the termination of term rewriting systems orderings between terms are needed, for functional programs we need orderings between objects of algebraic data types. Our method solves this problem and enables term orderings to be used for termination proofs of functional programs.
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...