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64
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 Schema", whichgenera ..."
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Cited by 755 (22 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 210 (47 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.
Termination of Term Rewriting: Interpretation and Type Elimination
 Journal of Symbolic Computation
, 1994
"... this paper we introduce the notion of a monotone algebra as the natural concept for semantical methods. Though we focus on `pure' TRS, the ideas are easily extended to conditional TRS, typed TRS and TRS modulo equations. We propose a classification of types of termination based upon the types of ord ..."
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Cited by 85 (13 self)
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this paper we introduce the notion of a monotone algebra as the natural concept for semantical methods. Though we focus on `pure' TRS, the ideas are easily extended to conditional TRS, typed TRS and TRS modulo equations. We propose a classification of types of termination based upon the types of orderings of the underlying monotone algebras. Some remarks and examples are not claimed to be new but are included for completeness and for illustrating the setting of monotone algebras.
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.
Termination of Nested and Mutually Recursive Algorithms
, 1996
"... This paper deals with automated termination analysis for functional programs. Previously developed methods for automated termination proofs of functional programs often fail for algorithms with nested recursion and they cannot handle algorithms with mutual recursion. We show that termination proofs ..."
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Cited by 39 (9 self)
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This paper deals with automated termination analysis for functional programs. Previously developed methods for automated termination proofs of functional programs often fail for algorithms with nested recursion and they cannot handle algorithms with mutual recursion. We show that termination proofs for nested and mutually recursive algorithms can be performed without having to prove the correctness of the algorithms simultaneously. Using this result, nested and mutually recursive algorithms do no longer constitute a special problem and the existing methods for automated termination analysis can be extended to nested and mutual recursion in a straightforward way. We give some examples of algorithms whose termination can now be proved automatically (including wellknown challenge problems such as McCarthy's f_91 function).
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.
Termination of ContextSensitive Rewriting
 Proc. of 8th International Conference on Rewriting Techniques and Applications, RTA'97, LNCS 1232:172186
, 1997
"... Contextsensitive term rewriting is a kind of term rewriting in which reduction is not allowed inside some fixed arguments of some function symbols. We introduce two new techniques for proving termination of contextsensitive rewriting. The first one is a modification of the technique of interpretat ..."
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Cited by 32 (0 self)
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Contextsensitive term rewriting is a kind of term rewriting in which reduction is not allowed inside some fixed arguments of some function symbols. We introduce two new techniques for proving termination of contextsensitive rewriting. The first one is a modification of the technique of interpretation in a wellfounded order, the second one is implied by a transformation in which contextsensitive termination of the original system can be concluded from termination of the transformed one. In combination with purely automatic techniques for proving ordinary termination, the latter technique is purely automatic too. 1 Introduction The function computing the factorial is usually defined as follows: fact(x) = if(x = 0; 1; x fact(x \Gamma 1)); together with some standard rules like if(true; x; y) = x and if(false; x; y) = y. Considered as a term rewriting system however, the rule fact(x) ! if(x = 0; 1; x fact(x \Gamma 1)) is not terminating. Apparently here general term rewriting doe...