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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 252 (49 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.
Proving and Disproving Termination of HigherOrder Functions
 IN: PROC. 5TH FROCOS
, 2005
"... The dependency pair technique is a powerful modular method for automated termination proofs of term rewrite systems (TRSs). We present two important extensions of this technique: First, we show how to prove termination of higherorder functions using dependency pairs. To this end, the dependency ..."
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Cited by 57 (19 self)
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The dependency pair technique is a powerful modular method for automated termination proofs of term rewrite systems (TRSs). We present two important extensions of this technique: First, we show how to prove termination of higherorder functions using dependency pairs. To this end, the dependency pair technique is extended to handle (untyped) applicative TRSs. Second, we introduce a method to prove nontermination with dependency pairs, while up to now dependency pairs were only used to verify termination. Our results lead to a framework for combining termination and nontermination techniques for firstand higherorder functions in a very flexible way. We implemented and evaluated our results in the automated termination prover AProVE.
A Collection of Examples for Termination of Term Rewriting Using Dependency Pairs
, 2001
"... This report contains a collection of examples to demonstrate the use and the power of the dependency pair technique developed by Arts and Giesl. This technique allows automated termination and innermost termination proofs for many term rewrite systems for which such proofs were not possible before. ..."
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Cited by 28 (11 self)
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This report contains a collection of examples to demonstrate the use and the power of the dependency pair technique developed by Arts and Giesl. This technique allows automated termination and innermost termination proofs for many term rewrite systems for which such proofs were not possible before.
A Technique for Automatically Proving Termination of Constructor Systems
, 1995
"... A technique is described to prove termination of constructor systems (CSs) automatically. The technique consists of three major steps. First, determine the dependency pairs of a constructor system. Second, find an equational theory in which the constructor system is contained, and third, prove that ..."
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Cited by 1 (1 self)
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A technique is described to prove termination of constructor systems (CSs) automatically. The technique consists of three major steps. First, determine the dependency pairs of a constructor system. Second, find an equational theory in which the constructor system is contained, and third, prove that no infinite chain w.r.t. the equational theory of these dependency pairs exists. The first step is easy and can be performed completely automatically. Here we first concentrate on the last step. We assume the equational theory given in the form of a complete TRS and present several general criteria on the syntax of the dependency pairs to prove that no infinite chain can exist with respect to the given equational theory. For these criteria no semantic unification is needed and they can be performed completely automatically. Second we demonstrate a technique to find a complete TRS automatically in case the CS that has to be proved terminating is of a special form. We combine all techniques to...
Detecting NonTermination of Term Rewriting Systems Using an Unfolding Operator
"... Abstract. In this paper, we present an approach to nontermination of term rewriting systems inspired by a technique that was designed in the context of logic programming. Our method is based on a classical unfolding operation together with semiunification and is independent of a particular reducti ..."
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Abstract. In this paper, we present an approach to nontermination of term rewriting systems inspired by a technique that was designed in the context of logic programming. Our method is based on a classical unfolding operation together with semiunification and is independent of a particular reduction strategy. We also describe a technique to reduce the explosion of rules caused by the unfolding process. The analyser that we have implemented is able to solve most of the nonterminating examples in the Termination Problem Data Base. 1