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 right-hand sides of rewrite rules, but introduce the notion of dependency pairs to compare left-hand sides with special subterms of the right-hand 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, well-known simplification orderings (such as the recursive path ordering, polynomial orderings, or the Knuth-Bendix 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.

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 well-known challenge problems such as McCarthy's f_91 function).

Context-sensitive 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 context-sensitive rewriting. The first one is a modification of the technique of interpretation in a well-founded order, the second one is implied by a transformation in which context-sensitive 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...

To prove termination of term rewriting systems (TRSs), several methods have been developed to synthesize suitable well-founded orderings automatically. However, virtually all orderings that are amenable to automation are so-called 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 well-founded 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...

We present a technique to prove innermost normalisation of term rewriting systems (TRSs) automatically. In contrast to previous methods, our technique is able to prove innermost normalisation of TRSs that are not terminating. Our technique can also be used for termination proofs of all TRSs where innermost normalisation implies termination, such as non-overlapping TRSs or locally confluent overlay systems. In this way, termination of many (also non-simply terminating) TRSs can be verified automatically. 1. Introduction Innermost rewriting, i.e. rewriting where only innermost redexes are contracted, can be used to model call-by-value computation semantics. For that reason, there has been an increasing interest in innermost normalisation (also called innermost termination), i.e. in proving that the length of every innermost reduction is finite. Techniques for proving innermost normalisation can for example be utilized for termination proofs of functional programs (modelled by TRSs) or o...

. 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...

Proving termination is a central problem in software development and formal methods for termination analysis are essential for program verification. However, since the halting problem is undecidable and totality of functions is not even semi-decidable, there is no procedure to prove or disprove the

We present a transformation technique called freezing to facilitate automatic termination proofs for left-linear term rewriting systems. The significant merits of this technique lie in its simplicity, its amenability to automation and its effectiveness, especially, when combined with other well-known methods such as recursive path orderings and polynomial interpretations. We prove that applying the freezing technique to a left-linear term rewriting system always terminates. We also show that many interesting TRSs in the literature can be handled with the help of freezing while they elude a lot of other approaches aiming for generating termination proofs automatically for term rewriting systems. We have mechanically verified all the left-linear examples presented in this paper.

In this paper we present a technique for automated termination proofs of term rewriting systems (TRS). More precisely, we introduce criteria for both strong and innermost normalisation and we show how these criteria can be checked automatically. In this way, termination of numerous (also nonsimply terminating) TRSs resp. innermost normalisation of many (also nonterminating) TRSs can be proved fully automatically. 1 Introduction One of the most interesting properties of a term rewriting system is termination (or strong normalisation), cf. e.g. [DJ90]. While in general this problem is undecidable [HL78], several methods for proving termination have been developed (e.g. path orderings [Pla78, Der82, DH95, Ste95b], Knuth-Bendix orderings [KB70, DKM90], semantic interpretations [Lan79, BL87, BL93, Ste94, Zan94, Gie95], transformation orderings [BD86, BL90, Ste95a], semantic labelling [Zan95] etc. --- for surveys see e.g. [Der87, Ste95b]). Moreover, termination properties of rewriting unde...

We present a technique to prove innermost normalisation of term rewriting systems (TRSs) automatically. In contrast to previous methods, our technique is able to prove innermost normalisation of TRSs that are not terminating.