Most systems for the automation of termination proofs using polynomial orderings are only semi-automatic, 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 semi-automatic approaches. Based on this technique we develop a method for the automated synthesis of a suited polynomial ordering.

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).

To prove the termination of a functional program there has to be a well-founded 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 well-founded 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 well-founded 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.

Semantic labelling is a powerful tool for proving termination of term rewrite systems. The usefulness of the extension to equational term rewriting described in Zantema [24] is however rather limited. In this paper we introduce a stronger version of equational semantical labelling, parameterized by three choices: (1) the order on the underlying algebra (partial order vs. quasi-order), (2) the relation between the algebra and the rewrite system (model vs. quasi-model), and (3) the labelling of the function symbols appearing in the equations (forbidden vs.

We present a new technique for proving AC-termination. We show that if certain conditions are met, AC-termination can be reduced to termination, i. e., termination of a TRS S modulo an AC-theory can be inferred from termination of another TRS R with no AC-theory involved. This is a new perspective and opens new possibilities to deal with AC-termination. 1 Introduction Termination of term rewriting systems (TRS's) is crucial for the use of rewriting in proofs and computations, and many theories have been developed in this field, especially for the case where function symbols do not obey any particular law or property. However, many interesting and useful systems have operators which are associative and commutative (AC), and most techniques developed for proving termination of TRS's do not carry over to rewriting modulo equational theories so that the theory developed to study termination of TRS's needs to be adapted to the equational case. Along these lines, a lot of work has been done...

. We investigate one of the classical problems of the theory of term rewriting, namely termination. We present an ordering for comparing higher-order terms that can be utilized for testing termination and decreasingness of higher-order conditional term rewriting systems. The ordering relies on a first-order interpretation of higher-order terms and a suitable extension of the RPO. 1 Motivation Term rewriting systems (TRSs) can be considered as a powerful theoretical model for reasoning about functional and logic programming in an abstract way, independently of a particular programming language. In such an approach to computer programming, logic and functional programs are represented by means of executable specifications essentially consisting of conditional equations. The operational semantics of these specifications is defined by term rewriting and equation solving, respectively. The extension of first-order logic to higher-order logic by means of (universally quantified) condi...

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