Termination Of Term Rewriting Systems Research Articles

Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect to termination proving power, Lucas managed to prove in 2006 that there are rewrite systems that can be shown polynomially terminating by polynomial interpretations with real (algebraic) coefficients, but cannot be shown polynomially terminating using polynomials with rational coefficients only. He also proved the corresponding statement regarding the use of rational coefficients versus integer coefficients. In this article we extend these results, thereby giving the full picture of the relationship between the aforementioned variants of polynomial interpretations. In particular, we show that polynomial interpretations with real or rational coefficients do not subsume polynomial interpretations with integer coefficients. Our results hold also for incremental termination proofs with polynomial interpretations.

Loops under Strategies ... Continued

While there are many approaches for automatically proving termination of term rewrite systems, up to now there exist only few techniques to disprove their termination automatically. Almost all of these techniques try to find loops, where the existence of a loop implies non-termination of the rewrite system. However, most programming languages use specific evaluation strategies, whereas loop detection techniques usually do not take strategies into account. So even if a rewrite system has a loop, it may still be terminating under certain strategies. Therefore, our goal is to develop decision procedures which can determine whether a given loop is also a loop under the respective evaluation strategy. In earlier work, such procedures were presented for the strategies of innermost, outermost, and context-sensitive evaluation. In the current paper, we build upon this work and develop such decision procedures for important strategies like leftmost-innermost, leftmost-outermost, (max-)parallel-innermost, (max-)parallel-outermost, and forbidden patterns (which generalize innermost, outermost, and context-sensitive strategies). In this way, we obtain the first approach to disprove termination under these strategies automatically.

Innermost Termination of Rewrite Systems by Labeling

Semantic labeling is a powerful transformation technique for proving termination of term rewrite systems. The semantic part is given by a model or a quasi-model of the rewrite rules. A variant of semantic labeling is predictive labeling where the quasi-model condition is only required for the usable rules. In this paper we investigate how semantic and predictive labeling can be used to prove innermost termination. Moreover, we show how to reduce the set of usable rules for predictive labeling even further, both in the termination and the innermost termination case.

Automating the dependency pair method

Developing automatable methods for proving termination of term rewrite systems that resist traditional techniques based on simplification orders has become an active research area in the past few years. The dependency pair method of Arts and Giesl is one of the most popular such methods. However, there are several obstacles that hamper its automation. In this paper we present new ideas to overcome these obstacles. We provide ample numerical data supporting our ideas.

Simulating Liveness by Reduction Strategies

We define a general framework to handle liveness and related properties by reduction strategies in abstract reduction and term rewriting. Classically, reduction strategies in rewriting are used to simulate the evaluation process in programming languages. The aim of our work is to use reduction strategies to also study liveness questions which are of high importance in practice (e.g., in protocol verification for distributed processes). In particular, we show how the problem of verifying liveness is related to termination of term rewrite systems (TRSs). Using our results, techniques for proving termination of TRSs can be used to verify liveness properties.

Termination of term rewriting using dependency pairs

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.

Distribution Elimination by Semantic Labelling : Methods for Proving Termination of Term Rewriting Systems.

Distribution Elimination by Semantic Labelling : Methods for Proving Termination of Term Rewriting Systems.

Generating polynomial orderings

There are various methods of proving termination of term rewriting systems (TRS). Most of these are based on reduction orderings, which are well-founded, compatible with the structure of terms and stable with respect to substitutions. The notion of reduction ordering allows the following characterization of termination of TRS: A TRS 9? terminates iff there exists a reduction ordering + such that I t r for each rule I + r of 9. Polynomial orderings are special reduction orderings, mapping terms into a well-founded set by attaching monotonic polynomials (so-called interpretations) to operators. They have been studied by Manna and Ness [12], Lankford [lo], Dershowitz [3], Huet and Oppen [8] and BenCherifa and Lescanne [l]. In contrast to the others, Dershowitz uses an arbitrary set of functions (polynomials) by requiring them to possess the subterm property, i.e. his ordering is a simplification ordering including the homeomorphic embedding relation. One of the main problems concerning polynomial orderings is the choice of adequate interpretations for a given TRS. The object of this paper is to describe an algorithm (being powerful in practice, see Section 5) based on the Simplex method for finding polynomial interpretations for a given TRS.

Termination by completion

This paper presents a completion procedure for proving termination of term rewrite systems. It works as follows. Given a term rewrite systemR supposed to terminate and a term rewrite systemT used to transformR, the completion builds an auxiliary term rewrite systemS, the system transformed ofR byT. The termination ofS andT associated with a property called local cooperation implies the termination ofR. If the process terminates this provides a mechanical proof of the termination ofR.