We investigate restricted termination and confluence properties of term rewritADing systems, in particular weak termination and innermost termination, and theirinterrelation. New criteria are provided which are sufficient for the equivalenceof innermost / weak termination and uniform termination of term rewriting sysADtems. These criteria provide interesting possibilities to infer completeness, i.e.termination plus confluence, from restricted termination and confluence properADties.Using these basic results we are also able to prove some new results aboutmodular termination of rewriting. In particular, we show that termination ismodular for some classes of innermost terminating and locally confluent termrewriting systems, namely for nonADoverlapping and even for overlay systems. Asan easy consequence this latter result also entails a simplified proof of the factthat completeness is a decomposable property of soADcalled constructor systems.Furthermore we show how to obtain similar results for even more general cases of(nonADdisjoint) combined systems with shared constructors and of certain hierarADchical combinations of systems with constructors. Interestingly, these modularityresults are obtained by means of a proof technique which itself constitutes a modADular approach.
Various methods for proving the termination of term rewriting systems havebeen suggested. Most of them are based on the notion of simplification ordering.In this paper, the theoretical time complexities (of the worst cases) of a collectionof well-known simplification orderings will be presented.
This report contains a collection of abstracts for talks given at the "Deduktionstreffen" held at Kaiserslautern, October 6 to 8, 1993. The topics of the talks range from theoretical aspects of term rewriting systems and higher order resolution to descriptions of practical proof systems in various applications. They are grouped together according the following classification: Distribution and Combination of Theorem Provers, Termination, Completion, Functional Programs, Inductive Theorem Proving, Automatic Theorem Proving, Proof Presentation. The Deduktionstreffen is the annual meeting of the Fachgruppe Deduktionssysteme in the Gesellschaft für Informatik (GI), the German association for computer science.
A general concept for combining planning with automatic theorem provingis introduced. From this a system architecture based on the notion of planningtrees, methods and sensors is developed. It is illustrated by examples taken fromthe domain of sorting algorithms.
We study deterministic conditional rewrite systems, i.e. conditional rewrite systemswhere the extra variables are not totally free but 'input bounded'. If such a systemR is quasi-reductive then !R is decidable and terminating. We develop a critical paircriterion to prove confluence if R is quasi-reductive and strongly deterministic. In thiscase we prove that R is logical, i.e./!R==R holds. We apply our results to proveHorn clause programs to be uniquely terminating.This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314, Project D4
Relating Innermost, Weak, Uniform and Modular Termination of Term Rewriting Systems
(1993)
