We present an inference system for clausal theorem proving w.r.t. various kinds of inductivevalidity in theories specified by constructor-based positive/negative-conditional equations. The reductionrelation defined by such equations has to be (ground) confluent, but need not be terminating. Our con-structor-based approach is well-suited for inductive theorem proving in the presence of partially definedfunctions. The proposed inference system provides explicit induction hypotheses and can be instantiatedwith various wellfounded induction orderings. While emphasizing a well structured clear design of theinference system, our fundamental design goal is user-orientation and practical usefulness rather thantheoretical elegance. The resulting inference system is comprehensive and relatively powerful, but requiresa sophisticated concept of proof guidance, which is not treated in this paper.This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt)
We study the combination of the following already known ideas for showing confluence ofunconditional or conditional term rewriting systems into practically more useful confluence criteria forconditional systems: Our syntactic separation into constructor and non-constructor symbols, Huet's intro-duction and Toyama's generalization of parallel closedness for non-noetherian unconditional systems, theuse of shallow confluence for proving confluence of noetherian and non-noetherian conditional systems, theidea that certain kinds of limited confluence can be assumed for checking the fulfilledness or infeasibilityof the conditions of conditional critical pairs, and the idea that (when termination is given) only primesuperpositions have to be considered and certain normalization restrictions can be applied for the sub-stitutions fulfilling the conditions of conditional critical pairs. Besides combining and improving alreadyknown methods, we present the following new ideas and results: We strengthen the criterion for overlayjoinable noetherian systems, and, by using the expressiveness of our syntactic separation into constructorand non-constructor symbols, we are able to present criteria for level confluence that are not criteria forshallow confluence actually and also able to weaken the severe requirement of normality (stiffened withleft-linearity) in the criteria for shallow confluence of noetherian and non-noetherian conditional systems tothe easily satisfied requirement of quasi-normality. Finally, the whole paper also gives a practically usefuloverview of the syntactic means for showing confluence of conditional term rewriting systems.
While most approaches to similarity assessment are oblivious of knowledge and goals, there is ample evidence that these elements of problem solving play an important role in similarity judgements. This paper is concerned with an approach for integrating assessment of similarity into a framework of problem solving that embodies central notions of problem solving like goals, knowledge and learning.
An important property and also a crucial point ofa term rewriting system is its termination. Transformation or-derings, developed by Bellegarde & Lescanne strongly based on awork of Bachmair & Dershowitz, represent a general technique forextending orderings. The main characteristics of this method aretwo rewriting relations, one for transforming terms and the otherfor ensuring the well-foundedness of the ordering. The centralproblem of this approach concerns the choice of the two relationssuch that the termination of a given term rewriting system can beproved. In this communication, we present a heuristic-based al-gorithm that partially solves this problem. Furthermore, we showhow to simulate well-known orderings on strings by transformationorderings.
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.
Orderings on polynomial interpretations of operators represent a powerful technique for proving thetermination of rewriting systems. One of the main problems of polynomial orderings concerns thechoice of the right interpretation for a given rewriting system. It is very difficult to develop techniquesfor solving this problem. Here, we present three new heuristic approaches: (i) guidelines for dealingwith special classes of rewriting systems, (ii) an algorithm for choosing appropriate special polynomialsas well as (iii) an extension of the original polynomial ordering which supports the generation ofsuitable interpretations. All these heuristics will be applied to examples in order to illustrate theirpractical relevance.
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 first show that ground term-rewriting systems can be completed in apolynomial number of rewriting steps, if the appropriate data structure for termsis used. We then apply this result to study the lengths of critical pair proofs innon-ground systems, and obtain bounds on the lengths of critical pair proofsin the non-ground case. We show how these bounds depend on the types ofinference steps that are allowed in the proofs.
We will answer a question posed in [DJK91], and will show that Huet's completion algorithm [Hu81] becomes incomplete, i.e. it may generate a term rewriting system that is not confluent, if it is modified in a way that the reduction ordering used for completion can be changed during completion provided that the new ordering is compatible with the actual rules. In particular, we will show that this problem may not only arise if the modified completion algorithm does not terminate: Even if the algorithm terminates without failure, the generated finite noetherian term rewriting system may be non-confluent. Most existing implementations of the Knuth-Bendix algorithm provide the user with help in choosing a reduction ordering: If an unorientable equation is encountered, then the user has many options, especially, the one to orient the equation manually. The integration of this feature is based on the widespread assumption that, if equations are oriented by hand during completion and the completion process terminates with success, then the generated finite system is a maybe non terminating but locally confluent system (see e.g. [KZ89]). Our examples will show that this assumption is not true.
Patdex is an expert system which carries out case-based reasoning for the fault diagnosis of complex machines. It is integrated in the Moltke workbench for technical diagnosis, which was developed at the university of Kaiserslautern over the past years, Moltke contains other parts as well, in particular a model-based approach; in Patdex where essentially the heuristic features are located. The use of cases also plays an important role for knowledge acquisition. In this paper we describe Patdex from a principal point of view and embed its main concepts into a theoretical framework.