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The reasoning power of human-oriented plan-based reasoning systems is primarilyderived from their domain-specific problem solving knowledge. Such knowledge is, how-ever, intrinsically incomplete. In order to model the human ability of adapting existingmethods to new situations we present in this work a declarative approach for represent-ing methods, which can be adapted by so-called meta-methods. Since apparently thesuccess of this approach relies on the existence of general and strong meta-methods,we describe several meta-methods of general interest in detail by presenting the prob-lem solving process of two familiar classes of mathematical problems. These examplesshould illustrate our philosophy of proof planning as well: besides planning with thecurrent repertoire of methods, the repertoire of methods evolves with experience inthat new ones are created by meta-methods which modify existing ones.
In this paper we generalize the notion of method for proofplanning. While we adopt the general structure of methods introducedby Alan Bundy, we make an essential advancement in that we strictlyseparate the declarative knowledge from the procedural knowledge. Thischange of paradigm not only leads to representations easier to under-stand, it also enables modeling the important activity of formulatingmeta-methods, that is, operators that adapt the declarative part of exist-ing methods to suit novel situations. Thus this change of representationleads to a considerably strengthened planning mechanism.After presenting our declarative approach towards methods we describethe basic proof planning process with these. Then we define the notion ofmeta-method, provide an overview of practical examples and illustratehow meta-methods can be integrated into the planning process.
Extending the planADbased paradigm for auto-mated theorem proving, we developed in previ-ous work a declarative approach towards rep-resenting methods in a proof planning frame-work to support their mechanical modification.This paper presents a detailed study of a classof particular methods, embodying variations ofa mathematical technique called diagonaliza-tion. The purpose of this paper is mainly two-fold. First we demonstrate that typical math-ematical methods can be represented in ourframework in a natural way. Second we illus-trate our philosophy of proof planning: besidesplanning with a fixed repertoire of methods,metaADmethods create new methods by modify-ing existing ones. With the help of three differ-ent diagonalization problems we present an ex-ample trace protocol of the evolution of meth-ods: an initial method is extracted from a par-ticular successful proof. This initial method isthen reformulated for the subsequent problems,and more general methods can be obtained byabstracting existing methods. Finally we comeup with a fairly abstract method capable ofdealing with all the three problems, since it cap-tures the very key idea of diagonalization.
Die Beweisentwicklungsumgebung Omega-Mkrp soll Mathematiker bei einer ihrer Haupttätigkeiten, nämlich dem Beweisen mathematischer Theoreme unterstützen. Diese Unterstützung muß so komfortabel sein, daß die Beweise mit vertretbarem Aufwand formal durchgeführt werden können und daß die Korrektheit der so erzeugten Beweise durch das System sichergestellt wird. Ein solches System wird sich nur dann wirklich durchsetzen, wenn die rechnergestützte Suche nach formalen Beweisen weniger aufwendig und leichter ist, als ohne das System. Um dies zu erreichen, ergeben sich verschiedene Anforderungen an eine solche Entwicklungsumgebung, die wir im einzelnen beschreiben. Diese betreffen insbesondere die Ausdruckskraft der verwendeten Objektsprache, die Möglichkeit, abstrakt über Beweispläne zu reden, die am Menschen orientierte Präsentation der gefundenen Beweise, aber auch die effiziente Unterstützung beim Füllen von Beweislücken. Das im folgenden vorgestellte Omega-Mkrp-System ist eine Synthese der Ansätze des vollautomatischen, des interaktiven und des planbasierten Beweisens und versucht erstmalig die Ergebnisse dieser drei Forschungsrichtungen in einem System zu vereinigen. Dieser Artikel soll eine Übersicht über unsere Arbeit an diesem System geben.
This paper outlines the microplanner of PROVERB , a system that generates multilingual text from machine-found mathematical proofs. The main representational vehicle is the text structure proposed by Meteer. Following Panaget, we also distinguish between the ideational and the textual semantic categories, and use the upper model to replace the former. Based on this, a further extension is made to support aggregation before realization decisions are made. While our the framework of our macroplanner is kept languageindependent, our microplanner draws on language specific linguistic sources such as realization classes and lexicon. Since English and German are the first two languages to be generated and because the sublanguage of our mathematical domain is relatively limited, the upper model and the textual semantic categories are designed to cope with both languages. Since the work reported is still in progress, we also discuss open problems we are facing.
Over the past thirty years there have been significant achievements in the field of auto-mated theorem proving with respect to the reasoning power of the inference engines.Although some effort has also been spent to facilitate more user friendliness of the de-duction systems, most of them failed to benefit from more recent developments in therelated fields of artificial intelligence (AI), such as natural language generation and usermodeling. In particular, no model is available which accounts both for human deductiveactivities and for human proof presentation. In this thesis, a reconstructive architecture issuggested which substantially abstracts, reorganizes and finally translates machine-foundproofs into natural language. Both the procedures and the intermediate representationsof our architecture find their basis in computational models for informal mathematicalreasoning and for proof presentation. User modeling is not incorporated into the currenttheory, although we plan to do so later.
This paper concerns a knowledge structure called method , within a compu-tational model for human oriented deduction. With human oriented theoremproving cast as an interleaving process of planning and verification, the body ofall methods reflects the reasoning repertoire of a reasoning system. While weadopt the general structure of methods introduced by Alan Bundy, we make anessential advancement in that we strictly separate the declarative knowledgefrom the procedural knowledge. This is achieved by postulating some stand-ard types of knowledge we have identified, such as inference rules, assertions,and proof schemata, together with corresponding knowledge interpreters. Ourapproach in effect changes the way deductive knowledge is encoded: A newcompound declarative knowledge structure, the proof schema, takes the placeof complicated procedures for modeling specific proof strategies. This change ofparadigm not only leads to representations easier to understand, it also enablesus modeling the even more important activity of formulating meta-methods,that is, operators that adapt existing methods to suit novel situations. In thispaper, we first introduce briefly the general framework for describing methods.Then we turn to several types of knowledge with their interpreters. Finally,we briefly illustrate some meta-methods.