Kaiserslautern - Fachbereich Informatik
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- Deduction (1)
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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.
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.
This report presents the main ideas underlyingtheOmegaGamma mkrp-system, an environmentfor the development of mathematical proofs. The motivation for the development ofthis system comes from our extensive experience with traditional first-order theoremprovers and aims to overcome some of their shortcomings. After comparing the benefitsand drawbacks of existing systems, we propose a system architecture that combinesthe positive features of different types of theorem-proving systems, most notably theadvantages of human-oriented systems based on methods (our version of tactics) andthe deductive strength of traditional automated theorem provers.In OmegaGamma mkrp a user first states a problem to be solved in a typed and sorted higher-order language (called POST ) and then applies natural deduction inference rules inorder to prove it. He can also insert a mathematical fact from an integrated data-base into the current partial proof, he can apply a domain-specific problem-solvingmethod, or he can call an integrated automated theorem prover to solve a subprob-lem. The user can also pass the control to a planning component that supports andpartially automates his long-range planning of a proof. Toward the important goal ofuser-friendliness, machine-generated proofs are transformed in several steps into muchshorter, better-structured proofs that are finally translated into natural language.This work was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D2, D3)
To prove difficult theorems in a mathematical field requires substantial know-ledge of that field. In this paper a frame-based knowledge representation formalismis presented, which supports a conceptual representation and to a large extent guar-antees the consistency of the built-up knowledge bases. We define a semantics ofthe representation by giving a translation into the underlaying logic.
To prove difficult theorems in a mathematical field requires substantial know-ledge of that field. In this thesis a frame-based knowledge representation formal-ism including higher-order sorted logic is presented, which supports a conceptualrepresentation and to a large extent guarantees the consistency of the built-upknowledge bases. In order to operationalize this knowledge, for instance, in anautomated theorem proving system, a class of sound morphisms from higher-orderinto first-order logic is given, in addition a sound and complete translation ispresented. The translations are bijective and hence compatible with a later proofpresentation.In order to prove certain theorems the comprehension axioms are necessary,(but difficult to handle in an automated system); such theorems are called trulyhigher-order. Many apparently higher-order theorems (i.e. theorems that arestated in higher-order syntax) however are essentially first-order in the sense thatthey can be proved without the comprehension axioms: for proving these theoremsthe translation technique as presented in this thesis is well-suited.
Planverfahren
(1999)
This paper describes a declarative approach forencoding the plan operators in proof planning,the so-called methods. The notion of methodevolves from the much studied concept of a tac-tic and was first used by A. Bundy. Signific-ant deductive power has been achieved withthe planning approach towards automated de-duction; however, the procedural character ofthe tactic part of methods hinders mechanicalmodification. Although the strength of a proofplanning system largely depends on powerfulgeneral procedures which solve a large class ofproblems, mechanical or even automated modi-fication of methods is necessary, since methodsdesigned for a specific type of problems willnever be general enough. After introducing thegeneral framework, we exemplify the mechan-ical modification of methods via a particularmeta-method which modifies methods by trans-forming connectives to quantifiers.
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.