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Sat, 04 Mar 2000 00:00:00 +0200Sat, 04 Mar 2000 00:00:00 +0200The Adaption of Proof Methods by Reformulation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/63
Manfred Kerber; Xiaorong Huang; Lassaad Cheikhrouhouarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/63Mon, 03 Apr 2000 00:00:00 +0200Adapting Methods to Novel Tasks in Proof Planning ?
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/255
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.Manfred Kerber; Xiaorong Huang; Michael Kohlhase; Jörn Richtsarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/255Mon, 03 Apr 2000 00:00:00 +0200Adapting the Diagonalization Method by Reformulations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/257
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.Manfred Kerber; Xiaorong Huang; Lassaad Cheikhrouhouarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/257Mon, 03 Apr 2000 00:00:00 +0200Planning Mathematical Proofs with Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/367
In this article we formally describe a declarative approach for encoding plan operatorsin proof planning, the so-called methods. The notion of method evolves from the much studiedconcept tactic and was first used by Bundy. While significant deductive power has been achievedwith the planning approach towards automated deduction, the procedural character of the tacticpart of methods, however, hinders mechanical modification. Although the strength of a proofplanning system largely depends on powerful general procedures which solve a large class ofproblems, mechanical or even automated modification of methods is nevertheless necessary forat least two reasons. Firstly methods designed for a specific type of problem will never begeneral enough. For instance, it is very difficult to encode a general method which solves allproblems a human mathematician might intuitively consider as a case of homomorphy. Secondlythe cognitive ability of adapting existing methods to suit novel situations is a fundamentalpart of human mathematical competence. We believe it is extremely valuable to accountcomputationally for this kind of reasoning.The main part of this article is devoted to a declarative language for encoding methods,composed of a tactic and a specification. The major feature of our approach is that the tacticpart of a method is split into a declarative and a procedural part in order to enable a tractableadaption of methods. The applicability of a method in a planning situation is formulatedin the specification, essentially consisting of an object level formula schema and a meta-levelformula of a declarative constraint language. After setting up our general framework, wemainly concentrate on this constraint language. Furthermore we illustrate how our methodscan be used in a Strips-like planning framework. Finally we briefly illustrate the mechanicalmodification of declaratively encoded methods by so-called meta-methods.Xiaorong Huang; Manfred Kerber; Lassaad Cheikhrouhou; Jörn Richts; Arthur Sehnarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/367Mon, 03 Apr 2000 00:00:00 +0200Reuse of Proofs by Meta-Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/250
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.Manfred Kerber; Xiaorong Huangpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/250Mon, 03 Apr 2000 00:00:00 +0200Die Beweisentwicklungsumgebung Omega-MKRK
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/251
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.Manfred Kerber; Xiaorong Huang; Michael Kohlhase; Erica Melis; Dan Nesmith; Jörn Richts; Jörg Siekmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/251Mon, 03 Apr 2000 00:00:00 +0200A Test for Evaluating the Practical Usefulness of Deduction Systems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/252
Manfred Kerber; Xiaorong Huang; Michael Kohlhase; Dan Nesmith; Jörn Richtspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/252Mon, 03 Apr 2000 00:00:00 +0200Guaranteeing Correctness through the Communication of Checkable Proofs(or: Would You Really Trust an Automated Reasoning System?)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/254
Manfred Kerber; Xiaorong Huang; Michael Kohlhase; Dan Nesmith; Jörn Richtspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/254Mon, 03 Apr 2000 00:00:00 +0200Methods - The Basic Units for Planning and Verifying Proofs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/338
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.Xiaorong Huang; Manfred Kerber; Michael Kohlhasepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/338Mon, 03 Apr 2000 00:00:00 +0200OMEGA MKRP - A Proof Development Environment
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/340
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)Xiaorong Huang; Manfred Kerber; Michael Kohlhase; Erica Melis; Dan Nesmith; Jörn Richts; Jörg Siekmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/340Mon, 03 Apr 2000 00:00:00 +0200Adaptation of Declaratively Represented Methods in Proof Planning
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/382
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.Xiaorong Huang; Manfred Kerber; Lassaad Cheikhrouhoupreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/382Mon, 03 Apr 2000 00:00:00 +0200Omega: Towards a Mathematical Assistant
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/428
-mega is a mixed-initiative system with the ultimate pur-pose of supporting theorem proving in main-stream mathematics andmathematics education. The current system consists of a proof plannerand an integrated collection of tools for formulating problems, provingsubproblems, and proof presentation.Christoph Benzmüller; Lassaad Cheikhrouhou; Detlef Fehrer; Armin Fiedler; Xiaorong Huang; Manfred Kerber; Michael Kohlhase; Karsten Konrad; Andreas Meierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/428Mon, 03 Apr 2000 00:00:00 +0200