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Mon, 03 Apr 2000 00:00:00 +0200Mon, 03 Apr 2000 00:00:00 +0200On the Representation of Mathematical Concepts and their Translation into First-Order Logic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/334
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.Manfred Kerberpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/334Mon, 03 Apr 2000 00:00:00 +0200Tactics for the Improvement of Problem Formulation in Resolution-Based Theorem Proving
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/335
We transform a user-friendly formulation of aproblem to a machine-friendly one exploiting the variabilityof first-order logic to express facts. The usefulness of tacticsto improve the presentation is shown with several examples.In particular it is shown how tactical and resolution theoremproving can be combined.Manfred Kerber; Axel Präckleinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/335Mon, 03 Apr 2000 00:00:00 +0200Analogical Reasoning with Typical Examples
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/337
Typical examples, that is, examples that are representative for a particular situationor concept, play an important role in human knowledge representation and reasoning.In real life situations more often than not, instead of a lengthy abstract characteriza-tion, a typical example is used to describe the situation. This well-known observationhas been the motivation for various investigations in experimental psychology, whichalso motivate our formal characterization of typical examples, based on a partial orderfor their typicality. Reasoning by typical examples is then developed as a special caseof analogical reasoning using the semantic information contained in the correspondingconcept structures. We derive new inference rules by replacing the explicit informa-tion about connections and similarity, which are normally used to formalize analogicalinference rules, by information about the relationship to typical examples. Using theseinference rules analogical reasoning proceeds by checking a related typical example,this is a form of reasoning based on semantic information from cases.Manfred Kerber; Erica Melis; Jörg Siekmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/337Mon, 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 +0200Reasoning with Assertions and Examples
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/350
The hallmark of traditional Artificial Intelligence (AI) research is the symbolic representation and processing of knowledge. This is in sharp contrast to many forms of human reasoning, which to an extraordinary extent, rely on cases and (typical) examples. Although these examples could themselves be encoded into logic, this raises the problem of restricting the corresponding model classes to include only the intended models.There are, however, more compelling reasons to argue for a hybrid representa-tion based on assertions as well as examples. The problems of adequacy, availability of information, compactness of representation, processing complexity, and last but not least, results from the psychology of human reasoning, all point to the same conclusion: Common sense reasoning requires different knowledge sources and hybrid reasoning principles that combine symbolic as well as semantic-based inference. In this paper we address the problem of integrating semantic representations of examples into automateddeduction systems. The main contribution is a formal framework for combining sentential with direct representations. The framework consists of a hybrid knowledge base, made up of logical formulae on the one hand and direct representations of examples on the other, and of a hybrid reasoning method based on the resolution calculus. The resulting hybrid resolution calculus is shown to be sound and complete.Manfred Kerber; Erica Melisarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/350Mon, 03 Apr 2000 00:00:00 +0200Some Aspects of Analogy in Mathematical Reasoning
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/353
An important research problem is the incorporation of "declarative" knowledge into an automated theorem prover that can be utilized in the search for a proof. An interesting pro-posal in this direction is Alan Bundy's approach of using explicit proof plans that encapsulatethe general form of a proof and is instantiated into a particular proof for the case at hand. Wegive some examples that show how a "declarative" highlevel description of a proof can be usedto find proofs of apparently "similiar" theorems by analogy. This "analogical" information isused to select the appropriate axioms from the database so that the theorem can be proved.This information is also used to adjust some options of a resolution theorem prover. In orderto get a powerful tool it is necessary to develop an epistemologically appropriate language todescribe proofs, for which a large set of examples should be used as a testbed. We presentsome ideas in this direction.Manfred Kerberarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/353Mon, 03 Apr 2000 00:00:00 +0200Mechanization of Strong Kleene Logic for Partial Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/359
Even though it is not very often admitted, partial functions do play asignificant role in many practical applications of deduction systems. Kleenehas already given a semantic account of partial functions using three-valuedlogic decades ago, but there has not been a satisfactory mechanization. Recentyears have seen a thorough investigation of the framework of many-valuedtruth-functional logics. However, strong Kleene logic, where quantificationis restricted and therefore not truth-functional, does not fit the frameworkdirectly. We solve this problem by applying recent methods from sorted logics.This paper presents a resolution calculus that combines the proper treatmentof partial functions with the efficiency of sorted calculi.Manfred Kerber; Michael Kohlhasepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/359Mon, 03 Apr 2000 00:00:00 +0200How to Prove Higher Order Theorems in First Order Logic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/364
In this paper we are interested in using a firstorder theorem prover to prove theorems thatare formulated in some higher order logic. Tothis end we present translations of higher or-der logics into first order logic with flat sortsand equality and give a sufficient criterion forthe soundness of these translations. In addi-tion translations are introduced that are soundand complete with respect to L. Henkin's gen-eral model semantics. Our higher order logicsare based on a restricted type structure in thesense of A. Church, they have typed functionsymbols and predicate symbols, but no sorts.Manfred Kerberarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/364Mon, 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 +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 +0200Proving Ground Completeness of Resolution by Proof Planning
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/391
A lot of the human ability to prove hard mathematical theorems can be ascribedto a problem-specific problem solving know-how. Such knowledge is intrinsicallyincomplete. In order to prove related problems human mathematicians, however,can go beyond the acquired knowledge by adapting their know-how to new relatedproblems. These two aspects, having rich experience and extending it by need, can besimulated in a proof planning framework: the problem-specific reasoning knowledge isrepresented in form of declarative planning operators, called methods; since these aredeclarative, they can be mechanically adapted to new situations by so-called meta-methods. In this contribution we apply this framework to two prominent proofs intheorem proving, first, we present methods for proving the ground completeness ofbinary resolution, which essentially correspond to key lemmata, and then, we showhow these methods can be reused for the proof of the ground completeness of lockresolution.Manfred Kerber; Arthur C. Sehnpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/391Mon, 03 Apr 2000 00:00:00 +0200An Integration of Mechanised Reasoning and Computer Algebra that Respects Explicit Proofs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/392
Mechanised reasoning systems and computer algebra systems have apparentlydifferent objectives. Their integration is, however, highly desirable, since in manyformal proofs both of the two different tasks, proving and calculating, have to beperformed. Even more importantly, proof and computation are often interwoven andnot easily separable. In the context of producing reliable proofs, the question howto ensure correctness when integrating a computer algebra system into a mechanisedreasoning system is crucial. In this contribution, we discuss the correctness prob-lems that arise from such an integration and advocate an approach in which thecalculations of the computer algebra system are checked at the calculus level of themechanised reasoning system. This can be achieved by adding a verbose mode to thecomputer algebra system which produces high-level protocol information that can beprocessed by an interface to derive proof plans. Such a proof plan in turn can beexpanded to proofs at different levels of abstraction, so the approach is well-suited forproducing a high-level verbalised explication as well as for a low-level machine check-able calculus-level proof. We present an implementation of our ideas and exemplifythem using an automatically solved extended example.Manfred Kerber; Michael Kohlhase; Volker Sorgepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/392Mon, 03 Apr 2000 00:00:00 +0200Useful Properties of a Frame-Based Representation of Mathematical Knowledge
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/397
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.Manfred Kerberpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/397Mon, 03 Apr 2000 00:00:00 +0200