## Fachbereich Informatik

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#### Keywords

- Case-Based Reasoning (2)
- EBG (1)
- Fallbasiertes Schliessen (1)
- Fallbasiertes Schließen (1)
- Similarity Assessment (1)
- analogical reasoning (1)
- analogy (1)
- automated proof planner (1)
- case-based reasoning (1)
- concept representation (1)

Proof planning is an alternative methodology to classical automated theorem prov-ing based on exhausitve search that was first introduced by Bundy [8]. The goal ofthis paper is to extend the current realm of proof planning to cope with genuinelymathematical problems such as the well-known limit theorems first investigated for au-tomated theorem proving by Bledsoe. The report presents a general methodology andcontains ideas that are new for proof planning and theorem proving, most importantlyideas for search control and for the integration of domain knowledge into a general proofplanning framework. We extend proof planning by employing explicit control-rules andsupermethods. We combine proof planning with constraint solving. Experiments showthe influence of these mechanisms on the performance of a proof planner. For instance,the proofs of LIM+ and LIM* have been automatically proof planned in the extendedproof planner OMEGA.In a general proof planning framework we rationally reconstruct the proofs of limittheorems for real numbers (IR) that were first computed by the special-purpose programreported in [6]. Compared with this program, the rational reconstruction has severaladvantages: It relies on a general-purpose problem solver; it provides high-level, hi-erarchical representations of proofs that can be expanded to checkable ND-proofs; itemploys declarative contol knowledge that is modularly organized.

Typical instances, that is, instances that are representative for a particular situ-ation or concept, play an important role in human knowledge representationand reasoning, in particular in analogical reasoning. This wellADknown obser-vation has been a motivation for investigations in cognitive psychology whichprovide a basis for our characterization of typical instances within conceptstructures and for a new inference rule for justified analogical reasoning withtypical instances. In a nutshell this paper suggests to augment the proposi-tional knowledge representation system by a non-propositional part consistingof concept structures which may have directly represented instances as ele-ments. The traditional reasoning system is extended by a rule for justifiedanalogical inference with typical instances using information extracted fromboth knowledge representation subsystems.

This paper addresses two modi of analogical reasoning. Thefirst modus is based on the explicit representation of the justificationfor the analogical inference. The second modus is based on the repre-sentation of typical instances by concept structures. The two kinds ofanalogical inferences rely on different forms of relevance knowledge thatcause non-monotonicity. While the uncertainty and non-monotonicity ofanalogical inferences is not questioned, a semantic characterization ofanalogical reasoning has not been given yet. We introduce a minimalmodel semantics for analogical inference with typical instances.

Many mathematical proofs are hard to generate forhumans and even harder for automated theoremprovers. Classical techniques of automated theoremproving involve the application of basic rules, of built-in special procedures, or of tactics. Melis (Melis 1993)introduced a new method for analogical reasoning inautomated theorem proving. In this paper we showhow the derivational analogy replay method is relatedand extended to encompass analogy-driven proof planconstruction. The method is evaluated by showing theproof plan generation of the Pumping Lemma for con-text free languages derived by analogy with the proofplan of the Pumping Lemma for regular languages.This is an impressive evaluation test for the analogicalreasoning method applied to automated theorem prov-ing, as the automated proof of this Pumping Lemmais beyond the capabilities of any of the current auto-mated theorem provers.

This paper addresses the decomposition of proofs as a means of constructingmethods in plan-based automated theorem proving. It shows also, howdecomposition can beneficially be applied in theorem proving by analogy.Decomposition is also useful for human-style proof presentation. We proposeseveral decomposition techniques that were found to be useful in automatedtheorem proving and give examples of their application.

This paper analyzes how mathematicians prove the-orems. The analysis is based upon several empiricalsources such as reports of mathematicians and math-ematical proofs by analogy. In order to combine thestrength of traditional automated theorem provers withhuman-like capabilities, the questions arise: Whichproblem solving strategies are appropriate? Which rep-resentations have to be employed? As a result of ouranalysis, the following reasoning strategies are recog-nized: proof planning with partially instantiated meth-ods, structuring of proofs, the transfer of subproofs andof reformulated subproofs. We discuss the represent-ation of a component of these reasoning strategies, aswell as its properties. We find some mechanisms neededfor theorem proving by analogy, that are not providedby previous approaches to analogy. This leads us to acomputational representation of new components andprocedures for automated theorem proving systems.

This paper shows how a new approach to theorem provingby analogy is applicable to real maths problems. This approach worksat the level of proof-plans and employs reformulation that goes beyondsymbol mapping. The Heine-Borel theorem is a widely known result inmathematics. It is usually stated in R 1 and similar versions are also truein R 2 , in topology, and metric spaces. Its analogical transfer was proposedas a challenge example and could not be solved by previous approachesto theorem proving by analogy. We use a proof-plan of the Heine-Boreltheorem in R 1 as a guide in automatically producing a proof-plan of theHeine-Borel theorem in R 2 by analogy-driven proof-plan construction.

This paper addresses a model of analogy-driven theorem proving that is more general and cognitively more adequate than previous approaches. The model works at the level ofproof-plans. More precisely, we consider analogy as a control strategy in proof planning that employs a source proof-plan to guide the construction of a proof-plan for the target problem. Our approach includes a reformulation of the source proof-plan. This is in accordance with the well known fact that constructing ananalogy in maths often amounts to first finding the appropriate representation which brings out the similarity of two problems, i.e., finding the right concepts and the right level of abstraction. Several well known theorems were processed by our analogy-driven proof-plan construction that could not be proven analogically by previous approaches.

This paper addresses analogy-driven auto-mated theorem proving that employs a sourceproof-plan to guide the search for a proof-planof the target problem. The approach presen-ted uses reformulations that go beyond symbolmappings and that incorporate frequently usedre-representations and abstractions. Severalrealistic math examples were successfully pro-cessed by our analogy-driven proof-plan con-struction. One challenge example, a Heine-Borel theorem, is discussed here. For this ex-ample the reformulaitons are shown step bystep and the modifying actions are demon-strated.