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- Case-Based Reasoning (2)
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- 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)

While most approaches to similarity assessment are oblivious of knowledge and goals, there is ample evidence that these elements of problem solving play an important role in similarity judgements. This paper is concerned with an approach for integrating assessment of similarity into a framework of problem solving that embodies central notions of problem solving like goals, knowledge and learning.

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.

The amount of user interaction is the prime cause of costs in interactiveprogram verification. This paper describes an internal analogy techniquethat reuses subproofs in the verification of state-based specifications. Itidentifies common patterns of subproofs and their justifications in orderto reuse these subproofs; thus significant savings on the number of userinteractions in a verification proof are achievable.

Planning for realistic problems in a static and deterministic environment with complete information faces exponential search spaces and, more often than not, should produce plans comprehensible for the user. This article introduces new planning strategies inspired by proof planning examples in order to tackle the search-space-problem and the structured-plan-problem. Island planning and refinement as well as subproblem refinement are integrated into a general planning framework and some exemplary control knowledge suitable for proof planning is given.

The paper addresses two problems of comprehensible proof presentation, the hierarchically structured presentation at the level of proof methods and different presentation styles of construction proofs. It provides solutions for these problems that can make use of proof plans generated by an automated proof planner.

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.

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.