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- Declarative and Procedural Knowledge (1)
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Most automated theorem provers suffer from the problem that theycan produce proofs only in formalisms difficult to understand even forexperienced mathematicians. Efforts have been made to transformsuch machine generated proofs into natural deduction (ND) proofs.Although the single steps are now easy to understand, the entire proofis usually at a low level of abstraction, containing too many tedioussteps. Therefore, it is not adequate as input to natural language gen-eration systems.To overcome these problems, we propose a new intermediate rep-resentation, called ND style proofs at the assertion level . After illus-trating the notion intuitively, we show that the assertion level stepscan be justified by domain-specific inference rules, and that these rulescan be represented compactly in a tree structure. Finally, we describea procedure which substantially shortens ND proofs by abstractingthem to the assertion level, and report our experience with furthertransformation into natural language.

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)

Over the past thirty years there have been significant achievements in the field of auto-mated theorem proving with respect to the reasoning power of the inference engines.Although some effort has also been spent to facilitate more user friendliness of the de-duction systems, most of them failed to benefit from more recent developments in therelated fields of artificial intelligence (AI), such as natural language generation and usermodeling. In particular, no model is available which accounts both for human deductiveactivities and for human proof presentation. In this thesis, a reconstructive architecture issuggested which substantially abstracts, reorganizes and finally translates machine-foundproofs into natural language. Both the procedures and the intermediate representationsof our architecture find their basis in computational models for informal mathematicalreasoning and for proof presentation. User modeling is not incorporated into the currenttheory, although we plan to do so later.

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.

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.

We argue in this paper that sophisticated mi-croplanning techniques are required even formathematical proofs, in contrast to the beliefthat mathematical texts are only schematicand mechanical. We demonstrate why para-phrasing and aggregation significantly en-hance the flexibility and the coherence ofthe text produced. To this end, we adoptedthe Text Structure of Meteer as our basicrepresentation. The type checking mecha-nism of Text Structure allows us to achieveparaphrasing by building comparable combi-nations of linguistic resources. Specified interms of concepts in an uniform ontologicalstructure called the Upper Model, our se-mantic aggregation rules are more compactthan similar rules reported in the literature.

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.