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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.

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.

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.

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.

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)

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.

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.

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.

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.

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.

Most automated theorem provers suffer from the problem thatthey can produce proofs only in formalisms difficult to understand even forexperienced mathematicians. Effort has been made to reconstruct naturaldeduction (ND) proofs from such machine generated proofs. Although thesingle steps in ND proofs are easy to understand, the entire proof is usuallyat a low level of abstraction, containing too many tedious steps. To obtainproofs similar to those found in mathematical textbooks, we propose a newformalism, called ND style proofs at the assertion level , where derivationsare mostly justified by the application of a definition or a theorem. Aftercharacterizing the structure of compound ND proof segments allowing asser-tion level justification, we show that the same derivations can be achieved bydomain-specific inference rules as well. Furthermore, these rules can be rep-resented compactly in a tree structure. Finally, we describe a system calledPROVERB , which substantially shortens ND proofs by abstracting them tothe assertion level and then transforms them into natural language.

Planning Argumentative Texts
(1999)

This paper presents PROVERB a text planner forargumentative texts. PROVERB's main feature isthat it combines global hierarchical planning and un-planned organization of text with respect to local de-rivation relations in a complementary way. The formersplits the task of presenting a particular proof intosubtasks of presenting subproofs. The latter simulateshow the next intermediate conclusion to be presentedis chosen under the guidance of the local focus.

This paper outlines an implemented system called PROVERB that explains machine -found natural deduction proofs in natural language. Different from earlier works, we pursue a reconstructive approach. Based on the observation that natural deduction proofs are at a too low level of abstraction compared with proofs found in mathematical textbooks, we define first the concept of so-called assertion level inference rules. Derivations justified by these rules can intuitively be understood as the application of a definition or a theorem. Then an algorithm is introduced that abstracts machine-found ND proofs using the assertion level inference rules. Abstracted proofs are then verbalized into natural language by a presentation module. The most significant feature of the presentation module is that it combines standard hierarchical text planning and techniques that locally organize argumentative texts based on the derivation relation under the guidance of a focus mechanism. The behavior of the system is demonstrated with the help of a concrete example throughout the paper.

This paper deals with the reference choices involved in thegeneration of argumentative text. A piece of argument-ative text such as the proof of a mathematical theoremconveys a sequence of derivations. For each step of de-rivation, the premises (previously conveyed intermediateresults) and the inference method (such as the applica-tion of a particular theorem or definition) must be madeclear. The appropriateness of these references cruciallyaffects the quality of the text produced.Although not restricted to nominal phrases, our refer-ence decisions are similar to those concerning nominalsubsequent referring expressions: they depend on theavailability of the object referred to within a context andare sensitive to its attentional hierarchy . In this paper,we show how the current context can be appropriatelysegmented into an attentional hierarchy by viewing textgeneration as a combination of planned and unplannedbehavior, and how the discourse theory of Reichmann canbe adapted to handle our special reference problem.

Most automated theorem provers suffer from the problemthat the resulting proofs are difficult to understand even for experiencedmathematicians. An effective communication between the system andits users, however, is crucial for many applications, such as in a mathematical assistant system. Therefore, efforts have been made to transformmachine generated proofs (e.g. resolution proofs) into natural deduction(ND) proofs. The state-of-the-art procedure of proof transformation fol-lows basically its completeness proof: the premises and the conclusionare decomposed into unit literals, then the theorem is derived by mul-tiple levels of proofs by contradiction. Indeterminism is introduced byheuristics that aim at the production of more elegant results. This inde-terministic character entails not only a complex search, but also leads tounpredictable results.In this paper we first study resolution proofs in terms of meaningful op-erations employed by human mathematicians, and thereby establish acorrespondence between resolution proofs and ND proofs at a more ab-stract level. Concretely, we show that if its unit initial clauses are CNFsof literal premises of a problem, a unit resolution corresponds directly toa well-structured ND proof segment that mathematicians intuitively un-derstand as the application of a definition or a theorem. The consequenceis twofold: First it enhances our intuitive understanding of resolutionproofs in terms of the vocabulary with which mathematicians talk aboutproofs. Second, the transformation process is now largely deterministicand therefore efficient. This determinism also guarantees the quality ofresulting proofs.

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.

This paper outlines the microplanner of PROVERB , a system that generates multilingual text from machine-found mathematical proofs. The main representational vehicle is the text structure proposed by Meteer. Following Panaget, we also distinguish between the ideational and the textual semantic categories, and use the upper model to replace the former. Based on this, a further extension is made to support aggregation before realization decisions are made. While our the framework of our macroplanner is kept languageindependent, our microplanner draws on language specific linguistic sources such as realization classes and lexicon. Since English and German are the first two languages to be generated and because the sublanguage of our mathematical domain is relatively limited, the upper model and the textual semantic categories are designed to cope with both languages. Since the work reported is still in progress, we also discuss open problems we are facing.