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Constructing an analogy between a known and already proven theorem(the base case) and another yet to be proven theorem (the target case) oftenamounts to finding the appropriate representation at which the base and thetarget are similar. This is a well-known fact in mathematics, and it was cor-roborated by our empirical study of a mathematical textbook, which showedthat a reformulation of the representation of a theorem and its proof is in-deed more often than not a necessary prerequisite for an analogical inference.Thus machine supported reformulation becomes an important component ofautomated analogy-driven theorem proving too.The reformulation component proposed in this paper is embedded into aproof plan methodology based on methods and meta-methods, where the latterare used to change and appropriately adapt the methods. A theorem and itsproof are both represented as a method and then reformulated by the set ofmetamethods presented in this paper.Our approach supports analogy-driven theorem proving at various levels ofabstraction and in principle makes it independent of the given and often acci-dental representation of the given theorems. Different methods can representfully instantiated proofs, subproofs, or general proof methods, and hence ourapproach also supports these three kinds of analogy respectively. By attachingappropriate justifications to meta-methods the analogical inference can oftenbe justified in the sense of Russell.This paper presents a model of analogy-driven proof plan construction andfocuses on empirically extracted meta-methods. It classifies and formally de-scribes these meta-methods and shows how to use them for an appropriatereformulation in automated analogy-driven theorem proving.

Following Buchberger's approach to computing a Gröbner basis of a poly-nomial ideal in polynomial rings, a completion procedure for finitely generatedright ideals in Z[H] is given, where H is an ordered monoid presented by a finite,convergent semi - Thue system (Sigma; T ). Taking a finite set F ' Z[H] we get a(possibly infinite) basis of the right ideal generated by F , such that using thisbasis we have unique normal forms for all p 2 Z[H] (especially the normal formis 0 in case p is an element of the right ideal generated by F ). As the orderingand multiplication on H need not be compatible, reduction has to be definedcarefully in order to make it Noetherian. Further we no longer have p Delta x ! p 0for p 2 Z[H]; x 2 H. Similar to Buchberger's s - polynomials, confluence criteriaare developed and a completion procedure is given. In case T = ; or (Sigma; T ) is aconvergent, 2 - monadic presentation of a group providing inverses of length 1 forthe generators or (Sigma; T ) is a convergent presentation of a commutative monoid ,termination can be shown. So in this cases finitely generated right ideals admitfinite Gröbner bases. The connection to the subgroup problem is discussed.

This case study examines in detail the theorems and proofs that are shownby analogy in a mathematical textbook on semigroups and automata, thatis widely used as an undergraduate textbook in theoretical computer scienceat German universities (P. Deussen, Halbgruppen und Automaten, Springer1971). The study shows the important role of restructuring a proof for findinganalogous subproofs, and of reformulating a proof for the analogical trans-formation. It also emphasizes the importance of the relevant assumptions ofa known proof, i.e., of those assumptions actually used in the proof. In thisdocument we show the theorems, the proof structure, the subproblems andthe proofs of subproblems and their analogues with the purpose to providean empirical test set of cases for automated analogy-driven theorem proving.Theorems and their proofs are given in natural language augmented by theusual set of mathematical symbols in the studied textbook. As a first step weencode the theorems in logic and show the actual restructuring. Secondly, wecode the proofs in a Natural Deduction calculus such that a formal analysisbecomes possible and mention reformulations that are necessary in order toreveal the analogy.

We provide an overview of UNICOM, an inductive theorem prover for equational logic which isbased on refined rewriting and completion techniques. The architecture of the system as well as itsfunctionality are described. Moreover, an insight into the most important aspects of the internalproof process is provided. This knowledge about how the central inductive proof componentof the system essentially works is crucial for human users who want to solve non-trivial prooftasks with UNICOM and thoroughly analyse potential failures. The presentation is focussedon practical aspects of understanding and using UNICOM. A brief but complete description ofthe command interface, an installation guide, an example session, a detailed extended exampleillustrating various special features and a collection of successfully handled examples are alsoincluded.

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.

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.

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.

There are well known examples of monoids in literature which do not admit a finite andcanonical presentation by a semi-Thue system over a fixed alphabet, not even over an arbi-trary alphabet. We introduce conditional Thue and semi-Thue systems similar to conditionalterm rewriting systems as defined by Kaplan. Using these conditional semi-Thue systems wegive finite and canonical presentations of the examples mentioned above. Furthermore weshow, that each finitely generated monoid with decidable word problem is embeddable in amonoid which has a finite canonical conditional presentation.

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.

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.

We present a framework for the integration of the Knuth-Bendix completion algorithm with narrowing methods, compiled rewrite rules, and a heuristic difference reduction mechanism for paramodulation. The possibility of embedding theory unification algorithms into this framework is outlined. Results are presented and discussed for several examples of equality reasoning problems in the context of an actual implementation of an automated theorem proving system (the Mkrp-system) and a fast C implementation of the completion procedure. The Mkrp-system is based on the clause graph resolution procedure. The thesis shows the indispensibility of the constraining effects of completion and rewriting for equality reasoning in general and quantifies the amount of speed-up caused by various enhancements of the basic method. The simplicity of the superposition inference rule allows to construct an abstract machine for completion, which is presented together with computation times for a concrete implementation.

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)

An important property and also a crucial point ofa term rewriting system is its termination. Transformation or-derings, developed by Bellegarde & Lescanne strongly based on awork of Bachmair & Dershowitz, represent a general technique forextending orderings. The main characteristics of this method aretwo rewriting relations, one for transforming terms and the otherfor ensuring the well-foundedness of the ordering. The centralproblem of this approach concerns the choice of the two relationssuch that the termination of a given term rewriting system can beproved. In this communication, we present a heuristic-based al-gorithm that partially solves this problem. Furthermore, we showhow to simulate well-known orderings on strings by transformationorderings.

This report presents a methodology to guide equational reasoningin a goal directed way. Suggested by rippling methods developed inthe field of inductive theorem proving we use attributes of terms andheuristics to determine bridge lemmas, i.e. lemmas which have tobe used during the proof of the theorem. Once we have found sucha bridge lemma we use the techniques of difference unification andrippling to enable its use.

This paper develops a sound and complete transformation-based algorithm forunification in an extensional order-sorted combinatory logic supporting constantoverloading and a higher-order sort concept. Appropriate notions of order-sortedweak equality and extensionality - reflecting order-sorted fij-equality in thecorresponding lambda calculus given by Johann and Kohlhase - are defined, andthe typed combinator-based higher-order unification techniques of Dougherty aremodified to accommodate unification with respect to the theory they generate. Thealgorithm presented here can thus be viewed as a combinatory logic counterpartto that of Johann and Kohlhase, as well as a refinement of that of Dougherty, andprovides evidence that combinatory logic is well-suited to serve as a framework forincorporating order-sorted higher-order reasoning into deduction systems aimingto capitalize on both the expressiveness of extensional higher-order logic and theefficiency of order-sorted calculi.

We consider the problem of verifying confluence and termination of conditionalterm rewriting systems (TRSs). For unconditional TRSs the critical pair lemmaholds which enables a finite test for confluence of (finite) terminating systems.And for ensuring termination of unconditional TRSs a couple of methods forconstructing appropiate well-founded term orderings are known. If however ter-mination is not guaranteed then proving confluence is much more difficult. Re-cently we have obtained some interesting results for unconditional TRSs whichprovide sufficient criteria for termination plus confluence in terms of restrictedtermination and confluence properties. In particular, we have shown that anyinnermost terminating and locally confluent overlay system is complete, i.e. ter-minating and confluent. Here we generalize our approach to the conditional caseand show how to solve the additional complications due to the presence of con-ditions in the rules. Our main result can be stated as follows: Any conditionalTRS which is an innermost terminating semantical overlay system such that all(conditional) critical pairs are joinable is complete.

We will answer a question posed in [DJK91], and will show that Huet's completion algorithm [Hu81] becomes incomplete, i.e. it may generate a term rewriting system that is not confluent, if it is modified in a way that the reduction ordering used for completion can be changed during completion provided that the new ordering is compatible with the actual rules. In particular, we will show that this problem may not only arise if the modified completion algorithm does not terminate: Even if the algorithm terminates without failure, the generated finite noetherian term rewriting system may be non-confluent. Most existing implementations of the Knuth-Bendix algorithm provide the user with help in choosing a reduction ordering: If an unorientable equation is encountered, then the user has many options, especially, the one to orient the equation manually. The integration of this feature is based on the widespread assumption that, if equations are oriented by hand during completion and the completion process terminates with success, then the generated finite system is a maybe non terminating but locally confluent system (see e.g. [KZ89]). Our examples will show that this assumption is not true.

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.

The team work method is a concept for distributing automated theoremprovers and so to activate several experts to work on a given problem. We haveimplemented this for pure equational logic using the unfailing KnuthADBendixcompletion procedure as basic prover. In this paper we present three classes ofexperts working in a goal oriented fashion. In general, goal oriented experts perADform their job "unfair" and so are often unable to solve a given problem alone.However, as a team member in the team work method they perform highly effiADcient, even in comparison with such respected provers as Otter 3.0 or REVEAL,as we demonstrate by examples, some of which can only be proved using teamwork.The reason for these achievements results from the fact that the team workmethod forces the experts to compete for a while and then to cooperate by exADchanging their best results. This allows one to collect "good" intermediate resultsand to forget "useless" ones. Completion based proof methods are frequently reADgarded to have the disadvantage of being not goal oriented. We believe that ourapproach overcomes this disadvantage to a large extend.

In 1978, Klop demonstrated that a rewrite system constructed by adding the untyped lambda calculus, which has the Church-Rosser property, to a Church-Rosser first-order algebraic rewrite system may not be Church-Rosser. In contrast, Breazu-Tannen recently showed that argumenting any Church-Rosser first-order algebraic rewrite system with the simply-typed lambda calculus results in a Church-Rosser rewrite system. In addition, Breazu-Tannen and Gallier have shown that the second-order polymorphic lambda calculus can be added to such rewrite systems without compromising the Church-Rosser property (for terms which can be provably typed). There are other systems for which a Church-Rosser result would be desirable, among them being X^t+SP+FIX, the simply-typed lambda calculus extended with surjective pairing and fixed points. This paper will show that Klop's untyped counterexample can be lifted to a typed system to demonstrate that X^t+SP+FIX is not Church-Rosser.