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How Mathematicians Prove Theorems
 IN PROC. OF THE ANNUAL CONFERENCE OF THE COGNITIVE SCIENCE SOCIETY
, 1994
"... This paper analyzes how mathematicians prove theorems. The analysis is based upon several empirical sources such as reports of mathematicians and mathematical proofs by analogy. In order to combine the strength of traditional automated theorem provers with humanlike capabilities, the questions aris ..."
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This paper analyzes how mathematicians prove theorems. The analysis is based upon several empirical sources such as reports of mathematicians and mathematical proofs by analogy. In order to combine the strength of traditional automated theorem provers with humanlike capabilities, the questions arise: Which problem solving strategies are appropriate? Which representations have to be employed? As a result of our analysis, the following reasoning strategies are recognized: proof planning with partially instantiated methods, structuring of proofs, the transfer of subproofs and of reformulated subproofs. We discuss the representation of a component of these reasoning strategies, as well as its properties. We find some mechanisms needed for theorem proving by analogy, that are not provided by previous approaches to analogy. This leads us to a computational representation of new components and procedures for automated theorem proving systems.
Adapting Methods to Novel Tasks in Proof Planning
 KI94: ADVANCES IN ARTIFICIAL INTELLIGENCE  PROCEEDINGS OF KI94, 18TH GERMAN ANNUAL CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 1994
"... In this paper we generalize the notion of method for proof planning. While we adopt the general structure of methods introduced by Alan Bundy, we make an essential advancement in that we strictly separate the declarative knowledge from the procedural knowledge. This change of paradigm not only leads ..."
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Cited by 8 (8 self)
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In this paper we generalize the notion of method for proof planning. While we adopt the general structure of methods introduced by Alan Bundy, we make an essential advancement in that we strictly separate the declarative knowledge from the procedural knowledge. This change of paradigm not only leads to representations easier to understand, it also enables modeling the important activity of formulating metamethods, that is, operators that adapt the declarative part of existing methods to suit novel situations. Thus this change of representation leads to a considerably strengthened planning mechanism. After presenting our declarative approach towards methods we describe the basic proof planning process with these. Then we define the notion of metamethod, provide an overview of practical examples and illustrate how metamethods can be integrated into the planning process.
Analogy in problem solving
 Handbook of Practical Reasoning: Computational and Theoretical Aspects
, 1998
"... When Konrad Lorenz was awarded the Nobel Prize for medicine in 1973 he delivered the lecture "Analogy as a Source of Knowledge" and acknowledged that "...this procedure (analogical ..."
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When Konrad Lorenz was awarded the Nobel Prize for medicine in 1973 he delivered the lecture "Analogy as a Source of Knowledge" and acknowledged that "...this procedure (analogical
Representing and reformulating diagonalization methods
, 1994
"... Abstract Finding an appropriate representation of planning operators is crucial for theorem provers that work with proof planning. We show a new representation of operators and demonstrate how diagonalization can be represented by operators. We explain how a diagonalization operator used in one proo ..."
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Abstract Finding an appropriate representation of planning operators is crucial for theorem provers that work with proof planning. We show a new representation of operators and demonstrate how diagonalization can be represented by operators. We explain how a diagonalization operator used in one proofplan can be analogically transferred to an operator used in another proofplan. Finally, we find an operator that is common to all the proofplans and thus might be considered as the Diagonal Method. This research was supported by the MaxKade Foundation Keywords: proof planning, analogy, knowledge representation 1 Introduction As pointed out by Bundy [3] and Bledsoe [1], using proofplans is often very helpful in automated deduction. In planning, operators are needed and therefore an appropriate representation of these operators is crucial for proof planning. The operators have the same function in proof planning as mathematical methods (in the following referred to as mmethods) have in human theorem proving. Since mmethods can be adapted to different proofs, it is also desirable to have mechanisms for adapting operators. To be employed by a humanoriented theorem prover, these operators should allow for representing logical proof methods, such as Indirect Proof, and mathematical methods, such as Cantor's Diagonal method. In this paper we examine whether the presented representation actually covers mathematician's methods and how the methods can be adapted for other proof plans. We do this by analyzing the wellknown Diagonal Method which is central and widely applicable in many mathematical proofs concerning computability and decidability, including G"odel's Incompleteness theorem for arithmetic, the Unsolvability of the halting problem, Rice's theorem (see [5]), and the Second Recursion theorem (see [5]). Although this mmethod seems to be clearly understood, not all proofs have an obvious common proof schema, and some proofs are difficult to generate in logical detail.
Analogy Makes Proofs Feasible
 AAAIWORKSHOP ON CASEBASED REASONING
, 1994
"... Many mathematical proofs are hard to generate for humans and even harder for automated theorem provers. Classical techniques of automated theorem proving involve the application of basic rules, of builtin special procedures, or of tactics. Melis (Melis 1993) introduced a new method for analogical re ..."
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Many mathematical proofs are hard to generate for humans and even harder for automated theorem provers. Classical techniques of automated theorem proving involve the application of basic rules, of builtin special procedures, or of tactics. Melis (Melis 1993) introduced a new method for analogical reasoning in automated theorem proving. In this paper we show how the derivational analogy replay method is related and extended to encompass analogydriven proof plan construction. The method is evaluated by showing the proof plan generation of the Pumping Lemma for context free languages derived by analogy with the proof plan of the Pumping Lemma for regular languages. This is an impressive evaluation test for the analogical reasoning method applied to automated theorem proving, as the automated proof of this Pumping Lemma is beyond the capabilities of any of the current automated theorem provers.
ΩMKRP: A Proof Development Environment
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] wi ..."
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Cited by 2 (1 self)
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In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] with the firstorder theorem prover mkrp [11]. An important finding was that although current automated theorem provers have evidently reached the power to solve nontrivial problems, they do not provide sufficient assistance for proving the theorems contained in such a textbook. On account of this, we believe that significantly more support for proof development can be provided by a system with the following two features:  The system must provide a comfortable humanoriented problemsolving environment. In particular, a human user should be able to specify the problem to be solved in a natural way and communicate on proof
Analogies between Proofs  A Case Study
 Fachbereich Informatik, Universitat des Saarlandes, Im Stadtwald
, 1993
"... ion of both problems (i.e., theorem and assumptions) 7.5.7.2.c and 5.7.2.c based on the meaning of the two respective definitions of homomorphism. The key is a reformulation of terms of the form f \Delta term(x) to terms Op(term(x)) for 7.5.7.2c and term1\Deltaterm2 to Op(term1,term2) with a functi ..."
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ion of both problems (i.e., theorem and assumptions) 7.5.7.2.c and 5.7.2.c based on the meaning of the two respective definitions of homomorphism. The key is a reformulation of terms of the form f \Delta term(x) to terms Op(term(x)) for 7.5.7.2c and term1\Deltaterm2 to Op(term1,term2) with a function variable Op. This reformulation affects the definitions of homomorphism within the relevant assumptions: 8f8x(f 2 F x 2 S ! OE(f \Delta x) = f \Delta OE(x)) becomes 8x(x 2 S ! OE(Op(x)) = Op(OE(x))) by the mapping f \Delta term )Op(term) 8x; y(x 2 S 0 y 2 S 0 ! OE(x \Delta y) = OE(x) \Delta OE(y)) becomes 8x; y(x 2 S 0 y 2 S 0 ! OE(Op 0 (x; y)) = Op 0 (OE(x); OE(y))) by the mapping term1\Deltaterm2 )Op(term1,term2). The reformulation affects also the corresponding terms within the whole proof. Certain subformulae and quantifiers become superfluous and, hence, can be omitted. As a result we obtain the theorems and reformulated proofs 7.5.7.2c 0 and 5.7.2.c 0 . 2. T...
Representing the Diagonalization Methods
, 1994
"... on methods are wellknown and play a central role in many mathematical proofs. In our context, they are interesting since they are all based on one simple technique, but they appear in many different forms and can be applied to different problems. Concretely, we will examine the following problems w ..."
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on methods are wellknown and play a central role in many mathematical proofs. In our context, they are interesting since they are all based on one simple technique, but they appear in many different forms and can be applied to different problems. Concretely, we will examine the following problems where diagonalization is used:  Cantor's theorem which states that the cardinality of any set is smaller than that of its powerset.  Uncountability of the set of real numbers (concretely, the interval [0; 1]).  Unsolvability of the halting problem.  Godel's incompleteness theorem. Note that these problems are of quite a different nature at first glance. We shall see, however, how a method used for the first proof can be adapted for the second problem by applying some metamethods. We are investigating if the same can be done for the other two. Some of the metamethods used in these adaptations change a me
Theorem Proving by Analogy  A Compelling Example
, 1995
"... This paper shows how a new approach to theorem proving by analogy is applicable to real maths problems. This approach works at the level of proofplans and employs reformulation that goes beyond symbol mapping. The HeineBorel theorem is a widely known result in mathematics. It is usually stated in ..."
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This paper shows how a new approach to theorem proving by analogy is applicable to real maths problems. This approach works at the level of proofplans and employs reformulation that goes beyond symbol mapping. The HeineBorel theorem is a widely known result in mathematics. It is usually stated in R¹ and similar versions are also true in R², in topology, and metric spaces. Its analogical transfer was proposed as a challenge example and could not be solved by previous approaches to theorem proving by analogy. We use a proofplan of the HeineBorel theorem in R¹ as a guide in automatically producing a proofplan of the HeineBorel theorem in R² by analogydriven proofplan construction.