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Proving Ground Completeness of Resolution by Proof Planning
, 1997
"... A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow ..."
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A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problemspecific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by socalled metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.
The Mechanization of the Diagonalization Proof Strategy
 FACHBEREICH INFORMATIK, UNIVERSITAT DES SAARLANDES, IM STADTWALD
, 1996
"... We present an empirical study of mathematical proofs by diagonalization, the aim is their mechanization based on proof planning techniques. We show that these proofs can be constructed according to a strategy that (i) finds an indexing relation, (ii) constructs a diagonal element, and (iii) makes th ..."
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We present an empirical study of mathematical proofs by diagonalization, the aim is their mechanization based on proof planning techniques. We show that these proofs can be constructed according to a strategy that (i) finds an indexing relation, (ii) constructs a diagonal element, and (iii) makes the implicit contradiction of the diagonal element explicit. Moreover we suggest how diagonal elements can be represented.
Analogy and Automated Reasoning
, 1999
"... We survey the state of the art in the use of analogy in automated reasoning. We are particularly interested in the work of Melis and Whittle (1998). A detailed analysis of their analogy mechanism is given. The problem domain that they tackled is inductive theorems. We examine whether their techni ..."
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We survey the state of the art in the use of analogy in automated reasoning. We are particularly interested in the work of Melis and Whittle (1998). A detailed analysis of their analogy mechanism is given. The problem domain that they tackled is inductive theorems. We examine whether their technique can be used in other problems domains, e.g. in the domain of limit theorems and group theory. 1 Introduction Analogy is a term used in many fields (e.g. mathematics, philosophy, etc.). In this paper we concentrate on the use of analogy in mathematical problem solving. In particular, we are interested in the use of analogy to solve related new problems, or when using analogy may reduce the search space for finding a solution to a problem. We refer to a problem which is used for analogical guidance to solve a new problem as a source problem. The new problem which is solved by the use of analogy is referred to as the target problem. One of the many useful advices that P'olya gave us is ...
The Diagonalization Method In Automatic Proof
, 1997
"... In proof planning, common patterns of mathematical reasoning are embodied in tactics. Each of these is associated with a method, specifying its preconditions and effects. Using methods, a plan for a proof can be constructed from tactics. This plan is then used to guide a theorem prover through the p ..."
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In proof planning, common patterns of mathematical reasoning are embodied in tactics. Each of these is associated with a method, specifying its preconditions and effects. Using methods, a plan for a proof can be constructed from tactics. This plan is then used to guide a theorem prover through the proof. We have implemented diagonalization, a common form of mathematical reasoning, in Clam, a proof planning system. The methods are based on a formalisation of diagonalization inspired by results of Cantor and Turing. Tested against several theorems these methods constructed plans for all of them. We have been successful in using these plans to generate informal, English proofs of the theorems. However, there is still scope to generalise the methods. Acknowledgements I would like to thank my supervisors Alan Bundy, Toby Walsh and Antony Maciocia for their invaluable guidance during this project. Thanks also to Manfred Kerber and Michael Kohlhase for their help. Thank you to my friends C...