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19
Comparing approaches to the exploration of the domain of residue classes
 ARTICLE SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
, 2002
"... We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proof techniques, which are implemented as strategies in a multistrategy ..."
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Cited by 25 (13 self)
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We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proof techniques, which are implemented as strategies in a multistrategy proof planner. The search space of the proof planner can be drastically reduced by employing computations of two computer algebra systems during the planning process. To test the eectiveness of our approach we carried out a large number of experiments and also compared it with some alternative approaches. In particular, we experimented with substituting computer algebra by model generation and by proving theorems with a first order equational theorem prover instead of a proof planner.
Automatic learning of proof methods in proof planning
 L. J. of the IGPL
, 2002
"... Our research interests in this project are in exploring how automated reasoning systems can learn theorem proving strategies. In particular, we are looking into how a proof planning system (Bundy, 1988) can automatically learn ..."
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Cited by 11 (4 self)
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Our research interests in this project are in exploring how automated reasoning systems can learn theorem proving strategies. In particular, we are looking into how a proof planning system (Bundy, 1988) can automatically learn
Classifying Isomorphic Residue Classes
 In Proceedings of the 8th International Workshop on Computer Aided Systems Theory (EuroCAST 2001), volume 2178 of LNCS
, 2001
"... We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proving techniques, which are implemented as strategies in a multistrategy p ..."
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Cited by 11 (8 self)
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We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proving techniques, which are implemented as strategies in a multistrategy proof planner. We show how these techniques help to successfully derive proofs in our domain and explain how the search space of the proof planner can be drastically reduced by employing computations of two computer algebra systems during the planning process. Moreover, we discuss the results of experiments we conducted which give evidence that with the help of the computer algebra systems the planner is able to solve problems for which it would fail to create a proof otherwise.
Employing Theory Formation to Guide Proof Planning
, 2002
"... The invention of suitable concepts to characterise mathematical structures is one of the most challenging tasks for both human mathematicians and automated theorem provers alike. We present an approach where automatic concept formation is used to guide nonisomorphism proofs in the residue class ..."
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Cited by 10 (6 self)
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The invention of suitable concepts to characterise mathematical structures is one of the most challenging tasks for both human mathematicians and automated theorem provers alike. We present an approach where automatic concept formation is used to guide nonisomorphism proofs in the residue class domain. The main idea behind the proof is to automatically identify discriminants for two given structures to show that they are not isomorphic. Suitable discriminants are generated by a theory formation system; the overall proof is constructed by a proof planner with the additional support of traditional automated theorem provers and a computer algebra system.
Automatic learning in proof planning
 ECAI2002: European Conference on Artificial Intelligence
, 2002
"... Abstract. In this paper we present a framework for automatedlearning within mathematical reasoning systems. In particular, this framework enables proof planning systems to automatically learnnew proof methods from well chosen examples of proofs which use a similar reasoning pattern to prove related ..."
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Cited by 8 (3 self)
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Abstract. In this paper we present a framework for automatedlearning within mathematical reasoning systems. In particular, this framework enables proof planning systems to automatically learnnew proof methods from well chosen examples of proofs which use a similar reasoning pattern to prove related theorems. Our frameworkconsists of a representation formalism for methods and a machine learning technique which can learn methods using this representationformalism. We present the implementation of this framework within the \Omega MEGA proof planning system, and some experiments we ran onthis implementation to evaluate the validity of our approach.
Randomization and HeavyTailed Behavior in Proof Planning
, 2000
"... Proof planning is the application of Artificial Intelligence planning techniques to prove mathematical theorems. While exploring the domain of the residue classes over the integers with the multistrategy proof planner Multi we found a class of hard problems on which proof planning showed a remarkab ..."
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Cited by 5 (4 self)
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Proof planning is the application of Artificial Intelligence planning techniques to prove mathematical theorems. While exploring the domain of the residue classes over the integers with the multistrategy proof planner Multi we found a class of hard problems on which proof planning showed a remarkable high degree of variance. On problems of the same complexity we either succeeded very quickly with short proofs or the proof planning process took significantly longer and resulted in a large proof. Recent work in Artificial Intelligence points out that the unpredictability in the running time of heuristic search procedures can often be explained by the phenomenon of heavytailed cost distributions. Because of the nonstandard nature of these heavytailed cost distributions the controled introduction of randomization into the search procedures and quick restarts of the randomized procedure can eliminate heavytailed behavior and can take advantage of short runs. In this report,...
ΩMEGA: Computer supported mathematics
 IN: PROCEEDINGS OF THE 27TH GERMAN CONFERENCE ON ARTIFICIAL INTELLIGENCE (KI 2004)
, 2004
"... The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated dedu ..."
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The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces – often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited. The shift
Adaptable mixedinitiative proof planning for educational interaction
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2004
"... Today, most theorem proving systems are either used by their developers or by a (small) group of particularly trained and skilled users. In order to make theorem proving functionalities useful for a larger clientele we have to ask “What does an envisioned group of users need?” For educational purpos ..."
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Today, most theorem proving systems are either used by their developers or by a (small) group of particularly trained and skilled users. In order to make theorem proving functionalities useful for a larger clientele we have to ask “What does an envisioned group of users need?” For educational purposes a theorem prover can be used in different scenarios and can serve students with different needs. Therefore, the user interface as well as the choice of functionalities of the underlying prover have to be adapted to the context and the learner. In this paper, we present proof planning as backengine for interactive proof exercises as well as an interaction console, which is part of our graphical user interface. Based on the proof planning situation, the console offers suggestions for proof steps to the learner. These suggestions can dynamically be adapted, e.g., to the user and to pedagogical criteria using pedagogical knowledge on the creation and presentation of suggestions.
UITP 2003 Preliminary Version Adaptable MixedInitiative Proof Planning for Educational Interaction
"... Abstract Today, most theorem proving systems are either used by their developers or by a (small) group of particularly trained and skilled users. In order to make theorem proving functionalities useful for a larger clientele we have to ask "What does an envisioned group of users need?&q ..."
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Abstract Today, most theorem proving systems are either used by their developers or by a (small) group of particularly trained and skilled users. In order to make theorem proving functionalities useful for a larger clientele we have to ask &quot;What does an envisioned group of users need?&quot; For educational purposes a theorem prover can be used in different scenarios and can serve students with different needs. Therefore, the user interface as well as the choice of functionalities of the underlying prover have to be adapted to the context and the learner. In this paper, we present proof planning as backengine for interactive proof exercises as well as an interaction console, which is part of our graphical user interface. Based on the proof planning situation, the console offers suggestions for proof steps to the learner. These suggestions can dynamically be adapted, e.g., to the user and to pedagogical criteria using pedagogical knowledge on the creation and presentation of suggestions. Key words: mathematics education, adaptive GUI, adaptive theorem proving 1 Motivation So far, the main goal of developing automated theorem proving systems has been to output true/false for a statement formulated in some logic or to deliver a proof object. Interactive theorem proving systems aim to support the proof construction done by a user in different ways, they restrict the search space (the choices) by making valid suggestions for proof steps, they suggest applicable lemmas, or they produce a whole subproof automatically. These functionalities are useful, e.g., for checking a student's proof for validity or for verifying a program. They are not particularly helpful, when the goal is to This is a preliminary version. The final version will be published in