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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 23 (11 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.
AgentOriented Integration of Distributed Mathematical Services
 Journal of Universal Computer Science
, 1999
"... Realworld applications of automated theorem proving require modern software environments that enable modularisation, networked interoperability, robustness, and scalability. These requirements are met by the AgentOriented Programming paradigm of Distributed Artificial Intelligence. We argue that ..."
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Cited by 20 (10 self)
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Realworld applications of automated theorem proving require modern software environments that enable modularisation, networked interoperability, robustness, and scalability. These requirements are met by the AgentOriented Programming paradigm of Distributed Artificial Intelligence. We argue that a reasonable framework for automated theorem proving in the large regards typical mathematical services as autonomous agents that provide internal functionality to the outside and that, in turn, are able to access a variety of existing external services. This article describes...
Exploring Properties of Residue Classes
, 2000
"... We report on an experiment in exploring properties of residue classes over the integers with the combined effort of a multistrategy proof planner and two computer algebra systems. An exploration module classifies a given set and a given operation in terms of the algebraic structure they form. It th ..."
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Cited by 18 (11 self)
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We report on an experiment in exploring properties of residue classes over the integers with the combined effort of a multistrategy proof planner and two computer algebra systems. An exploration module classifies a given set and a given operation in terms of the algebraic structure they form. It then calls the proof planner to prove or refute simple properties of the operation. Moreover, we use different proof planning strategies to implement various proving techniques: from naive testing of all possible cases to elaborate techniques of equational reasoning and reduction to known cases.
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 9 (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.
Making Conjectures about Maple Functions
 In: Proceedings of the Tenth Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, LNAI 2385
, 2002
"... One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand. ..."
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Cited by 9 (6 self)
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One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand.
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 9 (6 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.
Nontrivial Symbolic Computations in Proof Planning
 In Proc. of FroCoS 2000, LNCS 1794
, 2000
"... We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using ..."
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Cited by 7 (4 self)
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We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using a powerful computer algebra system during the planning process to do nontrivial symbolic computations. Results of these computations are checked during the refinement of a proof plan to a calculus level proof using a small, selfimplemented, system that gives us protocol information on its calculation. This protocol can be easily expanded into a checkable lowlevel calculus proof ensuring the correctness of the computation. We demonstrate our approach with the concrete implementation in the Omega system.
Employing External Reasoners in Proof Planning
 In CALCULEMUS 99, Electronic Notes in Theoretical Computer Science
, 1999
"... This paper describes a the integration of computer algebra systems and constraint solvers into proof planners. It shows how efficient external reasoners can be employed in proof planning and how the shortcuts of the external reasoners can be expanded to verifiable natural deduction proofs in the pro ..."
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Cited by 5 (4 self)
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This paper describes a the integration of computer algebra systems and constraint solvers into proof planners. It shows how efficient external reasoners can be employed in proof planning and how the shortcuts of the external reasoners can be expanded to verifiable natural deduction proofs in the proof planning framework. It illustrates the integration and cooperation of the external reasoners with an example from proof planning limit theorems.
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,...
Adaptive Course Generation and Presentation
, 2000
"... Today's interactive mathematics textbooks use a collection of predefined documents, typically organized as a network of HtML pages. This makes a reuse and a sound recombination of the encoded knowledge impossible and inhibits a radical adaption of course presentation and content to the user&ap ..."
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Cited by 3 (0 self)
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Today's interactive mathematics textbooks use a collection of predefined documents, typically organized as a network of HtML pages. This makes a reuse and a sound recombination of the encoded knowledge impossible and inhibits a radical adaption of course presentation and content to the user's needs. In order to avoid these drawbacks we have designed a webbased framework for dynamically producing interactive documents for learning mathematics called ID. The system design relies on the separation of knowledge representation from system functionalities. Salient features of our system are the individual generation of interactive documents based on general domain knowledge, userspecific preferences and the user's knowledge as well as the integration of external problem solving systems. The paper describes the distributed webbased architecture of our system and the principles of its components.