Non-trivial Symbolic Computations in Proof Planning

Abstract. Interactive exercise assistants support students in practicing exercises, and acquiring procedural skills. Many mathematical topics can be practiced in such assistants. Ideally, an interactive exercise assistant not only validates final answers, but also comments on intermediate steps submitted by a student, provides hints on how to proceed, and presents worked-out examples. For these purposes, fine control over the symbolic simplification procedures of the underlying mathematical machinery is needed. In this paper, we introduce views for mathematical expressions. A view defines an equivalence relation by choosing a canonical form of mathematical expressions. We use views to track and recognize intermediate answers, to help in presenting expressions to a user, and to control the granularity of the steps in worked-out examples. We develop the concept of a view, discuss the laws it satisfies, and show how views are composed, which means that they can be used for multiple exercise classes.

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“…For example, Sorge [16] uses similar techniques as we do in the proof planner Ωmega. No concept of views is introduced though.…”

Section: Related Workmentioning

confidence: 99%

Canonical Forms in Interactive Exercise Assistants

Heeren

Jeuring

2009

Lecture Notes in Computer Science

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“…More detailed introductions can be found in Melis and Meier (2000), Kerber et al (1998) and Sorge (2000), respectively.…”

Section: Proof Planning and Computer Algebramentioning

confidence: 99%

“…In particular, we use the mainstream CAS Maple (Redfern, 1999) and Gap (GAP, 1998), a system specialized on group theory. In this paper we are not concerned with the technical side of the integration since we exploit previous work, in particular Kerber et al (1998), that presents the integration of computer algebra into proof planning, and Sorge (2000), that exemplifies how the correctness of certain limited computations of a large-scale CAS such as Maple can be guaranteed within the proof planning framework. Instead we concentrate on the cooperation between the systems in the context of exploring residue class properties.…”

Section: Employing Computer Algebra In Proof Planningmentioning

confidence: 99%

“…This suggested employing multi-strategy proof planning as a tool for constructing proofs since it allows easy modelling of different human-oriented proof techniques by different strategies. Moreover, proof planning enables us to exploit the power of CASs in a sound way (Kerber et al, 1998;Sorge, 2000) in order to guide and shorten the proof process.…”

Section: Introductionmentioning

confidence: 99%

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Comparing Approaches to the Exploration of the Domain of Residue Classes

Meier

Pollet

Sorge

2002

Journal of Symbolic Computation

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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 multi-strategy 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 effectiveness 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.

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“…The believing approach trusts completely the results given by a computer algebra system and treats them as axioms (Ballarinet al, 1995). The skeptical approach does not trust completely the results given by a computer algebra oracle and treats them as witnesses that require formal verification (Harrison and Théry, 1993;Kerber et al, 1998;Sorge, 2000). This approach presupposes that the chosen oracle is good at finding the witnesses.…”

Section: Introductionmentioning

confidence: 99%

Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

Caprotti

Oostdijk

2001

Journal of Symbolic Computation

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This paper focuses on how to use Pocklington's criterion to produce efficient formal proof-objects for showing primality of large positive numbers. First, we describe a formal development of Pocklington's criterion, done using the proof assistant Coq. Then we present an algorithm in which computer algebra software is employed as oracle to the proof assistant to generate the necessary witnesses for applying the criterion. Finally, we discuss the implementation of this approach and tackle the proof of primality for some of the largest numbers expressible in Coq.

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Non-trivial Symbolic Computations in Proof Planning

Cited by 16 publications

References 13 publications

Canonical Forms in Interactive Exercise Assistants

Canonical Forms in Interactive Exercise Assistants

Comparing Approaches to the Exploration of the Domain of Residue Classes

Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

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