Automated geometry theorem proving using Buchberger's algorithm

Kutzler, B.; Stifter, Sabine

doi:10.1145/32439.32480

Cited by 39 publications

(12 citation statements)

References 7 publications

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“…For the machine proofs, the highly successful algebraic methods, such as Wu's method [16,24,25] and the Gröbner basis method [26][27][28], are not applicable to visual presentations for two reasons: (1) the basic quantities of these methods, the Cartesian coordinates of points, generally do not have geometric meanings; (2) the proof itself generally involves intensive polynomial computations, thus is not readable.…”

Section: Discussionmentioning

confidence: 99%

Visually Dynamic Presentation of Proofs in Plane Geometry

2009

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With dynamic mediums such as computer displays, we propose a new kind of visually dynamic presentation of proofs in plane geometry. In a single diagram for the proof, when the proof text goes on step by step with mouse clicks, the related geometry elements in the diagram are added, animated, or deleted dynamically with various visually dynamic effects. It solves not only the problem of identifying geometry elements in the proof text with those in the diagram, but also makes the proof more vividly visualized and intuitive. Our ongoing developing system "Java Geometry Expert" (JGEX) uses two methods to create such visually dynamic presentations: the manual input method and the automatic method. In this first part of the series of our work, we propose the main features of our visually dynamic presentation of proofs and present the manual input method to create such presentations. The manual input method mainly uses mouse clicks to create the dynamic geometry diagram and the proof text.

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Section: Discussionmentioning

confidence: 99%

Visually Dynamic Presentation of Proofs in Plane Geometry

2009

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“…Hundreds of difficult theorems have been proved by the computer programs based on this method. 1 Inspired by the success of Wu's method, around 1985-1986, three groups were successful in applying another algebraic method, the Gröbner basis method, to the same class of geometry theorems that Wu's method addresses [3][4][5].…”

Section: Algebraic Proofs In Ordered Geometries and Unordered Geometriesmentioning

confidence: 99%

Visually Dynamic Presentation of Proofs in Plane Geometry

2009

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We present the method for automated generation of visually dynamic presentations of plane geometry proofs based on the full-angle method. The proof generated by the full-angle method is organized hierarchically, thus it is particularly suitable for visual presentations. We also present the method for automated generation of visually dynamic presentation of proofs for the deductive database method with an additional new visual feature: given a geometrical configuration or a diagram, the final database (the fixpoint) in the deductive database method has numerous geometric properties organized into a few categories. By clicking each category, all properties of the configuration in this category are listed. And by clicking each of these properties, the corresponding geometry elements in the diagram blink or animate and, if needed, the proof of this property is generated.

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“…Impressive results by several authors using Wu's method (e.g. [3,4,5,6]) encouraged researchers to consider other algebraic methods, among which those based on Gröbner bases [7] proved to be the most relevant (see [8,9,10]). …”

Section: Introductionmentioning

confidence: 99%

Automatic deduction in (dynamic) geometry: Loci computation

Botana

Abánades

2014

Computational Geometry

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A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.

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Automated geometry theorem proving using Buchberger's algorithm

Cited by 39 publications

References 7 publications

Visually Dynamic Presentation of Proofs in Plane Geometry

Visually Dynamic Presentation of Proofs in Plane Geometry

Visually Dynamic Presentation of Proofs in Plane Geometry

Automatic deduction in (dynamic) geometry: Loci computation

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