Using Gröbner bases to reason about geometry problems

Kapur, Deepak

doi:10.1016/s0747-7171(86)80007-4

Cited by 104 publications

(27 citation statements)

References 5 publications

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“…Most of Chou's work has been summarized in an impressive monograph including more than 500 geometric theorems automatically proved by this method [9]. Kapur [34] developed a prover based on a radical membership test via Gröbner Basis [8] computations. Kutzler and Stifter [37] developed another prover also based on Gröbner techniques.…”

Section: Geometrymentioning

confidence: 99%

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

Sturm

2017

Math.Comput.Sci.

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Effective quantifier elimination procedures for first-order theories provide a powerful tool for generically solving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifier elimination procedures are based on a fixed set of admissible logical symbols with an implicitly fixed semantics. This admits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination for the reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifier elimination we are going to discuss recent results with a subtropical procedure for an existential fragment of the reals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems in chemistry and in the life sciences.

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Section: Geometrymentioning

confidence: 99%

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

Sturm

2017

Math.Comput.Sci.

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“…The relevance of degeneracy conditions was pointed out from the very first works in Automatic Deduction in Geometry (see, for instance, [25,26]). While an automatic procedure to identify general degeneracy conditions is still to be done, a method to efficiently treat the case of degeneracy conditions in the particular case of geometric loci is presented in section 4.…”

Section: Automatic Discovery and Locus Equationsmentioning

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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“…Recent systems also permit one to build proofs (e.g., [2]), or to check facts using an automated theorem prover (e.g., Geometry Expert [10], Cinderella [22], and Geometry Explorer [41]). Automated geometry theorem proving (consisting of several techniques such as Wu's method [42], Grobner basis method [19], and angle method [7]) is one of the most successful areas of automated reasoning. In contrast, we address a technically harder problem of synthesizing constructions, as opposed to the problem of producing correctness proof of a given construction.…”

Section: Interactive Geometry Systemsmentioning

confidence: 99%

Synthesizing geometry constructions

Gulwani

Korthikanti

Tiwari

2011

Proceedings of the 32nd ACM SIGPLAN Conference on Programming Language Design and Implementation

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In this paper, we study the problem of automatically solving ruler/compass based geometry construction problems. We first introduce a logic and a programming language for describing such constructions and then phrase the automation problem as a program synthesis problem. We then describe a new program synthesis technique based on three key insights: (i) reduction of symbolic reasoning to concrete reasoning (based on a deep theoretical result that reduces verification to random testing), (ii) extending the instruction set of the programming language with higher level primitives (representing basic constructions found in textbook chapters, inspired by how humans use their experience and knowledge gained from chapters to perform complicated constructions), and (iii) pruning the forward exhaustive search using a goal-directed heuristic (simulating backward reasoning performed by humans). Our tool can successfully synthesize constructions for various geometry problems picked up from high-school textbooks and examination papers in a reasonable amount of time. This opens up an amazing set of possibilities in the context of making classroom teaching interactive.

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Using Gröbner bases to reason about geometry problems

Cited by 104 publications

References 5 publications

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

Automatic deduction in (dynamic) geometry: Loci computation

Synthesizing geometry constructions

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