Randomized Xero Testing of Radical Expressions and Elementary Geometry Theorem Proving

Tulone, Daniela; Yap, Chee; Li, Chen

doi:10.1007/3-540-45410-1_5

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…Since this is a randomized algorithm one might develop a probability estimation on how certain one can be that the solutions is actually continuous. Usually this is done using the Schwartz-Zippel lemma (see for example Mitzenmacher and Upfal (2005), or including radical expressions: Tulone et al (2000)). We will defer the development of randomized continuity testing using infinitesimal deflection to future work and use Algorithm 2 as heuristic algorithm.…”

Section: Application To Singularities In Geometric Constructionsmentioning

confidence: 99%

Non-standard analysis in dynamic geometry

Strobel

2020

Journal of Symbolic Computation

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In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are smaller than every real number: the so called "infinitesimal" numbers. The theory is well known from non-standard analysis. This enables us to define projective objects that deviate only infinitesimally (in an appropriate metric) from each other. And we show that in most cases operations involving such objects that deviate infinitesimally also experience only infinitesimal change. Another focus will be where this property does not hold true and show that this usually involves discontinuity or degeneration.Furthermore, we will explore common projective concepts like projective transformations, cross-ratios and conics in a non-standard setting.

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Section: Application To Singularities In Geometric Constructionsmentioning

confidence: 99%

Non-standard analysis in dynamic geometry

Strobel

2020

Journal of Symbolic Computation

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“…For instance, assuming all numbers involved are rational, a number is probably null if its interval is sharp enough and contains zero, and the number's hashed value (result of a modular computation modulo a large prime) is zero. This method is efficient and simple in the rational case, but is less appealing in the algebraic case [47,49,47,50].…”

Section: Solutions At the Arithmetic Levelmentioning

confidence: 99%

“…This probabilistic principle, called proof by example, is also used for probabilistic proofs of geometric theorems [51][52][53]. It has been extended beyond polynomials [49,47,50], e.g., Tulone et al [49] extend it to radical expressions which occur in ruler and compass geometric constructions and related geometric theorems, such as Pascal's for a circle.…”

Section: Solutions At the Arithmetic Levelmentioning

confidence: 99%

Robustness and Randomness

Michelucci

Moreau

Foufou

Reliable Implementation of Real Number Algorithms: Theory and Practice

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Abstract. Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floatingpoint instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discusses approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust.

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A Probabilistic Algorithm for Verification of Geometric Theorems

Chen

Zeng

2019

Algorithmic Aspects in Information and Management

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Randomized Xero Testing of Radical Expressions and Elementary Geometry Theorem Proving

Cited by 7 publications

References 14 publications

Non-standard analysis in dynamic geometry

Non-standard analysis in dynamic geometry

Robustness and Randomness

A Probabilistic Algorithm for Verification of Geometric Theorems

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