Mechanical Geometry Theorem Proving

Approach your problems from the right end It isn't that they can't see the solution.

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Author: Shang-Ching Chou

Publisher: Springer

ISBN: 1402003307

Category: Computers

Page: 362

View: 175

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Mechanical Theorem Proving in Geometries

The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a ...

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Author: Wen-tsün Wu

Publisher: Springer Science & Business Media

ISBN: 9783709166390

Category: Computers

Page: 288

View: 462

There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations.

Mathematics Mechanization

This book is a collection of essays centred around the subject of mathematical mechanization.

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Author: Wen-tsün Wu

Publisher:

ISBN: 7030066863

Category: Artificial intelligence

Page: 420

View: 586

A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton s Principia

We mechanize, within the theorem prover Isabelle, procedures of the Principia that have often been regarded as logically vague by investigating and applying concepts from both mechanical geometry theorem proving (GTP) and Nonstandard ...

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Author: Jacques Fleuriot

Publisher: Springer Science & Business Media

ISBN: 9780857293299

Category: Mathematics

Page: 140

View: 196

Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague. In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.

Wu wen jun quan ji

本书是围绕作者命名的"数学机械化"这一中心议题而陆续发表的一系列论文的综述, 试图以构造性与算法化的方式来研究数学, 使数学推理机械化以至于自动化, ...

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Author: 吴文俊

Publisher:

ISBN: 7508855507

Category:

Page: 424

View: 999

本书是围绕作者命名的"数学机械化"这一中心议题而陆续发表的一系列论文的综述,试图以构造性与算法化的方式来研究数学,使数学推理机械化以至于自动化,由此减轻繁琐的脑力劳动.

On the Mechanical Proof of Geometry Theorems Involving Inequalities

We illustrate the practical value of these techniques by numerous examples of their use in conjunction with a Wu-Ritt prover.

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Author: University of Texas at Austin. Dept. of Computer Sciences

Publisher:

ISBN: OCLC:21695871

Category: Automatic theorem proving

Page: 35

View: 232

In this paper we explore two techniques for extending Wu-Ritt and Gröbner provers to handle propositions involving inequalities: reduction of polynomials to canonical form modulo a (polynomial) ideal, and the Rabinowitsch/Seidenberg device of converting (polynomial) inequalities to equations by introducing new variables. We illustrate the practical value of these techniques by numerous examples of their use in conjunction with a Wu-Ritt prover."

Mechanical Theorem Proving in Differential Geometry I Space Curves

With an improved version of Ritt-Wu's zero decomposition algorithm for differential polynomials, we present two approaches to mechanical proving of geometry theorems in differential geometry.

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Author: University of Texas at Austin. Department of Computer Sciences

Publisher:

ISBN: OCLC:123331182

Category:

Page:

View: 779

Mechanical Theorem Proving in Riemann Geometry

Abstract: "This paper studies the mechanical theorem proving in Riemann geometry using algebraic methods.

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Author: University of Texas at Austin. Dept. of Computer Sciences

Publisher:

ISBN: OCLC:23270753

Category: Automatic theorem proving

Page: 32

View: 941

Abstract: "This paper studies the mechanical theorem proving in Riemann geometry using algebraic methods. We establish a theorem which can reduce a geometry statement in Riemann geometry to several substatements which are much easier to prove. For a class of constructive geometry statements, we present a method to generate sufficient non-degenerate conditions in geometric form mechanically. We also prove that an irreducible constructive statement is generally true if and only if it is universally true under the non-degenerate conditions generated by our method. More than 20 theorems other than those in projective geometry have been proved by our program."

Selected Topics in Geometry with Classical Vs Computer Proving

When a basic book Mechanical Geometry Theorem Proving written by Chinese mathematician S. Ch. Chou appeared in 1987 [22], American mathematician Larry Woss wrote in its preface: “When computers were first conceived, then designed, ...

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Author: Pavel Pech

Publisher: World Scientific

ISBN: 9789812709424

Category: Mathematics

Page: 239

View: 962

This textbook presents various automatic techniques based on Gr”bner bases elimination to prove well-known geometrical theorems and formulas. Besides proving theorems, these methods are used to discover new formulas, solve geometric inequalities, and construct objects ? which cannot be easily done with a ruler and compass.Each problem is firstly solved by an automatic theorem proving method. Secondly, problems are solved classically ? without using computer where possible ? so that readers can compare the strengths and weaknesses of both approaches.

Machine Proofs in Geometry

[84] D. A. Cyrluk, R. M. Harris, & D. Kapur, GEOMETER, A Theorem Prover for Algebraic Geometry, Proc. of CADE-9, Argonne, ... [93] X. S. Gao, An Introduction to Wu's Method of Mechanical Geometry Theorem Proving, IFIP Transaction on ...

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Author: S-C Chou

Publisher: World Scientific

ISBN: 9789814502603

Category: Computers

Page: 480

View: 648

This book reports recent major advances in automated reasoning in geometry. The authors have developed a method and implemented a computer program which, for the first time, produces short and readable proofs for hundreds of geometry theorems. The book begins with chapters introducing the method at an elementary level, which are accessible to high school students; latter chapters concentrate on the main theme: the algorithms and computer implementation of the method. This book brings researchers in artificial intelligence, computer science and mathematics to a new research frontier of automated geometry reasoning. In addition, it can be used as a supplementary geometry textbook for students, teachers and geometers. By presenting a systematic way of proving geometry theorems, it makes the learning and teaching of geometry easier and may change the way of geometry education. Contents:Part I: The Theory of Machine Proof:Geometry PreliminariesThe Area MethodMachine Proof in Plane GeometryMachine Proof in Solid GeometryVectors and Machine ProofsPart II: Topics from Geometry:List of SymbolsBibliographyIndex Readership: Researchers in artificial intelligence, computer science and mathematics; students and teachers. keywords:

Learning and Geometry Computational Approaches

S.C. Chou, “A Geometry Theorem Prover for Macintoshes”, in Proceedings of CADE-11, Lecture Notes in Computer Science, Vol. 607, 686–689, 1992. S.C. Chou and X.S. Gao, “Mechanical Theorem Proving in Riemann Goemetry”, TR-903, ...

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Author: David Kueker

Publisher: Springer Science & Business Media

ISBN: 9781461240884

Category: Computers

Page: 212

View: 537

The field of computational learning theory arose out of the desire to for mally understand the process of learning. As potential applications to artificial intelligence became apparent, the new field grew rapidly. The learning of geo metric objects became a natural area of study. The possibility of using learning techniques to compensate for unsolvability provided an attraction for individ uals with an immediate need to solve such difficult problems. Researchers at the Center for Night Vision were interested in solving the problem of interpreting data produced by a variety of sensors. Current vision techniques, which have a strong geometric component, can be used to extract features. However, these techniques fall short of useful recognition of the sensed objects. One potential solution is to incorporate learning techniques into the geometric manipulation of sensor data. As a first step toward realizing such a solution, the Systems Research Center at the University of Maryland, in conjunction with the Center for Night Vision, hosted a Workshop on Learning and Geometry in January of 1991. Scholars in both fields came together to learn about each others' field and to look for common ground, with the ultimate goal of providing a new model of learning from geometrical examples that would be useful in computer vision. The papers in the volume are a partial record of that meeting.

Computer Mathematics Proceedings Of The Special Program At Nankai Institute Of Mathematics

(CH2] S.C. Chou, Mechanical Geometry Theorem Proving, D.Reidel Publishing Company 1988. (CG1] S.C. Chou and X.S. Gao, Ritt-Wu's Decomposition Algorithm and Geometry Thec rem Proving, 10th International Conference on Automated Deduction, ...

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Author: Wu Wen-tsun

Publisher: World Scientific

ISBN: 9789814552516

Category:

Page: 172

View: 990

The area of molecular imaging has matured over the past decade and is still growing rapidly. Many concepts developed for molecular biology and cellular imaging have been successfully translated to in vivo imaging of intact organisms. Molecular imaging enables the study of processes at a molecular level in their full biological context. Due to the high specificity of the molecular readouts the approach bears a high potential for diagnostics. It is fair to say that molecular imaging has become an indispensable tool for biomedical research and drug discovery and development today.This volume familiarizes the reader with the concepts of imaging and molecular imaging in particular. Basic principles of imaging technologies, reporter moieties for the various imaging modalities, and the design of targeted probes are described in the first part. The second part illustrates how these tools can be used to visualize relevant molecular events in the living organism. Topics covered include the studies of the biodistribution of reporter probes and drugs, visualization of the expression of biomolecules such as receptors and enzymes, and how imaging can be used for analyzing consequences of the interaction of a ligand or a drug with its molecular target by visualizing signal transduction, or assessing the metabolic, physiological, or structural response of the organism studied. The final chapter deals with visualization of cell migration, for example in the context of cell therapies.The second edition covers novel developments over recent years, in particular regarding imaging technologies (hybrid techniques) and novel reporter concepts. Novel biomedical applications have been included, where appropriate. All the chapters have been thoroughly reworked and the artwork updated.

Handbook of Automated Reasoning

11 CHOU S. ( 1988 ) , Mechanical Geometry Theorem Proving , D. Reidel Publishing Company , Dordrecht , Netherlands . CHOU S. ( 1990 ) , Automated reasoning in geometry using the CS and GB methods , in ' Proc . of ISSAC'90 , ( Tokyo ) ' ...

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Author: Alan J.A. Robinson

Publisher: Gulf Professional Publishing

ISBN: 0444829490

Category: Computers

Page: 2122

View: 709

Handbook of Automated Reasoning.

Theorem Proving in Higher Order Logics

Machine Proofs in Geometry. World Scientific, Singapore, 1994. 3. Shang-Ching Chou. Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, 1988. 4. G. E. Collins. Quantifier elimination for real closed fields by cylindrical ...

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Author: Konrad Slind

Publisher: Springer

ISBN: 9783540301424

Category: Computers

Page: 340

View: 847

This volume constitutes the proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004) held September 14–17, 2004 in Park City, Utah, USA. TPHOLs covers all aspects of theorem proving in higher-order logics as well as related topics in theorem proving and veri?cation. There were 42 papers submitted to TPHOLs 2004 in the full research ca- gory, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the c- ference and publication in this volume. In keeping with longstanding tradition, TPHOLs 2004 also o?ered a venue for the presentation of work in progress, where researchers invited discussion by means of a brief introductory talk and then discussed their work at a poster session. A supplementary proceedings c- taining papers about in-progress work was published as a 2004 technical report of the School of Computing at the University of Utah. The organizers are grateful to Al Davis, Thomas Hales, and Ken McMillan for agreeing to give invited talks at TPHOLs 2004. The TPHOLs conference traditionally changes continents each year in order to maximize the chances that researchers from around the world can attend.

Computing in Euclidean Geometry

S.C. Chou, "GEO-Prover - A Geometry Theorem Prover Developed at UT" in Proceedings of CADES, Oxford, 1986. S.C. Chou, "An Introduction to Wu's Method for Mechanical Theorem Proving in Geometry", Journal of Automated Reasoning, 4 (1988), ...

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Author: Ding-Zhu Du

Publisher: World Scientific

ISBN: 9789814505604

Category: Mathematics

Page: 400

View: 986

This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. The topics covered are: a history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra; triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and steiner trees. Each chapter is written by a leading expert in the field and together they provide a clear and authoritative picture of what computational Euclidean geometry is and the direction in which research is going. Contents:Mesh Generation and Optimal Triangulation (M Bern & D Eppstein)Machine Proofs of Geometry Theorems (S-C Chou & M Rathi)Randomized Geometric Algorithms (K L Clarkson)Voronoi Diagrams and Delauney Triangulations (S Fortune)The State of Art on Steiner Ratio Problems (D-Z Du & F Hwang)On the Development of Quantitative Geometry from Pythagoras to Grassmann (W-Y Hsiang)Computational Geometry and Topological Network Design (J M Smith & P Winter)Polar Forms and Triangular B-Spline Surfaces (H-P Seidel) Readership: Computer scientists and mathematicians. keywords:Computational Geometry;Triangulation;Machine Proof;Randomized Geometric Algorithm;Voronoi Diagram;Delaunay Triangulation;B-Spline;Polar Form;Steiner Tree;Analytic Geometry “D-Z Du and F Hwang have put to rest an optimization problem known as the Steiner ratio conjecture. Their solution closes the book on a problem that had frustrated a generation of geometers, but it also writes the first chapter of a new volume. The key to Du and Hwang's successful attack on the conjecture is a new method that has potential for solving a raft of other optimization problems.” SIAM News, USA “… the eight surveys are well organized. Each survey is preceded by a good introductory section with a rich bibliography. Both beginners and experts will benefit from this book.” Mathematical Reviews “The papers are not just summaries; the authors present new material or fresh points of view … I recommend the book to anyone who works in one of the areas surveyed or who is interested in the interaction of Euclidean geometry and computers.” IEEE Parallel & Distributed Technology

Computing in Euclidean Geometry

S.C. Chou, "GEO-Prover - A Geometry Theorem Prover Developed at UT" in Proceedings of CADES, Oxford, 1986. S.C. Chou, "An Introduction to Wu's Method for Mechanical Theorem Proving in Geometry", Journal of Automated Reasoning, 4 (1988), ...

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Author: Ding-Zhu Du

Publisher: World Scientific

ISBN: 9789814501637

Category: Computers

Page: 508

View: 142

This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. Topics covered include the history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra, triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and Steiner trees. This second edition contains three new surveys covering geometric constraint solving, computational geometry and the exact computation paradigm. Contents:On the Development of Quantitative Geometry from Phythagoras to Grassmann (W-Y Hsiang)Computational Geometry: A Retrospective (B Chazelle)Mesh Generation and Optimal Triangulation (M Bern & D Eppstein)Machine Proofs of Geometry Theorems (S-C Chou & M Rathi)Randomized Geometric Algorithms (K L Clarkson)The State of Art on Steiner Ratio Problems (D-Z Du & F Hwang)Voronoi Diagrams and Delaunay Triangulations (S Fortune)Geometric Constraint Solving in R2 and R3 (C M Hoffmann & P J Vermeer)Polar Forms and Triangular B-Spline Surfaces (H-P Seidel)Computational Geometry and Topological Network Design (J M Smith & P Winter)The Exact Computation Paradigm (C Yap & T Dubé) Readership: Computer scientists and mathematicians. keywords:Computational Geometry;Triangulation;Machine Proof;Randomized Geometric Algorithm;Voronoi Diagram;Delaunay Triangulation;B-Spline;Polar Form;Steiner Tree;Analytic Geometry;Exact Computation Review on First Edition: “The papers are not just summaries; the authors present new material or fresh points of view … I recommend the book to anyone who works in one of the areas surveyed or who is interested in the interaction of Euclidean geometry and computers.” IEEE Parallel & Distributed Technology