Mechanical Geometry Theorem Proving

ISBN-13: 9781402003301 / Angielski / Miękka / 2001 / 362 str.

Shang-Ching Chou; Chou Shang-Ching Chou

Opis

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.

Recenzje

`This work is, in my opinion, completely revolutionary. I believe that, by itself, the book will convince any mathematician in the world that the automation of mathematical reasoning is a profound and rewarding enterprise of extraordinary potential.'
Robert S. Boyer

Spis treści

I: Methods in Mechanical Geometry Theorem Proving.- 1. An Introduction to Wu’s Method.- 1. The Defects in Traditional Proofs.- 1.1. The Traditional Euclidean Proof.- 1.2. The Traditional Analytic Proof.- 2. Four Examples.- 3. A Summary of Wu’s Method.- 4. Pseudo Division and Successive Pseudo Division.- 5. A Simple Triangulation Procedure.- 6. Geometry Statements of Constructive Type.- 7. Further Discussion of Geometry Statements of Constructive Type.- 2. Ritt’s Characteristic Set Method.- 1. The Prerequisite in Algebra.- 2. Ascending Chains and Characteristic Sets.- 3. Irreducible Ascending Chains.- 4. A Complete Triangulation Procedure: Ritt’s Principle.- 5. Ritt’s Decomposition Algorithm.- 3. Algebra and Geometry.- 1. Axiomatic Geometries and Number Systems.- 1.1. Affine Geometry.- 1.2. Metric Geometry.- 1.3. Hilbert Geometry.- 1.4. Tarski Geometry.- 2. On the Algebraic formulation of Geometry Statements.- 2.1. Formulation Fl.- 2.2. Formulation F2.- 3. Formulation F3.- 3.1. The Generic Validity of a Geometry Statement.- 3.2. Identifying Nondegenerate Conditions.- 3.3. The Generic Validity of a Geometry Statement in an Arbitrary Field.- 4. The Complete Method of Wu.- 1. Ritt’s Principle Revised.- 2. Ritt’s Decomposition Algorithm Revised.- 3. Complete Method of Wu — Irreducible Cases.- 4. Complete Method of Wu — General Cases.- 5. Examples.- 5. Geometry Theorem Proving Using The Gröbner Basis Method.- 1. A Review of the Gröbner Basis Method.- 2. Proof Methods for Formulation F3.- 3. A Proof Method for Formulation Fl.- 4. Connections Between Characteristic Sets and Gröbner Bases.- 5. A Comparison of the Gröbner Basis Method with Wu’s Method.- 5.1. The Scope.- 5.2. The Efficiency.- References.- II: 512 Theorems Mechanically Proved.- Explanations.- 1. General Remarks.- 2. Algebraic Representations of Geometric Conditions.- Theorems Proved Mechanically by Wu’s Method.- 2. Algebraic Representations of Geometric Conditions.- Appendix. The Timing For the Gröbner Basis Method.- Index of Examples.