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  • 1
    ISSN: 1432-0541
    Keywords: Geometry theorem proving ; Provers ; Nondegenerate conditions ; Ritt's algorithms ; Wu's method ; The Gröbner basis method ; Algebraically (or real) closed field ; Algebraic geometry ; Irreducible variety ; Nondegenerate component ; Generally true ; Simson's theorem ; Pappus' theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we analyze the algebraic formulations of certain geometry statements appearing in recent literature related to mechanical geometry theorem proving and give several examples to show that one of these formulations can cause serious problems. We clarify a formulation which is essentially due to W. T. Wu and, in our opinion, is the most satisfactory.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0541
    Keywords: Polynomial, (Prime) ideal ; Generators ; (Irreducible) ascending chain ; (Irreducible) algebraic set ; Decomposition of an algebraic set ; Geometric configuration ; Nondegenerate component ; Geometry theorem proving
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In Ritt's method, a prime ideal is given by a characteristic set. A characteristic set of a prime ideal is generally not a set of generators of this ideal. In this paper we present a simple algorithm for constructing Gröbner bases of a prime ideal from its characteristic set. We give a method for finding new theorems in geometry as an application of this algorithm.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0622
    Keywords: Geometric modeling ; Parameterization ; Algebraic curves ; Resolvents ; Ritt-Wu's decomposition algorithm ; Gröbner bases
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics , Technology
    Notes: Abstract In this paper, by using the concept of resolvents of a prime ideal introduced by Ritt, we give methods for constructing a hypersurface which is birational to a given irreducible variety and birational transformations between the hypersurface and the variety. In the case of algebraic curves, this implies that for an irreducible algebraic curveC, we can construct a plane curve which is birational toC. We also present a method to find rational parametric equations for a plane curve if it exists. Hence we have a complete method of parameterization for rational algebraic curves.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1573-0670
    Keywords: automated theorem proving ; Euclidean traditional proofs ; volume method ; constructive geometry statements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract This paper presents a method of producing readable proofs for theorems in solid geometry. The method is for a class of constructive geometry statements about straight lines, planes, circles, and spheres. The key idea of the method is to eliminate points from the conclusion of a geometric statement using several (fixed) high-level basic propositions about the signed volumes of tetrahedrons and Pythagorean differences of triangles. We have implemented the algorithm, and more than 80 examples from solid geometry have been used to test the program. Our program is efficient and the proofs produced by it are generally short and readable.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1573-0670
    Keywords: Geometry theorem proving ; pseudo division ; Ritt's principle ; irreducible ascending chain ; generic point ; Feuerbach's theorem ; Thèbault's conjecture
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Wu's algebraic method for mechanically proving geometry theorems is presented at a level as elementary as possible with sufficient examples for further understanding the complete method.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1573-0670
    Keywords: Mechanical theorem proving ; Wu's method ; Ritt-Wu's decomposition algorithm ; statement of equation type ; generally true ; universally true ; space curve theory ; elementary mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We clarify the formulation problem of mechanical theorem proving in differential geometry and mechanics and propose two formulations. We present complete methods of mechanical theorem proving for the two formulations. We also introduce predicates and a language to translate geometry statements into differential polynomial equations. A program based on our methods has proved more than 100 nontrivial theorems in differential geometry and elementary mechanics including various classification theorems for space curves, Bertrand's Theorem, Newton's gravitational laws, etc.
    Type of Medium: Electronic Resource
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  • 7
    ISSN: 1573-0670
    Keywords: deductive database ; automated geometry theorem proving and discovering ; search strategies ; redundant deduction ; Skolemization ; structured database
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We report our effort to build a geometry deductive database, which can be used to find the fixpoint for a geometric configuration. The system can find all the properties of the configuration that can be deduced using a fixed set of geometric rules. To control the size of the database, we propose the idea of a structured deductive database. Our experiments show that this technique could reduce the size of the database by one hundred times. We propose the data-based search strategy to improve the efficiency of forward chaining. We also make clear progress in the problems of how to select good geometric rules, how to add auxiliary points, and how to construct numerical diagrams as models automatically. The program is tested with 160 nontrivial geometry configurations. For these geometric configurations, the program not only finds most of their well-known properties but also often gives unexpected results, some of which are possibly new. Also, the proofs generated by the program are generally short and totally geometric.
    Type of Medium: Electronic Resource
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  • 8
    ISSN: 1573-0670
    Keywords: Wu's method for geometry theorem proving ; Ritt's decomposition algorithm ; Gröbner bases ; Heron's formula for the area of a triangle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract A method for mechanical derivation of formulas and calculation of quantities in geometry is given with several examples to illustrate its applications.
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  • 9
    ISSN: 1573-0670
    Keywords: Differential polynomial ; weak ascending chain ; W-perm ; Ritt-Wu's principle ; quasi zero set ; Ritt-Wu's decomposition algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract This is the first paper of a series of three papers under the same title. It presents an improved version of Ritt-Wu's decomposition algorithm which is the basis of our methods of mechanical theorem proving and mechanical formula derivation in differential geometry and elementary mechanics. We improve the original algorithm in two aspects. First, by using the weak ascending chain and W-perm, the sizes of the differential polynomials occurring in the decomposition can be reduced. Second, by using a special reduction procedure, the number of branches in the decomposition can be controlled effectively. The improved version significantly enhances the efficiency of the original algorithm.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Journal of automated reasoning 2 (1986), S. 253-273 
    ISSN: 1573-0670
    Keywords: Rewrite rules ; Gröbner bases ; geometry theorem proving
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We investigate the application of rewrite rules to proving theorems from elementary geometry. We have proven 80 theorems, some of them quite difficult. There is a discussion of the formulation of the problem and degenerate conditions.
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