Abstract
The problem of automatically proving geometric theorems has gained a lot of attention in the last two years. Following the general approach of translating a given geometric theorem into an algebraic one, various powerful provers based on characteristic sets and Gröbner bases have been implemented by groups at Academia Sinica Bejing (China), U. Texas at Austin (USA), General Electric Schenectady (USA), and Research Institute for Symbolic Computation Linz (Austria). So far, fair comparisons of the various provers were not possible, because the underlying hardware and the underlying algebra systems differed greatly. This paper reports on the first uniform implementation of all of these provers in the computer algebra system and language SCRATCHPAD II. We summarize the recent achievements in the area of automated geometry theorem proving, shortly review the SCRATCHPAD II system, describe the implementation of the geometry theorem proving package, and finally give a computing time statistics of 24 examples.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
7 References
Buchberger B. 1965. An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal. PhD thesis. Univ. Innsbruck, Austria. (In German)
Buchberger B. 1985. Gröbner bases: An algorithmic method in polynomial ideal theory. In Multidimensional Systems Theory, ed. N. K. Bose, Dordrecht: D. Reidel Publ. Comp., 1:184–232
Buchberger B. 1987. Applications of Gröbner bases in non-linear computational geometry. Lect. Notes Comp. Sci. 296:52–80
Cerutti E., Davis P.J. 1969. FORMAC meets Pappus. Am. Math. Monthly, vol. 76, pp. 895–905
Chou S. C. 1984. Proving elementary geometry theorems using Wu's Algorithm. Contemp. Math., vol. 29, pp. 243–286
Chou S. C. 1985. Proving and discovering geometry theorems using Wu's method. PhD thesis. Univ. Texas, Austin. 142pp.
Chou S. C. 1986. A collection of geometry theorems proved mechanically. Inst. f. Comput. Sci. Tech. Rep. 1986–50. Univ. Texas, Austin
Chou S. C., Schelter W. F. 1986. Proving geometry theorems with rewrite rules. J. Autom. Reason. 2:253–73
Chou S. C., Yang J. G. 1987. On the algebraic formulation of geometry theorems. Inst. f. Comput. Sci. Tech. Rep. Univ. Texas, Austin
Coelho H., Pereira L.M. 1979. GEOM: A PROLOG geometry theorem prover. Laboratorio Nacional de Engenharia Civil mem. no. 525
Collins G.E. 1975. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In Lect. Notes Comput. Sci. 33:134–83
Gelernter H. 1959. Realization of a geometry theorem-proving machine. Proc. Int. Conf. Info. Process., Paris, 1959
Jenks R.D., Sutor R.S., Watt S.M. 1987. SCRATCHPAD II: An abstract datatype system for mathematical computation. Lect. Notes Comput. Sci. 296:12–37
Kapur D. 1986. Geometry theorem proving using Hilbert's Nullstellensatz. Proc. Symp. Symb. Algebraic Comput., Waterloo, Canada, July 21–23, 1986, ed. B.W. Char, pp. 202–8
Ko H. P., Hussain M. A. 1985. ALGE-Prover: An algebraic geometry theorem proving software. Gen. Elect. Tech. Rep. 85CRD139. 28pp.
Kutzler B. 1988. Algebraic methods for geometric theorem proving. PhD thesis. Univ. Linz, Austria
Kutzler B., Stifter S. 1986a. New approaches to computerized proofs of geometry theorems. Proc. Computers & Mathematics, Stanford, USA, July 1986, eds. R. Gebauer, J. Davenport
Kutzler B., Stifter S. 1986b. Automated geometry theorem proving using Buchberger's algorithm. Proc. Symp. Symb. Algebraic Comput., Waterloo, Canada, July 21–23, 1986, ed. B.W. Char, pp. 209–14
Kutzler B., Stifter S., 1986c. Collection of computerized proofs of geometry theorems. Res. Inst. Symbol. Comput. Tech. Rep. 86–12.0, Univ. Linz, Austria
Mayr H. 1988. Geometry theorem proving package in SCRATCHPAD II: A primer. Res. Inst. Symbol. Comput. Tech. Rep. 88–1.0, Univ. Linz, Austria
Ritt J. F. 1950. Differential Algebra, Colloq. Publ. vol. 33. New York: Am. Math. Soc. 184pp.
Wu W. T. 1978. On the decision problem and the mechanization of theorem proving in elementary geometry. Sci. Sinica 21:159–72 also Contemp. Math. 29:213–34
Wu W. T. 1984. Basic principles of mechanical theorem proving in elementary geometries. J. Syst.Sci. & Math. Sci. 4:207–35
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kusche, K., Kutzler, B., Mayr, H. (1989). Implementation of a geometry theorem proving package in SCRATCHPAD II. In: Davenport, J.H. (eds) Eurocal '87. EUROCAL 1987. Lecture Notes in Computer Science, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51517-8_123
Download citation
DOI: https://doi.org/10.1007/3-540-51517-8_123
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51517-3
Online ISBN: 978-3-540-48207-9
eBook Packages: Springer Book Archive