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Empirical explorations of the geometry theorem machine

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D. W. LovelandAuthors Info & Claims

IRE-AIEE-ACM '60 (Western): Papers presented at the May 3-5, 1960, western joint IRE-AIEE-ACM computer conference

Pages 143 - 149

https://doi.org/10.1145/1460361.1460381

Published: 03 May 1960 Publication History

Abstract

In early spring, 1959, an IBM 704 computer, with the assistance of a program comprising some 20 000 individual instructions, proved its first theorem in elementary Euclidean plane geometry. Since that time, the geometry theorem-proving machine (a particular state configuration of the IBM 704 specified by the afore-mentioned machine code) has found solutions to a large number of problems taken from high school textbooks and final examinations in plane geometry. Some of these problems would be considered quite difficult by the average high school student. In fact, it is doubtful whether any but the brightest students could have produced a solution for any of the latter group when granted the same amount of prior "training" afforded the geometry machine (i.e., the same vocabulary of geometric concepts and the same stock of previously proved theorems).

References

[1]

H. Gelernter, "Realization of a Geometry Theorem-Proving Machine", Proc. of the International Conference on Information Processing, Paris, 1959

[2]

More than fifty proofs are on file at the present time.

[3]

H. Gelernter and N. Rochester, "Intelligent Behavior in Problem-Solving Machines", IBM Journal of Research and Development 2 (1958): 336--345

Digital Library

[4]

M. L. Minsky, "Some Methods of Artificial Intelligence and Heuristic Programming", Symposium on the Mechanization of Thought Processes, Teddington, 1958

[5]

A. Newell, J. C. Shaw, and H. A. Simon, "Report on a General Problem-Solving Program", Proc. of the International Conference on Information Processing, Paris, 1959

[6]

A. Newell and J. C. Shaw, "Programming the Logic Theory Machine", Proc. of the Western Joint Computer Conference, (1957): 230--240

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[7]

H. Gelernter, J. R. Hansen, and C. L. Gerberich, "A FORTRAN-Compiled List Processing Language", Journal of the Association for Computing Machinery 7, April 1960

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[8]

A semantic heuristic is one based on an interpretation of the formal system rather than on the structure of the strings within that system.

[9]

A number of these proofs are reproduced in reference 1.

[10]

H. Gelernter, "A Note on Syntactic Symmetry and the Manipulation of Formal Systems by Machine", Information and Control 2 (1959): 80--89

[11]

See, for example, Appendix A of reference 1.

[12]

A. M. Turing, "Computing Machinery and Intelligence", Mind 59, (1950): 433

[13]

It may be argued (and undoubtedly, it will be argued) that the truly knowledgeable interrogator, cognizant of the decideability of geometry, would certainly avoid this area as well, perhaps preferring the manifestly undecideable parts of the predicate calculus or number theory to effect the distinction between man and machine. We recall here that our methods are independent of the decideability of the formal system, and, in fact, Wang and Gilmore have developed techniques that have produced proofs for theorems in the undecideable area of the predicate calculus.

[14]

H. Wang, "Toward Mechanical Mathematics", IBM Journal of Research and Development 4, January 1960: 2--22

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[15]

P. C. Gilmore, "A Proof Method for Quantification Theory", IBM Journal of Research and Development 4, January 1960: 28--35

Digital Library

Cited By

Zou JZhang XHe YZhu NLeng T(2024)FGeo-DRL: Deductive Reasoning for Geometric Problems through Deep Reinforcement LearningSymmetry10.3390/sym1604043716:4(437)Online publication date: 5-Apr-2024
https://doi.org/10.3390/sym16040437
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cover image ACM Conferences

IRE-AIEE-ACM '60 (Western): Papers presented at the May 3-5, 1960, western joint IRE-AIEE-ACM computer conference

May 1960

391 pages

ISBN:9781450378697

DOI:10.1145/1460361

Conference Chair:
R. M. Bennett
IBM Corp., San Jose, California

Copyright © 1960 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 03 May 1960

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Cited By

Zou JZhang XHe YZhu NLeng T(2024)FGeo-DRL: Deductive Reasoning for Geometric Problems through Deep Reinforcement LearningSymmetry10.3390/sym1604043716:4(437)Online publication date: 5-Apr-2024
https://doi.org/10.3390/sym16040437
Botana FRecio TVélez M(2024)On Using GeoGebra and ChatGPT for Geometric DiscoveryComputers10.3390/computers1308018713:8(187)Online publication date: 30-Jul-2024
https://doi.org/10.3390/computers13080187
Shengyuan YXiuqin Z(2024)Geo-Qwen: A Geometry Problem-Solving Method Based on Generative Large Language Models and Heuristic Reasoning2024 21st International Computer Conference on Wavelet Active Media Technology and Information Processing (ICCWAMTIP)10.1109/ICCWAMTIP64812.2024.10873683(1-9)Online publication date: 14-Dec-2024
https://doi.org/10.1109/ICCWAMTIP64812.2024.10873683
Aliyev Y(2024)Set Theory and Elementary Algebra in LEAN 4 Theorem Prover2024 IEEE 18th International Conference on Application of Information and Communication Technologies (AICT)10.1109/AICT61888.2024.10740429(1-6)Online publication date: 25-Sep-2024
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Quaresma PSantos VTeles J(2024)Proof exploration using dynamic geometry systems with integrated automated deduction capabilitiesInternational Journal of Mathematical Education in Science and Technology10.1080/0020739X.2024.2377724(1-25)Online publication date: 24-Jul-2024
https://doi.org/10.1080/0020739X.2024.2377724
Trinh TWu YLe QHe HLuong T(2024)Solving olympiad geometry without human demonstrationsNature10.1038/s41586-023-06747-5625:7995(476-482)Online publication date: 17-Jan-2024
https://doi.org/10.1038/s41586-023-06747-5
Ning MWang QHuang KHuang XEl Saddik AMei TCucchiara RBertini MTobon Vallejo DAtrey PHossain M(2023)A Symbolic Characters Aware Model for Solving Geometry ProblemsProceedings of the 31st ACM International Conference on Multimedia10.1145/3581783.3612570(7767-7775)Online publication date: 26-Oct-2023
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Janičić PNarboux J(2023)Automated generation of illustrated proofs in geometry and beyondAnnals of Mathematics and Artificial Intelligence10.1007/s10472-023-09857-y91:6(797-820)Online publication date: 3-Jul-2023
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Baeta NQuaresma P(2023)Towards a geometry deductive database proverAnnals of Mathematics and Artificial Intelligence10.1007/s10472-023-09839-091:6(851-863)Online publication date: 1-Dec-2023
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Quaresma P(2022)Evolution of Automated Deduction and Dynamic Constructions in GeometryMathematics Education in the Age of Artificial Intelligence10.1007/978-3-030-86909-0_1(3-22)Online publication date: 10-Mar-2022
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