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OTTER Proofs in Tarskian Geometry

  • Michael Beeson
  • Larry Wos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8562)

Abstract

We report on a project to use OTTER to find proofs of the theorems in Tarskian geometry proved in Szmielew’s part (Part I) of [9]. These theorems start with fundamental properties of betweenness, and end with the development of geometric definitions of addition and multiplication that permit the representation of models of geometry as planes over Euclidean fields, or over real-closed fields in the case of full continuity. They include the four challenge problems left unsolved by Quaife, who two decades ago found some OTTER proofs in Tarskian geometry (solving challenges issued in [15]).

Quaife’s four challenge problems were: every line segment has a midpoint; every segment is the base of some isosceles triangle; the outer Pasch axiom (assuming inner Pasch as an axiom); and the first outer connectivity property of betweenness. These are to be proved without any parallel axiom and without even line-circle continuity. These are difficult theorems, the first proofs of which were the heart of Gupta’s Ph. D. thesis under Tarski. OTTER proved them all in 2012. Our success, we argue, is due to improvements in techniques of automated deduction, rather than to increases in computer speed and memory.

The theory of Hilbert (1899) can be translated into Tarski’s language, interpreting lines as pairs of distinct points, and angles as ordered triples of non-collinear points. Under this interpretation, the axioms of Hilbert either occur among, or are easily deduced from, theorems in the first 11 (of 16) chapters of Szmielew. We have found Otter proofs of all of Hilbert’s axioms from Tarski’s axioms (i.e. through Satz 11.49 of Szmielew, plus Satz 12.11). Narboux and Braun have recently checked these same proofs in Coq.

Keywords

Inference Rule Automate Reasoning Isosceles Triangle Automate Deduction Proof Check 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Beeson, M.: The Tarski formalization project, http://www.michaelbeeson.com/research/FormalTarski/index.php
  2. 2.
    Braun, G., Narboux, J.: From Tarski to Hilbert. In: Ida, T., Fleuriot, J. (eds.) Automated Deduction in Geometry 2012 (2012)Google Scholar
  3. 3.
    Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien/New York (1998)zbMATHGoogle Scholar
  4. 4.
    Gupta, H.N.: Contributions to the Axiomatic Foundations of Geometry. Ph.D. thesis, University of California, Berkeley (1965)Google Scholar
  5. 5.
    Hilbert, D.: Foundations of Geometry (Grundlagen der Geometrie), 2nd English edn. Open Court, La Salle (1960), translated from the tenth German edition by Leo Unger. Original publication date (1899)Google Scholar
  6. 6.
    Pasch, M.: Vorlesung über Neuere Geometrie. Teubner, Leipzig (1882)Google Scholar
  7. 7.
    Pasch, M., Dehn, M.: Vorlesung über Neuere Geometrie. B. G. Teubner, Leipzig (1926), The 1st edn. (1882), which is the one digitized by Google Scholar, does not contain the appendix by DehnGoogle Scholar
  8. 8.
    Quaife, A.: Automated Development of Fundamental Mathematical Theories. Springer, Heidelberg (1992)zbMATHGoogle Scholar
  9. 9.
    Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie: Teil I: Ein axiomatischer Aufbau der euklidischen Geometrie. In: Teil II: Metamathematische Betrachtungen (Hochschultext). Springer (1983); reprinted 2012 by Ishi Press, with a new foreword by Michael BeesonGoogle Scholar
  10. 10.
    Tarski, A.: A decision method for elementary algebra and geometry. Tech. Rep. R-109, 2nd revised edn., reprinted in [3], pp. 24–84. Rand Corporation (1951)Google Scholar
  11. 11.
    Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarksi, A. (eds.) The Axiomatic Method, with Special Reference to Geometry and Physics. Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, December 26, 1957-January 4, 1958. Studies in Logic and the Foundations of Mathematics, pp. 16–29. North-Holland, Amsterdam (1959); available as a 2007 reprint, Brouwer Press, ISBN 1-443-72812-8Google Scholar
  12. 12.
    Tarski, A., Givant, S.: Tarski’s system of geometry. The Bulletin of Symbolic Logic 5(2), 175–214 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Veblen, O.: A system of axioms for geometry. Transactions of the American Mathematical Society 5, 343–384 (1904)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program. Journal of Automated Reasoning 16(3), 223–239 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wos, L.: Automated reasoning: 33 basic research problems. Prentice Hall, Englewood Cliffs (1988)Google Scholar
  16. 16.
    Wos, L.: Automated reasoning and the discovery of missing and elegant proofs. Rinton Press, Paramus (2003)zbMATHGoogle Scholar
  17. 17.
    Wos, L., Pieper, G.W.: A fascinating country in the world of computing. World Scientific (1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michael Beeson
    • 1
  • Larry Wos
    • 2
  1. 1.San José State UniversityUSA
  2. 2.Argonne National LaboratoryUSA

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