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The application of automated reasoning to questions in mathematics and logic

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Abstract

Sharply different from the well-known use of various computer programs for the numerical aspects of mathematics and logic is the newer and less familiar use of a program to address the logical reasoning aspects. In this article, we focus on such a program — the automated reasoning program OTTER — which is portable, available electronically by anonymous FTP, and usable on a wide variety of computers, even on personal computers. We discuss the types of assistance provided by OTTER, including proof finding, conjecture formulation, and object construction. With OTTER's assistance, we have answered a number of open questions taken from a variety of fields. We focus on such questions from combinatory logic, equivalential calculus, Robbins algebra, and finite semigroup theory. For those who enjoy a challenge, we also offer ten questions, including some that are still open, and an open question that eluded even Tarski.

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References

  1. S. Anantharaman, J. Hsiang and J. Mzali, SbReve2: A term rewriting laboratory with (AC)-unfailing completion, in:Proc. 3rd Int. Conf. on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 335 (Springer, Berlin, 1989) pp. 533–537.

    Google Scholar 

  2. H.P. Barendregt,The Lambda Calculus: Its Syntax and Semantics (North-Holland, Amsterdam, 1981).

    Google Scholar 

  3. R. Boyer and J. Moore,A Computational Logic (Academic Press, New York, 1979).

    Google Scholar 

  4. R. Boyer and J. Moore, Proof checking the RSA public key encryption algorithm, Amer. Math. Monthly 91 (1984) 181–189.

    Google Scholar 

  5. R. Boyer and J. Moore,A Computational Logic Handbook (Academic Press, San Diego, 1988).

    Google Scholar 

  6. E. Dijkstra, Dijkstra's reply to comments, in: A debate on teaching computing science, CACM 32 (December 1989) 1397–1414.

    Google Scholar 

  7. J.M. Font, A.J. Rodriguez and A. Torrens, Wajsberg algebras, Stochastica 8 (1984) 5–31.

    Google Scholar 

  8. C. Green, Theorem-proving by resolution as a basis for question-answering systems, in:Machine Intelligence 4, eds. B. Meltzer and D. Michie (Edinburgh University Press, Edinburgh, 1969) pp. 183–205.

    Google Scholar 

  9. L. Henkin, J. Monk and A. Tarski,Cylindric Algebras, Part I (North-Holland, Amsterdam, 1971).

    Google Scholar 

  10. E. Huntington, New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia Mathematica, Trans. AMS 35 (1933) 274–304.

    Google Scholar 

  11. J. Kalman, A shortest single axiom for the classical equivalential calculus, Notre Dame J. Formal Logic 19 (1978) 141–144.

    Google Scholar 

  12. D. Kapur and H. Zhang, An overview of RRL (Rewrite Rule Laboratory), in:Proc. 3rd Int. Conf. of Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 335 (Springer, Berlin, 1989) pp. 559–563.

    Google Scholar 

  13. D. Knuth and P. Bendix, Simple word problems in universal algebras, in:Computational Problems in Abstract Algebra, ed. J. Leech (Pergamon Press, New York, 1970) pp. 263–297.

    Google Scholar 

  14. J. Lukasiewicz, The equivalential calculus, in:Jan Lukasiewicz: Selected Works, ed. L. Borkowski (North-Holland, Amsterdam, 1970) pp. 250–277.

    Google Scholar 

  15. E. Lusk and R. McFadden, Using automated reasoning tools: A study of the semigroup F2B2, Semigroup Forum 36 (1987) 75–88.

    Google Scholar 

  16. A. Marien, private communication (1989).

  17. J. McCharen, R. Overbeek and L. Wos, Complexity and related enhancements for automated theorem-proving programs, Comput. Math. Appl. 2 (1976) 1–16.

    Google Scholar 

  18. W. McCune and L. Wos, A case study in automated theorem proving: finding sages in combinatory logic, J. Auto. Reasoning 3 (1987) 91–108.

    Google Scholar 

  19. W. McCune and L. Wos, The absence and the presence of fixed point combinators, Mathematics and Computer Science Division Preprint MCS-P87-0689, Argonne National Laboratory, Argonne, IL (1989).

    Google Scholar 

  20. W. McCune, OTTER 2.0 Users Guide, Technical Report ANL-90/9, Argonne National Laboratory, Argonne, IL (1990).

    Google Scholar 

  21. W. McCune, Experiments with discrimination tree indexing and path indexing for term retrieval, J. Auto. Reasoning, to appear.

  22. J. Peterson, The Possible Shortest Single Axioms for EC-Tautologies, Department of Mathematics Report Series No. 105, Auckland University, Auckland, New Zealand (1977).

    Google Scholar 

  23. G. Robinson and L. Wos, Paramodulation and theorem-proving in first-order theories with equality, in:Machine Intelligence 4, eds. B. Meltzer and D. Michie (Edinburgh University Press, Edinburgh, 1969) pp. 135–150.

    Google Scholar 

  24. J. Robinson, A machine-oriented logic based on the resolution principle, J. ACM 12 (1965) 23–41.

    Google Scholar 

  25. J. Robinson, Automatic deduction with hyper-resolution, Int. J. Comput. Math. 1 (1965) 227–234.

    Google Scholar 

  26. B. Smith, Reference Manual for the Environmental Theorem Prover, An Incarnation of AURA, Technical Report ANL-88-2, Argonne National Laboratory, Argonne, IL (1988).

    Google Scholar 

  27. R. Smullyan,To Mock a Mockingbird (Alfred A. Knopf, New York, 1985).

    Google Scholar 

  28. R. Smullyan, private correspondence (1987).

  29. R. Statman, private correspondence with Raymond Smullyan (1986).

  30. R. Statman, private correspondence (1990).

  31. M. Stickel, A Prolog technology theorem prover, J. Auto. Reasoning 4 (1988) 353–380.

    Google Scholar 

  32. A. Tarski, private communication (1980).

  33. R. Veroff and L. Wos, The linked inference principle, I: The formal treatment, Mathematics and Computer Science Division Preprint MCS-P160-0690, Argonne National Laboratory, Argonne, IL (1990).

    Google Scholar 

  34. S. Winker, private communication (1980).

  35. S. Winker, L. Wos and E. Lusk, Semigroups, antiautomorphisms, and involutions: A computer solution to an open problem, I, Math. Comput. 37 (1981) 533–545.

    Google Scholar 

  36. S. Winker, Generation and verification of finite models and counterexamples using an automated theorem prover answering two open questions, J. ACM 29 (1982) 273–284.

    Google Scholar 

  37. L. Wos, G. Robinson and D. Carson, Efficiency and completeness of the set of support strategy in theorem proving, J. ACM 12 (1965) 536–541.

    Google Scholar 

  38. L. Wos, G. Robinson, D. Carson and L. Shalla, The concept of demodulation in theorem proving, J. ACM 14 (1967) 698–709.

    Google Scholar 

  39. L. Wos and G. Robinson, Maximal models and refutation completeness: Semidecision procedures in automatic theorem proving, in:Word Problems: Decision Problems and the Burnside Problem in Group Theory, eds. W. Boone, F. Cannonito and R. Lyndon (North-Holland, New York, 1973) pp. 609–639.

    Google Scholar 

  40. L. Wos, R. Overbeek, E. Lusk and J. Boyle,Automated Reasoning: Introduction and Applications (Prentice-Hall, Englewood Cliffs, NJ, 1984).

    Google Scholar 

  41. L. Wos, S. Winker, R. Veroff, B. Smith and L. Henschen, A new use of an automated reasoning assistant: Open questions in equivalential calculus and the study of infinite domains, Art. Int. 22 (1984) 303–356.

    Google Scholar 

  42. L. Wos,Automated Reasoning: 33 Basic Research Problems (Prentice-Hall, Englewood Cliffs, NJ, 1987).

    Google Scholar 

  43. L. Wos and W. McCune, Searching for fixed point combinators by using automated theorem proving: a preliminary report, Technical Report ANL-88-10, Argonne National Laboratory, Argonne, IL (1988).

    Google Scholar 

  44. L. Wos, S. Winker, W. McCune, R. Overbeek, E. Lusk, R. Stevens and R. Butler, OTTER experiments pertinent to CADE-10, Technical Report ANL-89/39, Argonne National Laboratory, Argonne, IL (1989).

    Google Scholar 

  45. L. Wos, Meeting the challenge of fifty years of logic, J. Auto. Reasoning 6 (1990) 213–232.

    Google Scholar 

  46. L. Wos, R. Overbeek and E. Lusk, Subsumption, a sometimes undervalued procedure,Festschrift for JA. Robinson, ed. J.-L. Lassez, to appear.

  47. L. Wos, S. Winker, W. McCune, R. Overbeek, E. Lusk, R. Stevens and R. Butler, Automated reasoning contributes to mathematics and logic, in:Proc. 10th Int. Conf. on Automated Deduction, Lecture Notes in Artificial Intelligence, vol. 449 (Springer, New York, 1990) pp. 485–499.

    Google Scholar 

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 60439, Argonne, IL, USA

    Larry Wos & William McCune

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  1. Larry Wos
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  2. William McCune
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Additional information

This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38.

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Wos, L., McCune, W. The application of automated reasoning to questions in mathematics and logic. Ann Math Artif Intell 5, 321–369 (1992). https://doi.org/10.1007/BF01543481

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