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TPS: A theorem-proving system for classical type theory

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Abstract

This is description of TPS, a theorem-proving system for classical type theory (Church's typed λ-calculus). TPS has been designed to be a general research tool for manipulating wffs of first- and higher-order logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems that TPS can prove completely automatically are given to illustrate certain aspects of TPS's behavior and problems of theorem proving in higher-order logic.

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References

  1. Andrews, P. B.: Resolution in type theory, J. Symbolic Logic 36 (1971), 414–342.

    Article  MathSciNet  Google Scholar 

  2. Andrews, P. B.: Provability in elementary type theory, Z. Math. Logic und Grundlagen der Mathematik 20 (1974), 411–418.

    Google Scholar 

  3. Andrews, P. B.: Refutations by matings, IEEE Trans. Computers C-25 (1976), 801–807.

    MathSciNet  Google Scholar 

  4. Andrews, P. B.: Transforming matings into natural deduction proofs, in W. Bibel and R. Kowalski (eds), 5th Conf. Automated Deduction, Les Arcs, France, Lecture Notes in Computer Sci. 87, Springer-Verlag, 1980, pp. 281–292.

  5. Andrews, P. B.: Theorem proving via general matings. J. ACM 28 (1981), 193–214.

    Article  Google Scholar 

  6. Andrews, P. B., Miller, D. A., Cohen, E. L., and Pfenning, F.: Automating higher-order logic, in W. W. Bledsoe and D. W. Loveland (eds), Automated Theorem Proving: After 25 Years, Contemporary Mathematics Series, Vol. 29, Amer. Math. Soc., 1984, pp. 169–192.

  7. Andrews, P. B.: An Introduction to Mathematical Logic and Type Theory: To Truth through Proof, Academic Press, New York, 1986.

    MATH  Google Scholar 

  8. Andrews, P. B.: On connections and higher-order logic. J. Automated Reasoning 5 (1989), 257–291.

    Article  Google Scholar 

  9. Andrews, P. B., Issar, S., Nesmith, D., Pfenning, F., Xi, H., and Bishop, M.: TPS3 Facilities Guide for Programmers and Users, 1994.

  10. Andrews, P. B., Issar, S., Nesmith, D., Pfenning, F., Xi, H., and Bishop, M.: TPS3 Facilities Guide for Users, 1994.

  11. Bailin, S. C. and Barker-Plummer, D.: Z-match: An inference rule for incrementally elaborating set instantiations. J. Automated Reasoning 11 (1993), 391–428. Errata: JAR 12 (1994), 411–412.

    Article  MathSciNet  Google Scholar 

  12. Barker-Plummer, D.: Gazing: An approach to the problem of definition and lemma use, J. Automated, Reasoning 8 (1992), 311–344.

    Article  MathSciNet  Google Scholar 

  13. Bibel, W.: Automated Theorem Proving, Vieweg, Braunschweig, 1987.

    MATH  Google Scholar 

  14. Bishop, M., Nesmith, D., Pfenning, F., Issar, S., Andrews, P. B., and Xi, H.: TPS3 Programmer's Guide, 1994.

  15. Bledsoe, W. W. and Bruell, P.: A man-machine theorem-proving system, Artificial Intelligence 5 (1974), 51–72.

    Article  MathSciNet  Google Scholar 

  16. Bledsoe, W. W.: A maximal method for set variables in automatic theorem proving, in J. E.Hayes, D.Michie, and L. I.Mikulich (eds), Machine Intelligence, Vol. 9, Ellis Harwood Ltd., Chichester, and Wiley, New York, 1979, pp. 53–100.

    Google Scholar 

  17. Bledsoe, W. W. and Feng, G.: Set-var, J. Automated Reasoning 11 (1993), 293–314.

    Article  MathSciNet  Google Scholar 

  18. Boyer, R., Lusk, E., McCune, W., Overbeek, R., Stickel, M., and Wos, L.: Set theory in first-order logic: Clauses for Gödel's axioms, J. Automated Reasoning 2 (1986), 287–327.

    Article  Google Scholar 

  19. Church, A.: Introduction to Mathematical Logic, Princeton University Press, Princeton, NJ, 1956.

    MATH  Google Scholar 

  20. Church, A.: A formulation of the simple theory of types, J. Symbolic Logic 5 (1940), 56–68.

    Article  MathSciNet  Google Scholar 

  21. Constable, R. L. et al.: Implementing Mathematics with the Nuprl Proof Development System, Prentice Hall, 1986.

  22. Coquand, T. and Huet, G.: The calculus of constructions, Information and Computation 76 (1988), 95–120.

    Article  MathSciNet  Google Scholar 

  23. Farmer, W. M., Guttman, J. D., and Thayer, J.: IMPS: An interactive mathematical proof system, J. Automated Reasoning 11 (1993), 213–248.

    Article  Google Scholar 

  24. Gallier, J. H., Narendran, P., Raatz, S., and Snyder, W.: Theorem proving using equational matings and rigid E-unification. J. ACM 39 (1992), 377–429.

    Article  MathSciNet  Google Scholar 

  25. Goldson, D. and Reeves, S.: Using programs to teach logic to computer scientists, Notices Amer. Math. Soc. 40 (1993), 143–148.

    Google Scholar 

  26. Goldson, D., Reeves, S., and Bornat, R.: A review of several programs for the teaching of logic, Computer J. 36 (1993), 373–386.

    Article  Google Scholar 

  27. Gordon, M. J. C., Milner, A. J., and Wadsworth, C. P.: Edinburgh LCF, Lecture Notes Computer Sci. 78, Springer-Verlag, 1979.

  28. Gordon, M. J. C.: HOL: A proof generating system for higher-order logic, in G.Birtwistle and P. A.Subrahmanyam (eds), VLSI Specification, Verification, and Synthesis, Kluwer, Dordrecht, 1988, pp. 73–128.

    Google Scholar 

  29. Gordon, M. J. C. and Melham, T. F.(eds): Introduction to HOL: A Theorem-Proving Environment for Higher-Order Logic, Cambridge University Press, 1993.

  30. Helmink, L. and Ahn, R.: Goal directed proof construction in type theory, in G. Huet and G. Plotkin (eds), Logical Frameworks, Cambridge University Press, 1991, pp. 120–148.

  31. Huet, G. P.: A mechanization of type theory, in Proc. 3rd Int. Joint Conf. Artificial Intelligence, IJCAI, 1973, pp. 139–146.

  32. Huet, G. P.: A unification algorithm for typed λ-calculus, Theor. Comp. Sci. 1 (1975), 27–57.

    Article  MathSciNet  Google Scholar 

  33. Issar, S.: Path-focused duplication: A search procedure for general matings, in AAAI-90 Proc. 8th Nat. Conf. Artificial Intelligence, AAAI Press/The MIT Press, 1990, pp. 221–226.

  34. Issar, S.: Operational issues in automated theorem proving using matings, PhD Thesis, Carnegie Mellon University, 1991.

  35. Issar, S., Andrews, P. B., Pfenning, F., and Nesmith, D.: GRADER Manual, 1992.

  36. Kolodner, I. I.: Fixed points, Amer. Math. Monthly 71, (1964), 906.

    Article  MathSciNet  Google Scholar 

  37. Miller, D. A., Cohen, E. L., and Andrews, P. B.: A look at TPS, in D. W. Loveland (ed.), 6th Conf. Automated Deduction, Lecture Notes in Computer Sci. 138, Springer-Verlag, 1982, pp. 50–69.

  38. Miller, D. A.: Proofs in higher-order logic, PhD Thesis, Carnegie Mellon University, 1983.

  39. Miller, D. A.: Expansion tree proofs and their conversion to natural deduction proofs, in R. E. Shostak (ed.), 7th Int. Conf. Automated Deduction, Lecture Notes in Computer Sci. 170, Springer-Verlag, 1984, pp. 375–393.

  40. Miller, D. A.: A compact representation of proofs, Stud. Logica 46 (1987), 347–370.

    Article  Google Scholar 

  41. Milner, R.: A theory of type polymorphism in programming, J. Computer System Sci. 17 (1978), 348–375.

    Article  MathSciNet  Google Scholar 

  42. Nesmith, D., Bishop, M., Andrews, P. B., Issar, S., Pfenning, F., and Xi, H.: TPS User's Manual, 1994.

  43. Ohlbach, H. J.: A resolution calculus for modal logics, PhD Thesis, Department of Computer Science, University of Kaiserslautern, 1988.

  44. Paulson, L. C.: The foundation of a generic theorem prover, J. Automated Reasoning 5, (1989), 363–397.

    Article  MathSciNet  Google Scholar 

  45. Pfenning, F.: Analytic and non-analytic proofs, in R. E. Shostak (ed.), 7th Int. Conf. Automated Deduction, Lecture Notes Computer Sci. 170, Springer-Verlag, 1984, pp. 394–413.

  46. Pfenning, F.: Proof transformations in higher-order logic, PhD Thesis, Carnegie Mellon University, 1987.

  47. Pfenning, F. and Nesmith, D.: Presenting intuitive deductions via symmetric simplification, in M. E. Stickel (ed.), 10th Int. Conf. Automated Deduction, Lecture Notes in Artificial Intelligence 449, Springer-Verlag, 1990, pp. 336–350.

  48. Pfenning, F., Issar, S., Nesmith, D., Andrews, P. B., Xi, H., and Bishop, M.: ETPS User's Manual, 1994.

  49. Quaife, A.: Andrews' challenge problem revisited, AAR Newslet. 15 (1990), 3–7.

    Google Scholar 

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

    Article  Google Scholar 

  51. Russell, B.: The Principles of Mathematics, Cambridge University Press, 1903.

  52. Russell, B.: Mathematical logic as based on the theory of types, Amer. J. Math. 30 (1908), 222–262. Reprinted in [55], pp. 150–182.

    Article  MathSciNet  Google Scholar 

  53. Snyder, W.: Higher-order E-unification, in M. E. Stickel (ed.), 10th Int. Conf. Automated Deduction, Kaiserslautern, FRG, Lecture Notes in Artificial Intelligence 449, Springer-Verlag, 1990, pp. 573–587.

  54. Sutcliffe, G., Suttner, Ch., and Yemenis, T.: The TPTP problem library, in A. Bundy (ed.), Automated Deduction — CADE-12, 12th Int. Conf. Automated Deduction, Nancy, France, Lecture Notes in Artificial Intelligence 814, Springer-Verlag, 1994, pp. 252–266.

  55. vanHeijenoort, J.: From Frege to Gödel, Harvard University Press, Cambridge, MA, 1967.

    MATH  Google Scholar 

  56. Wallen, L. A.: Automated Deduction in Nonclassical Logics, The MIT Press, 1990.

  57. Whitehead, A. N. and Russell, B.: Principia Mathematica, 3 vols, Cambridge University Press, Cambridge, England, 1910–1913.

    Google Scholar 

  58. Wos, L.: The problem of definition expansion and contraction, J. Automated Reasoning 3 (1987), 433–435.

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Carnegie Mellon University, 15213, Pittsburgh, PA, USA

    Peter B. Andrews, Matthew Bishop & Hongwei Xi

  2. School of Computer Science, Carnegie Mellon University, 15213, Pittsburgh, PA, USA

    Sunil Issar

  3. Department of Computer Science, Carnegie Mellon University, 15213, Pittsburgh, PA, USA

    Frank Pfenning

Authors
  1. Peter B. Andrews
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  2. Matthew Bishop
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  3. Sunil Issar
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  4. Dan Nesmith
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  5. Frank Pfenning
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  6. Hongwei Xi
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Additional information

This material is based upon work supported by the National Science Foundation under grant CCR-9201893 and previous grants.

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Andrews, P.B., Bishop, M., Issar, S. et al. TPS: A theorem-proving system for classical type theory. Journal of Automated Reasoning 16, 321–353 (1996). https://doi.org/10.1007/BF00252180

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