Journal of Automated Reasoning

, Volume 27, Issue 4, pp 323–351 | Cite as

Nonstandard Analysis in ACL2

  • Ruben A. Gamboa
  • Matt Kaufmann
Article

Abstract

ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complex-rational numbers and excluding the real and complex irrationals. In conjunction with the arithmetic completion axioms, this numeric type system makes it possible to prove the nonexistence of specific irrational numbers, such as √2. This paper describes ACL2(r), a version of ACL2 with support for the real and complex numbers. The modifications are based on nonstandard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.

nonstandard analysis automated theorem proving with the reals 

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References

  1. 1.
    Ballantyne, A. and Bledsoe, W. W.: Automatic proofs of theorems in analysis using nonstandard techniques, J. Assoc. Comput. Mach. (JACM) 24(3) (1997), 353–371.Google Scholar
  2. 2.
    Ballantyne, A. M.: The metatheorist: Automatic proofs of theorems in analysis using nonstandard techniques, Part II, in R. S. Boyer (ed.), Automated Reasoning: Essays in Honor of Woody Bledsoe, Kluwer Academic Publishers, Dordrecht, 1991, pp. 61–75.Google Scholar
  3. 3.
    Bledsoe, W. W.: The UT natural deduction prover, Technical Report ATP-17B, University of Texas at Austin, 1983.Google Scholar
  4. 4.
    Bledsoe, W. W.: Some automatic proofs in analysis, Contemp. Math. 29 (1984).Google Scholar
  5. 5.
    Boyer, R. S., Goldschlag, D., Kaufmann, M., and Moore, J S.: Functional instantiation in first order logic, in V. Lifschitz (ed.), Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy, 1991, pp. 7–26.Google Scholar
  6. 6.
    Boyer, R. S. and Moore, J S.: A Computational Logic, Academic Press, Orlando, 1979.Google Scholar
  7. 7.
    Boyer, R. S. and Moore, J S.:A Computational Logic Handbook, Academic Press, San Diego, 1988.Google Scholar
  8. 8.
    Brock, B.,Kaufmann, M., and Moore, J S.: ACL2 theorems about commercial microprocessors, in M. Srivas and A. Camilleri (eds.), Formal Methods in Computer-Aided Design (FMCAD '96), 1996, pp. 275–293.Google Scholar
  9. 9.
    Diener, F. and Diener, M. (eds.): Nonstandard Analysis in Practice, Springer, 1995.Google Scholar
  10. 10.
    Dutertre, B.: Elements of mathematical analysis in PVS, in Proceedings of the Ninth International Conference on Theorem Proving in Higher-Order Logics (TPHOL '96), 1996.Google Scholar
  11. 11.
    Farmer, W. M., Guttman, J. D., and Thayer, F. J.: IMPS: An interactive mathematical proof system, J. Automated Reasoning 11(2) (1993), 213–248.Google Scholar
  12. 12.
    Fleuriot, J.: A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, Ph.D. Thesis, University of Cambridge, 1999.Google Scholar
  13. 13.
    Gamboa, R.: Square roots in ACL2: A study in sonata form, Technical Report CS-TR-96-34, University of Texas at Austin, 1996.Google Scholar
  14. 14.
    Gamboa, R.: Mechanically verifying the correctness of the fast Fourier transform in ACL2, in J. Rolim (ed.), Parallel and Distributed Processing, 1998, pp. 796–806.Google Scholar
  15. 15.
    Gamboa, R.: Mechanically Verifying Real-Valued Algorithms in ACL2, Ph.D. Thesis, The University of Texas at Austin, 1999.Google Scholar
  16. 16.
    Gamboa, R.: Continuity and differentiability in ACL2, in M. Kaufmann, P. Manolios and J S. Moore (eds.), Computer-Aided Reasoning: ACL2 Case Studies, Kluwer Academic Publishers, Dordrecht, 2000, Chapt. 18.Google Scholar
  17. 17.
    Harrison, J.: Theorem Proving with the Real Numbers, Ph.D. Thesis, University of Cambridge, 1996.Google Scholar
  18. 18.
    Jr., G. L. S.: Common LISP The Language, 2nd edn, Digital Press, Bedford, MA, 1990.Google Scholar
  19. 19.
    Kaufmann, M.: Modular proof: The fundamental theorem of calculus, in M. Kaufmann, P. Manolios and J S. Moore (eds.), Computer-Aided Reasoning: ACL2 Case Studies, Kluwer Academic Publishers, Dordrecht, 2000, Chapt. 6.Google Scholar
  20. 20.
    Kaufmann, M., Manolios, P., and Moore, J S.: Computer-Aided Reasoning: An Approach, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
  21. 21.
    Kaufmann, M. and Moore, J S.: ACL2: A computational logic for applicative common lisp, the user's manual. Available on theWorld-WideWeb at http://www.cs.utexas.edu/users/moore/acl2/acl2-doc.html.Google Scholar
  22. 22.
    Kaufmann, M. and Moore, J S.: A precise description of the ACL2 logic, Available on the World-Wide Web at http://www.cs.utexas.edu/users/moore/publications/km97a.ps. Z.Google Scholar
  23. 23.
    Kaufmann, M. and Moore, J S.: Design goals for ACL2, Technical Report 101, Computational Logic, Inc., 1994. See URL http://www.cs.utexas.edu/users/moore/publications/-acl2-paper s.html#Overviews.Google Scholar
  24. 24.
    Kaufmann, M. and Moore, J S.: Structured theory development for a mechanized logic, J. Automated Reasoning 26(1) (2001), 161–203.Google Scholar
  25. 25.
    Keisler, H. J.: Elementary Calculus, Prindle, Weber and Schmidt, Boston, 1976.Google Scholar
  26. 26.
    Moore, J S., Lynch, T., and Kaufmann, M.: A mechanically checked proof of the AMD5K86 floating-point division program, IEEE Trans. Comp. 47(9) (1998), 913–926.Google Scholar
  27. 27.
    Nelson, E.: On-line books: Internal set theory, Available on the World-Wide Web at http://www.math.princeton.edu/nelson/books.html.Google Scholar
  28. 28.
    Nelson, E.: Internal set theory, Bull. Amer. Math. Soc. 83 (1977), 1165–1198.Google Scholar
  29. 29.
    Owre, S., Rajan, S., Rushby, J. M., Shankar, N., and Srivas, M. K.: PVS: Combining specification, proof checking, and model checking, in R. Alur and T. A. Henzinger (eds.), Computer-Aided Verification, CAV '96, Lecture Notes in Comput. Sci. 1102, New Brunswick, NJ, 1996, pp. 411–414.Google Scholar
  30. 30.
    Robert, A.: Non-Standard Analysis, Wiley, 1988.Google Scholar
  31. 31.
    Robinson, A.: Non-Standard Analysis, Princeton University Press, 1996.Google Scholar
  32. 32.
    Rudnicki, P.: An overview of the MIZAR project, in Proceedings of the 1992 Workshop on Types for Proofs and Programs, 1992.Google Scholar
  33. 33.
    Russinoff, D.: A mechanically checked proof of IEEE compliance of a register-transferlevel specification of the AMD-K7 floating-point multiplication, division, and square root instructions, London Math. Soc. J. Comput. Math. 1 (1998), 148–200.Google Scholar
  34. 34.
    Russinoff, D.: A mechanically checked proof of correctness of the AMD-K5 floating-point square root microcode, Formal Methods in System Design 14 (1999), 75–125.Google Scholar
  35. 35.
    Trybulec, A.: The Mizar-QC/6000 logic information language, Bull. Assoc. Literary and Linguistic Computing (LLAC) 6(2) (1978).Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Ruben A. Gamboa
    • 1
  • Matt Kaufmann
    • 2
  1. 1.Loop One, Inc.AustinU.S.A.
  2. 2.Advanced Micro Devices, Inc.AustinU.S.A.

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