Proof Generation in the Touchstone Theorem Prover

  • George C. Necula
  • Peter Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1831)

Abstract

The ability of a theorem prover to generate explicit derivations for the theorems it proves has major benefits for the testing and maintenance of the prover. It also eliminates the need to trust the correctness of the prover at the expense of trusting a much simpler proof checker. However, it is not always obvious how to generate explicit proofs in a theorem prover that uses decision procedures whose operation does not directly model the axiomatization of the underlying theories. In this paper we describe the modifications that are necessary to support proof generation in a congruence-closure decision procedure for equality and in a Simplex-based decision procedure for linear arithmetic. Both of these decision procedures have been integrated using a modified Nelson-Oppen cooperation mechanism in the Touchstone theorem prover, which we use to produce proof-carrying code. Our experience with designing and implementing Touchstone is that proof generation has a relatively low cost in terms of design complexity and proving time and we conclude that the software-engineering benefits of proof generation clearly outweighs these costs.

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References

  1. Ack54.
    Ackermann, W.: Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1954)MATHGoogle Scholar
  2. AS80.
    Aspvall, B., Shiloach, Y.: A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality. SIAM Journal on Computing 9(4), 827–845 (1980)MATHCrossRefMathSciNetGoogle Scholar
  3. Ble74.
    Bledsoe, W.W.: The Sup-Inf method in Presurger arithmetic. Technical report. University of Texas Math Dept. (December 1974)Google Scholar
  4. BM79.
    Boyer, R., Moore, J.S.: A Computational Logic. Academic Press, London (1979)MATHGoogle Scholar
  5. Bou92.
    Boulton, R.J.: A lazy approach to fully-expansive theorem proving. In: International Workshop on Higher Order Logic Theorem Proving and its Applications, Leuven, Belgium, pp. 19–38. North- Holland, Amsterdam (1992) IFIP TransactionsGoogle Scholar
  6. Bou93.
    Boulton, R.J.: Efficiency in a Fully-Expansive Theorem Prover. PhD thesis. University of Cambridge (December 1993)Google Scholar
  7. Bou95.
    Boulton, R.J.: Combining decision procedures in the HOL system. In: Schubert, E.T., Alves-Foss, J., Windley, P. (eds.) HUG 1995. LNCS, vol. 971, pp. 75–89. Springer, Heidelberg (1995)Google Scholar
  8. DLNS98.
    Detlefs, D.L., Rustan, K., Leino, M., Nelson, G., Saxe, J.B.: Extended static checking. SRC Research Report 159, Compaq Systems Research Center, 130 Lytton Ave., Palo Alto (December 1998)Google Scholar
  9. DST80.
    Downey, P.J., Sethi, R., Tarjan, R.E.: Variations on the common subexpressions problem. Journal of the ACM 27(4), 758–771 (1980)MATHCrossRefMathSciNetGoogle Scholar
  10. Gor85.
    Gordon, M.: HOL: A machine oriented formulation of higher-order logic. Technical Report 85. University of Cambridge, Computer Laboratory (July 1985)Google Scholar
  11. HHP93.
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)MATHMathSciNetGoogle Scholar
  12. Mil91.
    Miller, D.: A logic programming language with lambda-abstraction, function variables, and simple unification. Journal of Logic and Computation 1(4), 497–536 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. MNPS91.
    Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51, 125–157 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. Nec97.
    Necula, G.C.: Proof-carrying code. In: The 24th Annual ACM Symposium on Principles of Programming Languages, pp. 106–119. ACM, New York (1997)CrossRefGoogle Scholar
  15. Nec98.
    Necula, G.C.: Compiling with Proofs. PhD thesis, Carnegie Mellon University (September 1998), Also available as CMU-CS-98-154Google Scholar
  16. Nel81.
    Nelson, G.: Techniques for program verification. Technical Report CSL- 81-10, Xerox Palo Alto Research Center (1981)Google Scholar
  17. NO79.
    Nelson, G., Oppen, D.: Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems 1(2), 245–257 (1979)MATHCrossRefGoogle Scholar
  18. NO80.
    Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. Journal of the Association for Computing Machinery 27(2), 356–364 (1980)MATHMathSciNetGoogle Scholar
  19. ORS92.
    Owre, S., Rushby, J.M., Shankar, N.: PVS: A prototype verification system. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992)Google Scholar
  20. Pau94.
    Paulson, L.C.: Isabelle: A generic theorem prover. LNCS, vol. 828, p. 321. Springer, Heidelberg (1994)MATHGoogle Scholar
  21. Pfe91.
    Pfenning, F.: Logic programming in the LF logical framework. In: Huet, G., Plotkin, G. (eds.) Logical Frameworks, pp. 149–181. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  22. Pfe94.
    Pfenning, F.: Elf: A meta-language for deductive systems (system description). In: Bundy, A. (ed.) CADE 1994. LNCS (LNAI), vol. 814, pp. 811–815. Springer, Heidelberg (1994)Google Scholar
  23. Pra77.
    Pratt, V.R.: Two easy theories whose combination is hard(1977) (unpublished manuscript)Google Scholar
  24. SD99.
    Stump, A., Dill, D.L.: Generating proofs from a decision procedure. In: Pnueli, A., Traverso, P. (eds.) Proceedings of the FLoC Workshop on Run-Time Result Verifiication, Trento, Italy (July 1999)Google Scholar
  25. Sho81.
    Shostak, R.: Deciding linear inequalities by computing loop residues. Journal of the ACM 28(4), 769–779 (1981)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • George C. Necula
    • 1
  • Peter Lee
    • 2
  1. 1.Electrical Engineering and Computer Science DepartmentUniversity of CaliforniaBerkeleyUSA
  2. 2.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

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