A Proof-Producing Decision Procedure for Real Arithmetic

  • Sean McLaughlin
  • John Harrison
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3632)

Abstract

We present a fully proof-producing implementation of a quantifier elimination procedure for real closed fields. To our knowledge, this is the first generally useful proof-producing implementation of such an algorithm. While many problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving.

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References

  1. 1.
    Barrett, C., Berezin, S.: CVC lite: A new implementation of the cooperating validity checker category B. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 515–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Blum, M.: Program result checking: A new approach to making programs more reliable. In: Lingas, A., Carlsson, S., Karlsson, R. (eds.) ICALP 1993. LNCS, vol. 700, pp. 1–14. Springer, Heidelberg (1993)Google Scholar
  3. 3.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 36, Springer, Heidelberg (1998)Google Scholar
  4. 4.
    Boulton, R.J.: Efficiency in a fully-expansive theorem prover. Technical Report 337, University of Cambridge Computer Laboratory, New Museums Site, Pembroke Street, Cambridge, CB2 3QG, UK, Author’s PhD thesis (1993)Google Scholar
  5. 5.
    Caviness, B.F., Johnson, J.R.: Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and monographs in symbolic computation. Springer, Heidelberg (1998)MATHGoogle Scholar
  6. 6.
    Cohen, P.J.: Decision procedures for real and p-adic fields. Communications in Pure and Applied Mathematics 22, 131–151 (1969)MATHCrossRefGoogle Scholar
  7. 7.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) Second GI Conference on Automata Theory and Formal Languages, Kaiserslautern. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1976)Google Scholar
  8. 8.
    Davis, M.: A computer program for Presburger’s algorithm. In: Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, Institute for Defense Analyses, Princeton, NJ, pp. 215–233 (1957); Reprinted in [78], pp. 41–48Google Scholar
  9. 9.
    Engeler, E.: Foundations of Mathematics: Questions of Analysis, Geometry and Algorithmics, Original German edn., Metamathematik der Elementarmathematik, Series Hochschultext. Springer, Heidelberg (1993)Google Scholar
  10. 10.
    Gårding, L.: Some Points of Analysis and Their History, volume 11 of University Lecture Series. American Mathematical Society / Higher Education Press (1997)Google Scholar
  11. 11.
    Gordon, M.J.C., Melham, T.F.: Introduction to HOL: a theorem proving environment for higher order logic. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  12. 12.
    Gordon, M., Wadsworth, C.P., Milner, R.: Edinburgh LCF: A Mechanised Logic of Computation. LNCS, vol. 78. Springer, Heidelberg (1979)Google Scholar
  13. 13.
    Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995), Available on the Web as, http://www.cl.cam.ac.uk/users/jrh/papers/reflect.dvi.gz
  14. 14.
    Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  15. 15.
    Harrison, J.: Theorem Proving with the Real Numbers. Springer-Verlag, Revised version of author’s PhD thesis (1998)Google Scholar
  16. 16.
    Harrison, J.: Complex quantifier elimination in HOL. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. Published as Informatics Report Series EDI-INF-RR-0046, pp. 159–174. University of Edinburgh (2001), Available on the Web at http://www.informatics.ed.ac.uk/publications/report/0046.html
  17. 17.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators II. In: Grundlehren der mathematischen Wissenschaften, vol. 257, Springer, Heidelberg (1983)Google Scholar
  18. 18.
    Hunt, W.A., Krug, R.B., Moore, J.: Linear and nonlinear arithmetic in ACL2. In: Geist, D. (ed.) CHARME 2003. LNCS, vol. 2860, pp. 319–333. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Kreisel, G., Krivine, J.-L.: Elements of mathematical logic: model theory, revised 2nd edn. (1971), 1st edn. (1967). Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1971); Translation of the French ‘Eléments de logique mathématique, théorie des modeles’ published by Dunod, Paris (1964)Google Scholar
  20. 20.
    Kumar, R., Kropf, T., Schneider, K.: Integrating a first-order automatic prover in the HOL environment. In: Archer, M., Joyce, J.J., Levitt, K.N., Windley, P.J. (eds.) Proceedings of the 1991 International Workshop on the HOL theorem proving system and its Applications, University of California at Davis, Davis CA, USA, pp. 170–176. IEEE Computer Society Press, Los Alamitos (1991)CrossRefGoogle Scholar
  21. 21.
    Mahboubi, A., Pottier, L.: Elimination des quantificateurs sur les réels en Coq. In: Journées Francophones des Langages Applicatifs (JFLA), available on the Web (2002), from http://pauillac.inria.fr/jfla/2002/actes/index.html08-mahboubi.ps
  22. 22.
    Michaux, C., Ozturk, A.: Quantifier elimination following Muchnik. Université de Mons-Hainaut, Institute de Mathématique, Preprint 10, http://w3.umh.ac.be/math/preprints/src/Ozturk020411.pdf
  23. 23.
    Necula, G.C.: Proof-carrying code. In: Conference record of POPL 1997: the 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 106–119. ACM Press, New York (1997)CrossRefGoogle Scholar
  24. 24.
    Parrilo, P.: Semidefinite programming relaxations for semialgebraic problems. Available from the Web (2001), at http://citeseer.nj.nec.com/parrilo01semidefinite.html
  25. 25.
    Paulson, L.C.: Isabelle: a generic theorem prover. LNCS, vol. 828. Springer, Heidelberg (1994); (With contributions by Tobias Nipkow)MATHCrossRefGoogle Scholar
  26. 26.
    Schoutens, H.: Muchnik’s proof of Tarski-Seidenberg. Notes (2001), available from http://www.math.ohio-state.edu/~schoutens/PDF/Muchnik.pdf
  27. 27.
    Seidenberg, A.: A new decision method for elementary algebra. Annals of Mathematics 60, 365–374 (1954)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Siekmann, J., Wrightson, G.: Automation of Reasoning — Classical Papers on Computational Logic (1957-1966), vol. I. Springer, Heidelberg (1983)Google Scholar
  29. 29.
    Sturm, C.: Mémoire sue la résolution des équations numériques. Mémoire des Savants Etrangers 6, 271–318 (1835)Google Scholar
  30. 30.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, Previous version published as a technical report by the RAND Corporation, (1948); prepared for publication by J. C. C. McKinsey. Reprinted In: [5], pp. 24–84 (1951)Google Scholar
  31. 31.
    Vorobjov, N.N.: Deciding consistency of systems of polynomial in exponent inequalities in subexponential time. In: Mora, T., Traverso, C. (eds.) Proceedings of the MEGA-90 Symposium on Effective Methods in Algebraic Geometry, Castiglioncello, Livorno, Italy. Progress in Mathematics, vol. 94, pp. 491–500. Birkhäuser, Basel (1990)Google Scholar
  32. 32.
    Weispfenning, V.: Quantifier elimination for real algebra — the quadratic case and beyond. Applicable Algebra in Engineering Communications and Computing 8, 85–101 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sean McLaughlin
    • 1
  • John Harrison
    • 2
  1. 1.Computer Science DepartmentCarnegie-Mellon UniversityPittsburghUSA
  2. 2.Intel Corporation, JF1-13HillsboroUSA

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