Advertisement

Effective Quantifier Elimination for Presburger Arithmetic with Infinity

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5743)

Abstract

We consider Presburger arithmetic extended by infinity. For this we give an effective quantifier elimination and decision procedure which implies also the completeness of our extension. The asymptotic worst-case complexity of our procedure is bounded by a function that is triply exponential in the input word length, which is known to be a tight bound for regular Presburger arithmetic. Possible application areas include quantifier elimination and decision procedures for Boolean algebras with cardinality constraints, which have recently moved into the focus of computer science research for software verification, and deductive database queries.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du premier congrès de Mathématiciens des Pays Slaves, Warsaw, Poland, pp. 92–101 (1929)Google Scholar
  2. 2.
    Cooper, D.C.: Theorem proving in arithmetic without multiplication. Machine Intelligence 7, 91–99 (1972)zbMATHGoogle Scholar
  3. 3.
    Ferrante, J., Rackoff, C.W.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer, Berlin (1979)zbMATHGoogle Scholar
  4. 4.
    Fischer, M.J., Rabin, M.: Super-exponential complexity of Presburger arithmetic. SIAM-AMS Proceedings 7, 27–41 (1974)Google Scholar
  5. 5.
    Reddy, C.R., Loveland, D.W.: Presburger arithmetic with bounded quantifier alternation. In: STOC 1978: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pp. 320–325. ACM, New York (1978)CrossRefGoogle Scholar
  6. 6.
    Pugh, W.: The omega test: a fast and practical integer programming algorithm for dependence analysis. In: Supercomputing 1991: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, pp. 4–13. ACM, New York (1991)CrossRefGoogle Scholar
  7. 7.
    Oppen, D.C.: A \(2^{2^{2^{pn}}}\) upper bound on the complexity of Presburger arithmetic. J. Comput. Syst. Sci. 16(3), 323–332 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Weispfenning, V.: The complexity of almost linear Diophantine problems. Journal of Symbolic Computation 10(5), 395–403 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lasaruk, A., Sturm, T.: Weak quantifier elimination for the full linear theory of the integers. A uniform generalization of Presburger arithmetic. Applicable Algebra in Engineering, Communication and Computing 18(6), 545–574 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lasaruk, A., Sturm, T.F.: Weak integer quantifier elimination beyond the linear case. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 275–294. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Davis, M.: Final report on mathematical procedures for decision problems. Technical report, Institute for Advanced Study, Princeton, NJ (October 1954), Under Technical Supervision of Commanding General, Aberdeen Proving Ground. Work Performed During Period 1, to 31, Under Contract No. DA-36-034-ORD-1645. Department of Army Project No. 599-01-004 (1954)Google Scholar
  12. 12.
    Luckham, D.C., German, S.M., von Henke, F.W., Karp, R.A., Milne, P.W., Oppen, D.C., Polak, W., Scherlis, W.L.: Stanford Pascal verifier user manual. Technical report, Stanford University, Stanford, CA, USA (1979)Google Scholar
  13. 13.
    Revesz, P.Z.: Quantifier-elimination for the first-order theory of boolean algebras with linear cardinality constraints. In: Benczúr, A.A., Demetrovics, J., Gottlob, G. (eds.) ADBIS 2004. LNCS, vol. 3255, pp. 1–21. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Kuncak, V., Nguyen, H.H., Rinard, M.: An algorithm for deciding BAPA: Boolean algebra with presburger arithmetic. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 260–277. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Kuncak, V., Nguyen, H.H., Rinard, M.: Deciding Boolean algebra with Presburger arithmetic. Journal of Automated Reasoning 36(3), 213–239 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kuncak, V.: Quantifier-free Boolean algebra with Presburger arithmetic is NP-complete. Technical Report TR-2007-001, MIT Computer Science and AI Lab, Cambridge, MA (January 2007)Google Scholar
  17. 17.
    Weispfenning, V.: Quantifier elimination and decision procedures for valued fields. In: Mueller, G.H., Richter, M.M. (eds.) Models and Sets. Proceedings of the Logic Colloquium held in Aachen, July 18–23, 1983 Part I. Lecture Notes in Mathematics (LNM), vol. 1103, pp. 419–472. Springer, Heidelberg (1984)Google Scholar
  18. 18.
    Monk, J.D.: Mathematical Logic. Graduate Texts in Mathematics, vol. 37. Springer, Heidelberg (1976)CrossRefzbMATHGoogle Scholar
  19. 19.
    Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76(4), 447–470 (1915)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Skolem, T.: Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen. Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse 6, 1–36 (1920)zbMATHGoogle Scholar
  21. 21.
    Malcev, A.: Untersuchungen aus dem Gebiete der mathematischen Logik. Rec. Math. [Mat. Sbornik] N.S. 1(43)(3), 323–336 (1936)zbMATHGoogle Scholar
  22. 22.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation 24(2), 209–231 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1&2), 3–27 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. The Computer Journal 36(5), 450–462 (1993); Special issue on computational quantifier eliminationMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing 8(2), 85–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sturm, T.: Linear problems in valued fields. Journal of Symbolic Computation 30(2), 207–219 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sturm, T., Weispfenning, V.: Quantifier elimination in term algebras. The case of finite languages. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2002, Institut für Informatik, Technische Universität München, Garching, Germany, pp. 285–300 (2002)Google Scholar
  28. 28.
    Dolzmann, A.: Algorithmic Strategies for Applicable Real Quantifier Elimination. Doctoral dissertation, Universität Passau, 94030 Passau, Germany (July 2000)Google Scholar
  29. 29.
    Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)CrossRefGoogle Scholar
  30. 30.
    Sturm, T., Weber, A., Abdel-Rahman, E.O., El Kahoui, M.: Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Mathematics in Computer Science 2(3), 493–515 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Weispfenning, V.: Simulation and optimization by quantifier elimination. Journal of Symbolic Computation 24(2), 189–208 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.FORWISSUniversität PassauPassauGermany
  2. 2.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain

Personalised recommendations