Skip to main content

Effective Quantifier Elimination for Presburger Arithmetic with Infinity

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-04103-7_18
  • Chapter length: 18 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   99.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-04103-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   129.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. Cooper, D.C.: Theorem proving in arithmetic without multiplication. Machine Intelligence 7, 91–99 (1972)

    MATH  Google Scholar 

  3. Ferrante, J., Rackoff, C.W.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer, Berlin (1979)

    MATH  Google Scholar 

  4. Fischer, M.J., Rabin, M.: Super-exponential complexity of Presburger arithmetic. SIAM-AMS Proceedings 7, 27–41 (1974)

    Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Weispfenning, V.: The complexity of almost linear Diophantine problems. Journal of Symbolic Computation 10(5), 395–403 (1990)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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. 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. 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)

    CrossRef  Google Scholar 

  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)

    CrossRef  Google Scholar 

  15. Kuncak, V., Nguyen, H.H., Rinard, M.: Deciding Boolean algebra with Presburger arithmetic. Journal of Automated Reasoning 36(3), 213–239 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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. 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. Monk, J.D.: Mathematical Logic. Graduate Texts in Mathematics, vol. 37. Springer, Heidelberg (1976)

    CrossRef  MATH  Google Scholar 

  19. Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76(4), 447–470 (1915)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  21. Malcev, A.: Untersuchungen aus dem Gebiete der mathematischen Logik. Rec. Math. [Mat. Sbornik] N.S. 1(43)(3), 323–336 (1936)

    MATH  Google Scholar 

  22. Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation 24(2), 209–231 (1997)

    MathSciNet  CrossRef  MATH  Google Scholar 

  23. Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1&2), 3–27 (1988)

    MathSciNet  CrossRef  MATH  Google Scholar 

  24. Loos, R., Weispfenning, V.: Applying linear quantifier elimination. The Computer Journal 36(5), 450–462 (1993); Special issue on computational quantifier elimination

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  26. Sturm, T.: Linear problems in valued fields. Journal of Symbolic Computation 30(2), 207–219 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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. Dolzmann, A.: Algorithmic Strategies for Applicable Real Quantifier Elimination. Doctoral dissertation, Universität Passau, 94030 Passau, Germany (July 2000)

    Google Scholar 

  29. Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  31. Weispfenning, V.: Simulation and optimization by quantifier elimination. Journal of Symbolic Computation 24(2), 189–208 (1997)

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lasaruk, A., Sturm, T. (2009). Effective Quantifier Elimination for Presburger Arithmetic with Infinity. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2009. Lecture Notes in Computer Science, vol 5743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04103-7_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04103-7_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04102-0

  • Online ISBN: 978-3-642-04103-7

  • eBook Packages: Computer ScienceComputer Science (R0)