Advertisement

On Intersection Problems for Polynomially Generated Sets

  • Wong Karianto
  • Aloys Krieg
  • Wolfgang Thomas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4052)

Abstract

Some classes of sets of vectors of natural numbers are introduced as generalizations of the semi-linear sets, among them the ‘simple semi-polynomial sets.’ Motivated by verification problems that involve arithmetical constraints, we show results on the intersection of such generalized sets with semi-linear sets, singling out cases where the non-emptiness of intersection is decidable. Starting from these initial results, we list some problems on solvability of arithmetical constraints beyond the semi-linear ones.

Keywords

Quadratic Equation Polynomially Generate Intersection Problem Acceptance Condition Pushdown Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bruyére, V., Dall’Olio, E., Raskin, J.F.: Durations, parametric model-checking in timed automata with Presburger arithmetic. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 687–698. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Dal Zilio, S., Lugiez, D.: XML schema, tree logic and sheaves automata. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 246–263. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Dang, Z., Ibarra, O.H., Bultan, T., Kemmerer, R.A., Su, J.: Binary reachability analysis of discrete pushdown timed automata. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 69–84. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas, and languages. Pacific J. Math. 16, 285–296 (1966)MATHMathSciNetGoogle Scholar
  5. 5.
    Grunewald, F., Segal, D.: On the integer solutions of quadratic equations. J. Reine Angew. Math. 569, 13–45 (2004)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)MATHGoogle Scholar
  7. 7.
    Ibarra, O.H., Bultan, T., Su, J.: Reachability Analysis for Some Models of Infinite-State Transition Systems. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 183–198. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Karianto, W.: Parikh automata with pushdown stack. In: Diploma thesis. RWTH Aachen (2004) http://www-i7.informatik.rwth-aachen.de
  9. 9.
    Klaedtke, F., Rueß, H.: Monadic second-order logics with cardinalities. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 681–696. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Lugiez, D.: Counting and equality constraints for multitree automata. In: Gordon, A.D. (ed.) ETAPS 2003 and FOSSACS 2003. LNCS, vol. 2620, pp. 328–342. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Matiyasevich, Y.V.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)Google Scholar
  12. 12.
    Parikh, R.J.: On context-free languages. J. ACM 13, 570–581 (1966)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Seidl, H., Schwentick, T., Muscholl, A.: Numerical document queries. In: Proc. PODS 2003, pp. 155–166. ACM Press, New York (2003)CrossRefGoogle Scholar
  14. 14.
    Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Siegel, C.L.: Zur Theorie der quadratischen Formen. Nachrichten der Akademie der Wissenschaften in Göttingen, II, Mathematisch-Physikalische Klasse 3, 21–46 (1972)Google Scholar
  16. 16.
    Xie, G., Dang, Z., Ibarra, O.H.: A solvable class of quadratic Diophantine equations with applications to verification of infinite-state systems. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 668–680. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Wong Karianto
    • 1
  • Aloys Krieg
    • 2
  • Wolfgang Thomas
    • 1
  1. 1.Lehrstuhl für Informatik 7RWTH AachenGermany
  2. 2.Lehrstuhl A für MathematikRWTH AachenGermany

Personalised recommendations