Advertisement

The Complexity of Membership Problems for Circuits over Sets of Integers

  • Stephen D. Travers
Conference paper
  • 475 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)

Abstract

We investigate the complexity of membership problems for \(\{\cup,\cap,\!\bar{\quad},+,\times\}\)-circuits computing sets of integers. These problems are a natural modification of the membership problems for circuits computing sets of natural numbers studied by McKenzie and Wagner (2003). We show that there are several membership problems for which the complexity in the case of integers differs significantly from the case of the natural numbers: Testing membership in the subset of integers produced at the output of a {∪,+,×}-circuit is NEXPTIME-complete, whereas it is PSPACE-complete for the natural numbers. As another result, evaluating \(\{\!\bar{\quad},+\}\)-circuits is shown to be P-complete for the integers and PSPACE-complete for the natural numbers. The latter result extends work by McKenzie and Wagner (2003) in nontrivial ways. Furthermore, evaluating {×}-circuits is shown to be \({\rm NL}\land\oplus{\rm L}\)-complete, and several other cases are resolved.

Keywords

Natural Number Boolean Formula Membership Problem Input Gate Output Gate 
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.
    Allender, E.: Making computation count: Arithmetic Circuits in the Nineties, in the Complexity Theory Column. SIGACT NEWS 28(4), 2–15 (1997)CrossRefGoogle Scholar
  2. 2.
    Balcázar, J.L., Lozano, A., Torán, J.: The complexity of algorithmic problems in succinct instances. In: Computer Science, Plenum, New York (1992)Google Scholar
  3. 3.
    Crandall, R.E., Mayer, E.W., Papadopoulos, J.S.: The Twenty-Fourth Fermat Number is Composite. Math. Comput. 72, 1555–1572 (2003)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Goldschlager, L.M.: The monotone and planar circuit value problems are logspace complete for P. SIGACT NEWS 9, 25–29 (1977)CrossRefGoogle Scholar
  5. 5.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford University Press, Oxford (1979)zbMATHGoogle Scholar
  6. 6.
    Immerman, N.: Nondeterministic space is closed under complementation. SIAM Journal on Computing 17, 935–938 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    McKenzie, P., Wagner, K.W.: The Complexity of Membership Problems for Circuits over Sets of Natural Numbers. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 571–582. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  9. 9.
    Petersen, H.: Bemerkungen zu ganzzahligen Ausdrücken, Private Communication (2004) Google Scholar
  10. 10.
    Savitch, W.J.: Maze recognizing automata and nondeterministic tape complexity. Journal of Computer and System Sciences 7, 389–403 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Stockmeyer, L.J., Meyer, A.R.: Word Problems Requiring Exponential Time. In: Proceedings of the 5th ACM Symposium on the Theory of Computing, pp. 1–9 (1973)Google Scholar
  12. 12.
    Szelepcsényi, R.: The method of forced enumeration for nondeterministic automata. AI 26, 279–284 (1984)Google Scholar
  13. 13.
    Wagner, K.W.: The complexity of problems concerning graphs with regularities. In: Chytil, M.P., Koubek, V. (eds.) MFCS 1984. LNCS, vol. 176, pp. 544–552. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  14. 14.
    Yang, K.: Integer circuit evaluation is PSPACE-complete. In: Proceedings 15th Conference on Computational Complexity, pp. 204–211 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Stephen D. Travers
    • 1
  1. 1.Theoretische InformatikBayerische Julius-Maximilians Universität WürzburgWürzburgGermany

Personalised recommendations