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The Complexity of Membership Problems for Circuits over Sets of Natural Numbers

  • Pierre McKenzie
  • Klaus W. Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2607)

Abstract

The problem of testing membership in the subset of the natural numbers produced at the output gate of a ∪, ∩,- ,+, * combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case ∪, ∩,+, * is shown NEXPTIME-complete, the cases ∪, ∩,- , *, ∪, ∩, *, ∪, ∩,+ are shown PSPACE-complete, the case ∪,+ is shown NP-complete, the case ∩,+ is shown C=L-complete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for union-intersection-concatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one’s choosing. Our results extend in nontrivial ways past work by Stockmeyer and Meyer (1973), Wagner (1984) and Yang (2000).

Classification:

Computational complexity 

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References

  1. [AAD00]
    M. Agrawal, E. Allender, and S. Datta, On TC0, AC0, and arithmetic circuits, J. Computer and System Sciences 60 (2000), pp. 395–421.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [All97]
    E. Allender, Making computation count: Arithmetic circuits in the Nineties, in the Complexity Theory Column, SIGACT NEWS 28 (4) (1997) pp. 2–15.CrossRefGoogle Scholar
  3. [ARZ99]
    E. Allender, K. Reinhardt, S. Zhou, Isolation, matching, and counting: Uniform and nonuniform upper bounds, J. Computer and System Sciences 59(1999), pp. 164–181.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [ABH01]
    E. Allender, D. Barrington, and W. Hesse, Uniform constant-depth threshold circuits for division and iterated multiplication, Proceedings 16th Conference on Computational Complexity, 2001, pp. 150–159.Google Scholar
  5. [AJMV98]
    E. Allender, J. Jiao, M. Mahajan and V. Vinay, Non-commutative arithmetic circuits: depth-reduction and depth lower bounds, Theoretical Computer Science Vol. 209 (1/2,2) (1998), pp. 47–86.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [BS96]
    E. Bach, J. Shallit, Algorithmic Number Theory, Volume I: Efficient Algorithms, MIT Press 1996.Google Scholar
  7. [BM95]
    M. Beaudry and P. McKenzie, Circuits, matrices and nonassociative computation, J. Computer and System Sciences 50 (1995), pp. 441–455.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [BMPT97]
    M. Beaudry, P. McKenzie, P. Péladeau, D. Thérien, Finite monoids: from word to circuit evaluation, SIAM J. Computing 26 (1997), pp. 138–152.zbMATHCrossRefGoogle Scholar
  9. [BCGR92]
    S. Buss, S. Cook, A. Gupta, V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Computing 21 (1992), pp. 755–780.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Bu87]
    S. R. Buss, The boolean formula value problemis in ALOGTIME, Proceedings 19th ACM Symp, on the Theory of Computing, 1987, pp. 123–131.Google Scholar
  11. [CMTV98]
    H. Caussinus, P. McKenzie, D. Thérien, H. Vollmer, Nondeterministic NC1 computation, J. Computer and System Sciences, 57 (2), 1998, pp. 200–212.zbMATHCrossRefGoogle Scholar
  12. [CSV84]
    A. K. Chandra, L. Stockmeyer, U. Vishkin, Constant depth reducibility, SIAM Journal on Computing, 13, 1984, pp. 423–439.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [CDL]
    A. Chiu, G. Davida, and B. Litow, NC 1 division, available at http://www.cs.jcu.edu.au/ bruce/papers/cr00 3.ps.gz
  14. [Go77]
    L. M. Goldschlager, The monotone and planar circuit value problems are logspace complete for P, SIGACT News, 9, 1977, pp. 25–29.CrossRefGoogle Scholar
  15. [He01]
    W. Hesse, Division in uniform TC0, Proceedings of the 28th International Colloquium on Automata, Languages, and Programming 2001, Lecture Notes in Computer Science 2076, pp. 104–114Google Scholar
  16. [Ga84]
    J. von zur Gathen, Parallel algorithms for algebraic problems, SIAM J. on Computing 13(4), (1984), pp. 802–824.Google Scholar
  17. [GHR95]
    R. Greenlaw, J. Hoover and L. Ruzzo, Limits to parallel computation, P-completeness theory, Oxford University Press, 1995, 311 pages.Google Scholar
  18. [Og98]
    M. Ogihara, The PL hierarchy collapses, SIAM Journal of Computing, 27, 1998, pp. 1430–1437.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [SM73]
    L. J. Stockmeyer, A. R. Meyer, Word Problems Requiring Exponential Time, Proceedings 5th ACM Symposium on the Theory of Computing, 1973, pp. 1–9.Google Scholar
  20. [Vo00]
    H. Vollmer, Circuit complexity, Springer, 2000.Google Scholar
  21. [Wa84]
    K. W. Wagner, The complexity of problems concerning graphs with regularities, Proceedings 11th Mathematical Foundations of Computer Science 1984, Lecture Notes in Computer Science 176, pp. 544–552. Full version as TR N/84/52, Friedrich-Schiller-Universität Jena, 1984.zbMATHGoogle Scholar
  22. [Ya00]
    K. Yang, Integer circuit evaluation is PSPACE-complete, Proceedings 15th Conference on Computational Complexity, 2000, pp. 204–211.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Pierre McKenzie
    • 1
  • Klaus W. Wagner
    • 2
  1. 1.Informatique et recherche opérationnelleUniversité de MontréalMontréalCanada
  2. 2.Theoretische InformatikBayerische Julius-Maximilians-Universität WürzburgWürzburgGermany

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