Advertisement

Linear Circuits, Two-Variable Logic and Weakly Blocked Monoids

  • Christoph Behle
  • Andreas Krebs
  • Mark Mercer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets Open image in new window satisfying a certain closure property, a one-to-one correspondence between \(FO[<,\ensuremath{\mathfrak{P}}]\)-uniform linear circuits, two-variable formulae with \(\ensuremath{\mathfrak{P}}\) predicates, and weak block products of monoids. In particular, we consider the case of linear TC0, majority quantifiers, and finitely typed monoids. This correspondence will hold for any numerical predicate set which is FO[ < ]-closed and whose predicates do not depend on the input length.

Keywords

Closure Property Input Gate Output Gate Binary Predicate Majority 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.
    Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1. J. Comp. System Sci. 38, 150–164 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barrington, D.A., Immerman, N., Straubing, H.: On uniformity within NC 1. J. Comp. System Sci. 41, 274–306 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barrington, D., Immerman, N., Lautemann, C., Schweickardt, N., Thérien, D.: The Crane Beach Conjecture. In: Proc. of the 16th IEEE Symposium On Logic in Computer Science, pp. 187–196. IEEE Computer Society Press, Los Alamitos (2001)CrossRefGoogle Scholar
  4. 4.
    Barrington, D., Thérien, D.: Finite Monoids and the Fine Structure of NC 1. Journal of ACM 35(4), 941–952 (1988)CrossRefGoogle Scholar
  5. 5.
    Behle, C., Krebs, A., Reifferscheid, S.: A 5 not in FO+MOD+MAJ2[reg], http://www-fs.informatik.uni-tuebingen.de/publi/a5notinltc0.pdf (to appear)
  6. 6.
    Behle, C., Lange, K.-J.: FO[ < ]-Uniformity. In: IEEE Conference on Compuatational Complexity (2006)Google Scholar
  7. 7.
    Furst, M., Saxe, J.B., Sipser, M.: Parity circuits and the polynomial-time hierarchy. In: Proc. 22th IEEE Symposium on Foundations of Computer Science, pp. 260–270 (1981)Google Scholar
  8. 8.
    Krebs, A., Lange, K.-J., Reifferscheid, St.: In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Koucký, M., Lautemann, C., Poloczek, S., Thérien, D.: Circuit lower bounds via Ehrenfeucht-Frai̇ssé games. In: Proc. 21st Conf. on Compuatational Complexity (CCC’06) (2006)Google Scholar
  10. 10.
    Lange, K.-J.: Some results on majority quantifiers over words. In: Proc. of the 19th IEEE Conference on Computational Complexity, pp. 123–129. IEEE Computer Society Press, Los Alamitos (2004)CrossRefGoogle Scholar
  11. 11.
    Lautemann, C., McKenzie, P., Schwentick, T., Vollmer, H.: The descriptive complexity approach to LOGCFL. J. Comp. System Sci. 62, 629–652 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lawson, M.: Finite Automata. Chapman & Hall/CRC (2004)Google Scholar
  13. 13.
    Rhodes, J., Tilson, B.: The Kernel of Monoid Morphisms. J. Pure Applied Alg. 62, 227–268 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Roy, A., Straubing, H.: Definability of Languages by Generalized First-Order Formulas over (N,+). In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, Springer, Heidelberg (to appear 2006)CrossRefGoogle Scholar
  15. 15.
    Ruhl, M.: Counting and addition cannot express deterministic transitive closure. In: Proc. of 14th IEEE Symposium On Logic in Computer Science, pp. 326–334. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  16. 16.
    Schweikardt, N.: On the Expressive Power of First-Order Logic with Built-In Predicates. In: Dissertation, Universität Mainz (2001)Google Scholar
  17. 17.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser (1994)Google Scholar
  18. 18.
    Straubing, H., Thérien, D., Thomas, W.: Regular languages defined by generalize quantifiers. Information and Computation 118, 289–301 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Straubing, H., Thérien, D.: Regular Languages Defined by Generalized First-Order Formulas with a Bounded Number of Bound Variables. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 551–562. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Straubing, H., Thérien, D.: Weakly Iterated Block Products of Finite Monoids. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 91–104. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  21. 21.
    Thérien, D., Wilke, T.: Over Words, Two Variables are as Powerful as One Quantifier Alternation. In: Proc. 30th ACM Symposium on the Theory of Computing, pp. 256–263 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Christoph Behle
    • 1
  • Andreas Krebs
    • 1
  • Mark Mercer
    • 2
  1. 1.WSI - University of Tuebingen, Sand 13, 72076 TuebingenGermany
  2. 2.McGill University, MontrealCanada

Personalised recommendations