Advertisement

An Algebraic Point of View on the Crane Beach Property

  • Clemens Lautemann
  • Pascal Tesson
  • Denis Thérien
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)

Abstract

A letter e ∈Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The Crane Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in first-order with arbitrary numerical predicates (\({\bf FO}[\mathit{Arb}]\)) is in fact FO [<] definable and is thus a regular, star-free language. More generally, we say that a logic or a computational model has the Crane Beach property if the only languages with neutral letter that it can define/compute are regular.

We develop an algebraic point of view on the Crane Beach properties using the program over monoid formalism which has proved of importance in circuit complexity. Using recent communication complexity results we establish a number of Crane Beach results for programs over specific classes of monoids. These can be viewed as Crane Beach theorems for classes of bounded-width branching programs. We also apply this to a standard extension of FO using modular-counting quantifiers and show that the boolean closure of this logic’s Σ1 fragment has the CBP.

Keywords

Communication Complexity Expressive Power Regular Language Boolean Circuit Algebraic Point 
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. Comput. Syst. Sci. 38(1), 150–164 (1989)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barrington, D.A.M., Immerman, N., Lautemann, C., Schweikardt, N., Thérien, D.: First-order expressibility of languages with neutral letters or: The Crane Beach conjecture. J. Comput. Syst. Sci. 70(2), 101–127 (2005)MATHCrossRefGoogle Scholar
  3. 3.
    Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC1. J. Comput. Syst. Sci. 41(3), 274–306 (1990)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barrington, D.A.M., Straubing, H.: Superlinear lower bounds for bounded-width branching programs. J. Comput. Syst. Sci. 50(3), 374–381 (1995)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Barrington, D.A.M., Straubing, H., Thérien, D.: Non-uniform automata over groups. Information and Computation 89(2), 109–132 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Barrington, D.A.M., Thérien, D.: Finite monoids and the fine structure of NC 1. Journal of the ACM 35(4), 941–952 (1988)CrossRefGoogle Scholar
  7. 7.
    Behle, C., Lange, K.-J.: FO-uniformity. In: Proc. 21st Conf.  on Computational Complexity (CCC 2006) (2006)Google Scholar
  8. 8.
    Benedikt, M., Libkin, L.: Expressive power: The finite case. In: Constraint Databases, pp. 55–87 (2000)Google Scholar
  9. 9.
    Chandra, A.K., Furst, M.L., Lipton, R.J.: Multi-party protocols. In: Proc.  15th ACM Symp. on Theory of Computing (STOC 1983), pp. 94–99 (1983)Google Scholar
  10. 10.
    Chattopadhyay, A., Krebs, A., Koucký, M., Szegedy, M., Tesson, P., Thérien, D.: Functions with bounded multiparty communication complexity (submitted, 2006)Google Scholar
  11. 11.
    Gavaldà, R., Thérien, D.: Algebraic characterizations of small classes of boolean functions. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Gurevich, Y., Lewis, H.: A logic for constant-depth circuits. Information and Control 61(1), 65–74 (1984)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  15. 15.
    Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)MATHGoogle Scholar
  16. 16.
    McKenzie, P., Péladeau, P., Thérien, D.: NC1: The automata theoretic viewpoint. Computational Complexity 1, 330–359 (1991)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Péladeau, P.: Formulas, regular languages and boolean circuits. Theor. Comput. Sci. 101(1), 133–141 (1992)MATHCrossRefGoogle Scholar
  18. 18.
    Pin, J.-E.: Syntactic semigroups. In: Handbook of language theory, vol. 1, ch.10, pp. 679–746. Springer, Heidelberg (1997)Google Scholar
  19. 19.
    Raymond, J.-F., Tesson, P., Thérien, D.: An algebraic approach to communication complexity. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 29–40. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  20. 20.
    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, pp. 489–499. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Proc. 19th ACM STOC, pp. 77–82 (1986)Google Scholar
  22. 22.
    Straubing, H.: Finite Automata, Formal Logic and Circuit Complexity. Boston, Birkhauser (1994)Google Scholar
  23. 23.
    Straubing, H.: When can one monoid simulate another? In: Algorithmic Problems in Groups and Semigroups, pp. 267–288. Birkhäuser (2000)Google Scholar
  24. 24.
    Straubing, H.: Languages defined by modular quantifiers. Information and Computation 166, 112–132 (2001)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Straubing, H., Thérien, D., Thomas, W.: Regular languages defined by generalized quantifiers. Information and Computation 118, 289–301 (1995)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Tesson, P.: Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, McGill University (2003)Google Scholar
  27. 27.
    Tesson, P., Thérien, D.: The computing power of programs over finite monoids. Journal of Automata, Languages and Combinatorics 7(2), 247–258 (2002)MATHMathSciNetGoogle Scholar
  28. 28.
    Tesson, P., Thérien, D.: Diamonds are forever: the variety DA. In: Semigroups, Algorithms, Automata and Languages. WSP (2002)Google Scholar
  29. 29.
    Tesson, P., Thérien, D.: Complete classifications for the communication complexity of regular languages. Theory of Computing Systems 38(2), 135–159 (2005)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tesson, P., Thérien, D.: Restricted two-variable sentences, circuits and communication complexity. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 526–538. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  31. 31.
    Tesson, P., Thérien, D.: Bridges between algebraic automata theory and complexity theory. The Complexity Column, Bull. EATCS 88, 37–64 (2006)MATHGoogle Scholar
  32. 32.
    Tesson, P., Thérien, D.: Logic meets algebra: the case of regular languages (submitted, 2006)Google Scholar
  33. 33.
    Thomas, W.: Languages, Automata and Logic, vol. III, ch. 7, pp. 389–455. Springer, Heidelberg (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Clemens Lautemann
    • 1
  • Pascal Tesson
    • 1
  • Denis Thérien
    • 2
  1. 1.Département d’Informatique et de Génie LogicielUniversité Laval 
  2. 2.School of Computer ScienceMcGill University 

Personalised recommendations