Skip to main content
SpringerLink
Log in

An Algebraic Point of View on the Crane Beach Property

  • Conference paper
Book cover Computer Science Logic (CSL 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4207))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  3. Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC1. J. Comput. Syst. Sci. 41(3), 274–306 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barrington, D.A.M., Straubing, H.: Superlinear lower bounds for bounded-width branching programs. J. Comput. Syst. Sci. 50(3), 374–381 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barrington, D.A.M., Straubing, H., Thérien, D.: Non-uniform automata over groups. Information and Computation 89(2), 109–132 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  7. Behle, C., Lange, K.-J.: FO-uniformity. In: Proc. 21st Conf.  on Computational Complexity (CCC 2006) (2006)

    Google Scholar 

  8. Benedikt, M., Libkin, L.: Expressive power: The finite case. In: Constraint Databases, pp. 55–87 (2000)

    Google Scholar 

  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. Chattopadhyay, A., Krebs, A., Koucký, M., Szegedy, M., Tesson, P., Thérien, D.: Functions with bounded multiparty communication complexity (submitted, 2006)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  12. Gurevich, Y., Lewis, H.: A logic for constant-depth circuits. Information and Control 61(1), 65–74 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  15. Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  16. McKenzie, P., Péladeau, P., Thérien, D.: NC1: The automata theoretic viewpoint. Computational Complexity 1, 330–359 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Péladeau, P.: Formulas, regular languages and boolean circuits. Theor. Comput. Sci. 101(1), 133–141 (1992)

    Article  MATH  Google Scholar 

  18. Pin, J.-E.: Syntactic semigroups. In: Handbook of language theory, vol. 1, ch.10, pp. 679–746. Springer, Heidelberg (1997)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Straubing, H.: Finite Automata, Formal Logic and Circuit Complexity. Boston, Birkhauser (1994)

    Google Scholar 

  23. Straubing, H.: When can one monoid simulate another? In: Algorithmic Problems in Groups and Semigroups, pp. 267–288. Birkhäuser (2000)

    Google Scholar 

  24. Straubing, H.: Languages defined by modular quantifiers. Information and Computation 166, 112–132 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Straubing, H., Thérien, D., Thomas, W.: Regular languages defined by generalized quantifiers. Information and Computation 118, 289–301 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tesson, P.: Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, McGill University (2003)

    Google Scholar 

  27. Tesson, P., Thérien, D.: The computing power of programs over finite monoids. Journal of Automata, Languages and Combinatorics 7(2), 247–258 (2002)

    MATH  MathSciNet  Google Scholar 

  28. Tesson, P., Thérien, D.: Diamonds are forever: the variety DA. In: Semigroups, Algorithms, Automata and Languages. WSP (2002)

    Google Scholar 

  29. Tesson, P., Thérien, D.: Complete classifications for the communication complexity of regular languages. Theory of Computing Systems 38(2), 135–159 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  31. Tesson, P., Thérien, D.: Bridges between algebraic automata theory and complexity theory. The Complexity Column, Bull. EATCS 88, 37–64 (2006)

    MATH  Google Scholar 

  32. Tesson, P., Thérien, D.: Logic meets algebra: the case of regular languages (submitted, 2006)

    Google Scholar 

  33. Thomas, W.: Languages, Automata and Logic, vol. III, ch. 7, pp. 389–455. Springer, Heidelberg (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département d’Informatique et de Génie Logiciel, Université Laval,  

    Clemens Lautemann & Pascal Tesson

  2. School of Computer Science, McGill University,  

    Denis Thérien

Authors
  1. Clemens Lautemann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pascal Tesson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Denis Thérien
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Rovira i Virgili University, Tarragona, Spain

    Zoltán Ésik

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lautemann, C., Tesson, P., Thérien, D. (2006). An Algebraic Point of View on the Crane Beach Property. In: Ésik, Z. (eds) Computer Science Logic. CSL 2006. Lecture Notes in Computer Science, vol 4207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11874683_28

Download citation

  • DOI: https://doi.org/10.1007/11874683_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-45458-8

  • Online ISBN: 978-3-540-45459-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation