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)


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.


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.


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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 

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