Paper: Definability with Bounded Number of Bound Variables (at LICS 1987)

Authors: Neil Immerman Dexter C. Kozen

Abstract

A theory satisfies the k-variable property if every first-order formula is equivalent to a formula with at most k bound variables (possibly reused). Gabbay has shown that a fixed time structure satisfies the k-variable property for some k if and only if there exists a finite basis for the temporal connectives over that structure. We give a model-theretic method for establishing the k--variable property, involving a restricted Ehrenfeucht-Fraisse game in which each player has only k pebbles. We use the method to unify and simplify results in the literature for linear orders. We also establish new k-variable properties for various theories of bounded-degree trees, and in each case obtain tight upper and lower bounds on k. This gives the first finite basis theorems for branching-time models.

BibTeX

  @InProceedings{ImmermanKozen-DefinabilitywithBou,
    author = 	 {Neil Immerman and Dexter C. Kozen},
    title = 	 {Definability with Bounded Number of Bound Variables},
    booktitle =  {Proceedings of the Second Annual IEEE Symposium on Logic in Computer Science (LICS 1987)},
    year =	 {1987},
    month =	 {June}, 
    pages =      {236--244},
    location =   {Ithaca, NY, USA}, 
    publisher =	 {IEEE Computer Society Press}
  }