Definability with Bounded Number of Bound Variables

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-theoretic 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.