Deenability with Bounded Number of Bound Variables

A theory satisses the k-variable property if every rst-order formula is equivalent to a formula with at most k bound variables (possibly reused). Gabbay has shown that a model of temporal logic satisses the k-variable property for some k if and only if there exists a nite basis for the temporal connectives over that model. 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 rst nite basis theorems for branching-time models of temporal logic.