First-order properties of trees, star-free expressions, and aperiodicity

Abstract

We characterize the first-order definable sets of finite trees in terms of appropriate star-free tree expressions and show that for sets of trees first-order definability is strictly weaker than aperiodicity. These two theorems show how far the results of McNaughton and Schutzenberger on starfree sets of words (stating the equivalence between first-order definability, star-freeness, and aperiodicity) are transferable to the context of trees. Both results of the paper rely on the method of the Ehrenfeucht-Fraisse-game.