Fixpoint Logic vs. Infinitary Logic in Finite-Model Theory

In recent years several extensions of first-order logic have been investigated in the context of finite-model theory. Fixpoint logic and the infinitary logic L∞ω with a finite number of variables have turned out to be of particular importance. The study of fixpoint logic generated interactions with both database theory and complexity theory, while the infinitary logic L∞ω proved to be a useful tool for analyzing the expressive power of fixpoint logic. In addition to being a proper extension of fixpoint logic, L∞ω enjoys a game-theoretic characterization and possesses interesting structural properties, such as the 0-1 law. In this paper we pursue further the study of the relationship between L∞ω and fixpoint logic. We observe that equivalence of two finite structures with respect to L∞ω is expressible in fixpoint logic. As a first application of this, we obtain a normal-form theorem for L∞ω on finite structures. We then focus on the relative expressive power of first-order logic, fixpoint logic, and L∞ω on arbitrary classes of finite structures. Our second main result characterizes when L∞ω collapses to first-order logic on an arbitrary class of finite structures. This resolves affirmatively a conjecture of G.L. McColm.