Implicit Definability and Infinitary Logic in Finite Model Theory

We study the relationship between the infinitary logic L ∞ω ω with finitely many variables and implicit definability in effective fragments of L ∞ω ω on finite structures. We show that fixpoint logic has strictly less expressive power than first-order implicit definability. We also establish that the separation of fixpoint logic from a certain restriction of first-order implicit definability to L ∞ω ω is equivalent to the separation of PTIME from UP ∩ co-UP. Finally, we delineate the relationship between partial fixpoint logic and implicit definability in partial fixpoint logic on finite structures.