Computational Complexity of Ehrenfeucht-Fraïssé Games on Finite Structures

Abstract

We show that deciding the winner of the r-moves Ehrenfeucht-Fraïssé game on two finite structures A and B, over any fixed signature Σ that contains at least one binary and one ternary relation, is PSPACE complete. We consider two natural modifications of the EF game, the one-sided r-moves EF game, where the spoiler can choose from the first structure A only, and therefore the duplicator wins only if B satisfies all the existential formulas of rank at most r that A satisfies; and the k-alternations r-moves EF game (for each fixed k), where the spoiler can choose from either structure, but he can switch structure at most k times, and therefore the duplicator wins iff A and B satisfy the same first order formulas of rank at most r and quantifier alternation at most k (defined in the paper). We show that deciding the winner in both the one-sided EF game and the k-alternations EF game is also PSPACE complete.