An Ehrenfeucht-Fräısse ́ Game for Lω1ω

Ehrenfeucht-Fräısse ́ games are very useful in studying separation and equivalence results in logic. The standard finite Ehrenfeucht-Fräısse ́ game characterizes equivalence in first order logic. The standard Ehrenfeucht-Fräısse ́ game in infinitary logic characterizes equivalence in L∞ω. The logic Lω1ω is the extension of first order logic with countable conjunctions and disjunctions. There was no Ehrenfeucht-Fräısse ́ game for Lω1ω in the literature. In this paper we develop an Ehrenfeucht-Fräısse ́ Game for Lω1ω. This game is based on a game for propositional and first order logic introduced by Hella and Väänänen. Unlike the standard Ehrenfeucht-Fräısse ́ games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of connectives in logic. We prove the adequacy theorem for this game. We also apply the new game to prove complexity results about infinite binary strings.