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Ehrenfeucht-Fraisse games are very useful in studying separation andequivalence results in logic. The standard finite Ehrenfeucht-Fraisse gamecharacterizes equivalence in first order logic. The standardEhrenfeucht-Fraisse game in infinitary logic characterizes equivalence in$L_{\infty\omega}$. The logic $L_{\omega_1\omega}$ is the extension of firstorder logic with countable conjunctions and disjunctions. There was noEhrenfeucht-Fraisse game for $L_{\omega_1\omega}$ in the literature.  In this paper we develop an Ehrenfeucht-Fraisse Game for$L_{\omega_1\omega}$. This game is based on a game for propositional and firstorder logic introduced by Hella and Vaananen. Unlike the standardEhrenfeucht-Fraisse games which are modeled solely after the behavior ofquantifiers, this new game also takes into account the behavior of connectivesin logic. We prove the adequacy theorem for this game. We also apply the newgame to prove complexity results about infinite binary strings.

An Ehrenfeucht-Fra\"{i}ss\'{e} Game for $L_{\omega_1\omega}$