  Ehrenfeucht-Fraisse games are very useful in studying separation and
equivalence results in logic. The standard finite Ehrenfeucht-Fraisse game
characterizes equivalence in first order logic. The standard
Ehrenfeucht-Fraisse game in infinitary logic characterizes equivalence in
$L_{\infty\omega}$. The logic $L_{\omega_1\omega}$ is the extension of first
order logic with countable conjunctions and disjunctions. There was no
Ehrenfeucht-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 first
order logic introduced by Hella and Vaananen. Unlike the standard
Ehrenfeucht-Fraisse 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.
