An EhrenfeuchtFraïssé Game for $L_{\omega_1\omega}$
Abstract
EhrenfeuchtFraisse games are very useful in studying separation and equivalence results in logic. The standard finite EhrenfeuchtFraisse game characterizes equivalence in first order logic. The standard EhrenfeuchtFraisse 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 EhrenfeuchtFraisse game for $L_{\omega_1\omega}$ in the literature. In this paper we develop an EhrenfeuchtFraisse 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 EhrenfeuchtFraisse 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2012
 arXiv:
 arXiv:1212.0108
 Bibcode:
 2012arXiv1212.0108V
 Keywords:

 Mathematics  Logic;
 03C75
 EPrint:
 22 pages, 1 figure