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

TL;DR

This paper introduces a novel Ehrenfeucht-Fra"{i}ssé} game tailored for the logic $L_{oldsymbol{ ext{ extomega}}_1 ext{ extomega}}$, extending the tool's applicability to infinitary logic with connectives, and demonstrates its effectiveness through complexity results.

Contribution

It develops the first Ehrenfeucht-Fra"{i}ssé} game for $L_{oldsymbol{ ext{ extomega}}_1 ext{ extomega}}$, incorporating connectives and proving its adequacy.

Findings

The game characterizes equivalence in $L_{oldsymbol{ ext{ extomega}}}_1 ext{ extomega}$.

Proves the adequacy theorem for the new game.

Applies the game to analyze complexity of infinite binary strings.

Abstract

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.