# What are some other uses for Ehrenfeucht-Fraïssé games?

Let $\mathfrak{A} = \langle A, \dots \rangle$ and $\mathfrak{B} = \langle B, \dots \rangle$ be structures for a signature $\mathscr{L}$. For each ordinal $\gamma$ we define a game of perfect information between two players, Spoiler and Duplicator:

For each $\xi < \gamma$ Spoiler picks an element from $A$ or $B$, and Duplicator responds with an element from the other set. Together they determine a pair $(a_{\xi}, b_{\xi})$ in $A \times B$. After $\gamma$ rounds, Duplicator wins if there is an isomorphism, carrying each $a_{\xi}$ to $b_{\xi}$, from the substructure of $\mathfrak{A}$ generated by the $a_{\xi}$ sequence to the substructure of $\mathfrak{B}$ generated by the $b_{\xi}$ sequence. Else Spoiler wins.

This is the Ehrenfeucht-Fraïssé game $G_{\gamma}(\mathfrak{A},\mathfrak{B})$ of length $\gamma$ on $(\mathfrak{A}, \mathfrak{B})$. Note, as François did in the comments, that in some contexts this term designates a different game.

If $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic, then clearly Duplicator can win the game for any $\gamma$. Moreover, if $\mathfrak{A}$ and $\mathfrak{B}$ are finite or countable, then they are isomorphic iff Duplicator has a winning strategy for $G_{\omega}(\mathfrak{A},\mathfrak{B})$. Such games may therefore be used toward establishing isomorphisms.

Ehrenfeucht-Fraïssé games can also show that a structural property isn't expressible in a first order language. Given a class $K$ of finite structures for a relational signature $\mathscr{L}$, to show that there is no first order $\mathscr{L}$ theory $\Gamma$ whose class of finite models is precisely $K$, it suffices to show that for each $n$ there are $\mathscr{L}$ structures $\mathfrak{A}_n \in K$ and $\mathfrak{B}_n \in \neg K$ such that Duplicator can win $G_n(\mathfrak{A}_n,\mathfrak{B}_n)$. This is because if Duplicator can win, then $\mathfrak{A}_n$ and $\mathfrak{B}_n$ verify the same sentences of quantifier rank $\leq n$. (I forget whether this works when $\mathscr{L}$ has function symbols.)

What are some lesser known uses for these games, or for minor variations on them? I have a special interest in models of set theory.

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This is not how EF games are defined for ordinals $\gamma \geq \omega$. The plays of an EF game are always finite. At each move, Spoiler picks an ordinal smaller than the previous one he played, or smaller than $\gamma$ if this is the first move. The game ends when Spoiler picks the ordinal 0, which is bound to happen in finitely many steps. –  François G. Dorais Aug 12 '11 at 21:13
Also, Duplicator wins $G_\omega(\mathfrak{A},\mathfrak{B})$ if and only if $\mathfrak{A}$ and $\mathfrak{B}$ are elementarily equivalent. When $\mathfrak{A}$ and $\mathfrak{B}$ are finite or countable and Duplicator wins the $G_\gamma(\mathfrak{A},\mathfrak{B})$ game for every ordinal $\gamma$, then $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic. –  François G. Dorais Aug 12 '11 at 21:15
François, I just copied the definition from pp. 95 to 96 of Hodges's treatise on model theory. Theorem 3.2.3 there states that, in the finite or countable case, Duplicator wins $G_\omega(\mathfrak{A}, \mathfrak{B})$ iff $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic. Perhaps you're thinking of what on p. 102 Hodges calls "unnested" EF games. Corrollary 3.3.3 states that $\mathfrak{A}$ and $\mathfrak{B}$ are elementarily equivalent iff Duplicator wins every such game of finite length. This fits with your second comment. –  Cole Leahy Aug 12 '11 at 22:06
I would believe it if you told me that unnested EF games are more useful than the ones I described. Do you care to elaborate? –  Cole Leahy Aug 12 '11 at 22:06
I figured out the history here. On the one hand, the EF games I just described were introduced by Barwise. On the other hand, Karp studied the game $G_\omega(\mathfrak{A},\mathfrak{B})$ as you describe it. (Longer games have been studied by various later authors.) It appears that both variants have been called EF games... –  François G. Dorais Aug 12 '11 at 22:07