Ehrenfeucht–Fraïssé game

The main idea behind the game is that we have two structures, and two players – Spoiler and Duplicator. Duplicator wants to show that the two structures are elementarily equivalent (satisfy the same first-order sentences), whereas Spoiler wants to show that they are different. The game is played in rounds. A round proceeds as follows: Spoiler chooses any element from one of the structures, and Duplicator chooses an element from the other structure. In simplified terms, the Duplicator's task is to always pick an element "similar" to the one that the Spoiler has chosen, whereas the Spoiler's task is to choose an element for which no "similar" element exists in the other structure. Duplicator wins if there exists an isomorphism between the eventual substructures chosen from the two different structures; otherwise, Spoiler wins.

The game lasts for a fixed number of steps ${\displaystyle \gamma }$ (which is an ordinal – usually a finite number or ${\displaystyle \omega }$).

Suppose that we are given two structures ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$, each with no function symbols and the same set of relation symbols, and a fixed natural number n. We can then define the Ehrenfeucht–Fraïssé game ${\displaystyle G_{n}({\mathfrak {A}},{\mathfrak {B}})}$ to be a game between two players, Spoiler and Duplicator, played as follows:

1. The first player, Spoiler, picks either a member ${\displaystyle a_{1}}$ of ${\displaystyle {\mathfrak {A}}}$ or a member ${\displaystyle b_{1}}$ of ${\displaystyle {\mathfrak {B}}}$.
2. If Spoiler picked a member of ${\displaystyle {\mathfrak {A}}}$, Duplicator picks a member ${\displaystyle b_{1}}$ of ${\displaystyle {\mathfrak {B}}}$; otherwise, Duplicator picks a member ${\displaystyle a_{1}}$ of ${\displaystyle {\mathfrak {A}}}$.
3. Spoiler picks either a member ${\displaystyle a_{2}}$ of ${\displaystyle {\mathfrak {A}}}$ or a member ${\displaystyle b_{2}}$ of ${\displaystyle {\mathfrak {B}}}$.
4. Duplicator picks an element ${\displaystyle a_{2}}$ or ${\displaystyle b_{2}}$ in the model from which Spoiler did not pick.
5. Spoiler and Duplicator continue to pick members of ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$ for ${\displaystyle n-2}$ more steps.
6. At the conclusion of the game, we have chosen distinct elements ${\displaystyle a_{1},\dots ,a_{n}}$ of ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle b_{1},\dots ,b_{n}}$ of ${\displaystyle {\mathfrak {B}}}$. We therefore have two structures on the set ${\displaystyle \{1,\dots ,n\}}$, one induced from ${\displaystyle {\mathfrak {A}}}$ via the map sending ${\displaystyle i}$ to ${\displaystyle a_{i}}$, and the other induced from ${\displaystyle {\mathfrak {B}}}$ via the map sending ${\displaystyle i}$ to ${\displaystyle b_{i}}$. Duplicator wins if these structures are the same; Spoiler wins if they are not.

For each n we define a relation ${\displaystyle {\mathfrak {A}}\ {\overset {n}{\sim }}\ {\mathfrak {B}}}$ if Duplicator wins the n-move game ${\displaystyle G_{n}({\mathfrak {A}},{\mathfrak {B}})}$. These are all equivalence relations on the class of structures with the given relation symbols. The intersection of all these relations is again an equivalence relation ${\displaystyle {\mathfrak {A}}\sim {\mathfrak {B}}}$.

It is easy to prove that if Duplicator wins this game for all finite n, that is, ${\displaystyle {\mathfrak {A}}\sim {\mathfrak {B}}}$, then ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$ are elementarily equivalent. If the set of relation symbols being considered is finite, the converse is also true.

If a property ${\displaystyle Q}$ is true of ${\displaystyle {\mathfrak {A}}}$ but not true of ${\displaystyle {\mathfrak {B}}}$, but ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$ can be shown equivalent by providing a winning strategy for Duplicator, then this shows that ${\displaystyle Q}$ is inexpressible in the logic captured by this game.

The back-and-forth method used in the Ehrenfeucht–Fraïssé game to verify elementary equivalence was given by Roland Fraïssé in his thesis;[2][3] it was formulated as a game by Andrzej Ehrenfeucht.[4] The names Spoiler and Duplicator are due to Joel Spencer.[5] Other usual names are Eloise [sic] and Abelard (and often denoted by ${\displaystyle \exists }$ and ${\displaystyle \forall }$) after Heloise and Abelard, a naming scheme introduced by Wilfrid Hodges in his book Model Theory, or alternatively Eve and Adam.

Chapter 1 of Poizat's model theory text[6] contains an introduction to the Ehrenfeucht–Fraïssé game, and so do Chapters 6, 7, and 13 of Rosenstein's book on linear orders.[7] A simple example of the Ehrenfeucht–Fraïssé game is given in one of Ivars Peterson's MathTrek columns.[8]

Phokion Kolaitis' slides[9] and Neil Immerman's book chapter[10] on Ehrenfeucht–Fraïssé games discuss applications in computer science, the methodology for proving inexpressibility results, and several simple inexpressibility proofs using this methodology.

Ehrenfeucht–Fraïssé games are the basis for the operation of derivative on modeloids. Modeloids are certain equivalence relations and the derivative provides for a generalization of standard model theory.

• Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007). Finite model theory and its applications. Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer-Verlag. ISBN 978-3-540-00428-8. Zbl 1133.03001.

• Six Lectures on Ehrenfeucht-Fraïssé games at MATH EXPLORERS' CLUB, Cornell Department of Mathematics.
• Modeloids I, Miroslav Benda, Transactions of the American Mathematical Society, Vol. 250 (Jun 1979), pp. 47 – 90 (44 pages)