Ehrenfeucht–Fraïssé game

In the mathematical discipline of model theory, the Ehrenfeucht-Fraïssé game is a technique for determining whether two structures are elementarily equivalent. The main application of Ehrenfeucht-Fraïssé games is in proving the inexpressibility of certain properties in first-order logic. Indeed, Ehrenfeucht-Fraïssé games provide a complete methodology for proving inexpressibility results for first-order logic. In this role, these games are of particular importance in finite model theory and its applications in computer science (specifically Computer Aided Verification and database theory), since Ehrenfeucht-Fraïssé games are one of the such few techniques from model theory that remain valid in the context of finite models. (Other widely used techniques for proving inexpressibility results, such as the compactness theorem, are not valid in the finite.)

Ehrenfeucht-Fraïssé like games can also be defined for other logics, e.g. pebble games for finite variable logics and fixpoint logics.

Definition

Suppose that we are given two structures $mathfrak\{A\}$ and $mathfrak\{B\}$, 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 $G\_n(mathfrak\{A\},mathfrak\{B\})$ to be a game between two players, Spoiler and Duplicator, played as follows:
# The first player, Spoiler, picks either a member $a\_1$ of $mathfrak\{A\}$ or a member $b\_1$ of $mathfrak\{B\}$.
# If Spoiler picked a member of $mathfrak\{A\}$, Duplicator picks a member $b\_1$ of $mathfrak\{B\}$; otherwise, Spoiler picked a member of $mathfrak\{B\}$, and Duplicator picks a member $a\_1$ of $mathfrak\{A\}$.
# Spoiler picks either a member $a\_2\; e\; a\_1$ of $mathfrak\{A\}$ or a member $b\_2\; e\; b\_1$ of $mathfrak\{B\}$.
# Duplicator picks whichever of $a\_2$ or $b\_2$ Spoiler did not pick.
# Spoiler and Duplicator continue to pick members of $mathfrak\{A\}$ and $mathfrak\{B\}$ for $n-2$ more steps.
# At the conclusion of the game, we have chosen distinct elements $a\_1,\; dots,\; a\_n$ of $mathfrak\{A\}$ and $b\_1,\; dots,\; b\_n$ of $mathfrak\{B\}$. We therefore have two structures on the set $\{1,\; dots,n\}$, one induced from $mathfrak\{A\}$ via the map sending $i$ to $a\_i$, and the other induced from $mathfrak\{B\}$ via the map sending $i$ to $b\_i$. Duplicator wins if these structures are the same; Spoiler wins if they are not.

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

History

The back-and-forth method used in the Ehrenfeucht-Fraïsségame to verify elementary equivalence was given by Roland Fraïssé in his thesis ["Sur une nouvelle classification des systèmes de relations", Roland Fraïssé, "Comptes Rendus" 230 (1950), 1022–1024.] , ["Sur quelques classifications des systèmes de relations", Roland Fraïssé, thesis, Paris, 1953;published in "Publications Scientifiques de l'Université d'Alger", series A 1 (1954), 35–182.] ;it was formulated as a game by Andrzej Ehrenfeucht. [An application of gamesto the completeness problem for formalized theories, A. Ehrenfeucht, "Fundamenta Mathematicae" 49 (1961), 129–141.] The names Spoiler and Duplicator are due to Joel Spencer. [ [http://plato.stanford.edu/entries/logic-games/ Stanford Encyclopedia of Philosophy, entry on Logic and Games.] ]

Further reading

Chapter 1 of Poizat's model theory text ["A Course in Model Theory", Bruno Poizat, tr. Moses Klein, New York: Springer-Verlag, 2000.] contains an introduction to the Ehrenfeucht-Fraïssé game, and so do Chapters 6, 7, and 13 of Rosenstein's book on linear orders. ["Linear Orderings", Joseph G. Rosenstein, New York: Academic Press, 1982.] A simple example of the Ehrenfeucht-Fraïssé game is given in [ [http://www.maa.org/mathland/mathtrek_3_19_01.html Example of the Ehrenfeucht-Fraïssé game.] ] .

Phokion Kolaitis' slides on Ehrenfeucht-Fraïssé games discuss applications in computer science, the methodology theorem for proving inexpressibility results, and several simple inexpressibility proofs using this methodology [ [http://www.cs.ucsc.edu/~kolaitis/talks/essllif.ps Course on combinatorial games in finite model theory by Phokion Kolaitis (.ps file)] ] .

References