- Main idea
- Definition
- History
- Further reading
- References
- External links
In the mathematical discipline of model theory, the Ehrenfeucht–Fraïssé game (also called back-and-forth games)
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 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, do not work in finite models.
Ehrenfeucht–Fraïssé like games can also be defined for other logics, such as fixpoint logics[1] and pebble games for finite variable logics; extensions are powerful enough to characterise definability in existential second-order logic.
Main idea
The main idea behind the game is that we have two structures, and two players (defined below). One of the players wants to show that the two structures are different, whereas the other player wants to show that they are elementarily equivalent (satisfy the same first-order sentences). The game is played in turns and rounds; A round proceeds as follows: First the first player (Spoiler) chooses any element from one of the structures, and the other player chooses an element from the other structure. The other player's task is to always pick an element that is "similar" to the one that Spoiler chose. The second player (Duplicator) wins if there exists an isomorphism between the elements chosen in the two different structures.
The game lasts for a fixed number of steps (
) (an ordinal, but usually a finite number or 
).
Definition
Suppose that we are given two structures 
and 
, 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 
to be a game between two players, Spoiler and Duplicator,
played as follows:
- The first player, Spoiler, picks either a member
of or a member 
of . - If Spoiler picked a member of , Duplicator picks a member of ; otherwise, Duplicator picks a member of .
- Spoiler picks either a member
of or a member 
of . - Duplicator picks an element or in the model from which Spoiler did not pick.
- Spoiler and Duplicator continue to pick members of and for
more steps. - At the conclusion of the game, we have chosen distinct elements
of and 
of . We therefore have two structures on the set 
, one induced from via the map sending 
to 
, and the other induced from via the map sending to 
. Duplicator wins if these structures are the same; Spoiler wins if they are not.
For each n we define a relation 
if Duplicator wins the n-move game . 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 
.
It is easy to prove that if Duplicator wins this game for all n, that is, , then and are elementarily equivalent. 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–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 
and 
) after Heloise and Abelard, a naming scheme introduced by Wilfrid Hodges in his book Model Theory, or alternatively Eve and Adam.
Further reading
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 theorem 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. [https://www.jstor.org/stable/1998979?seq=1#metadata_info_tab_contents Modeloids] are certain equivalence relations and the derivative provides for a generalization of standard model theory.
References
2. ^Sur une nouvelle classification des systèmes de relations, Roland Fraïssé, Comptes Rendus 230 (1950), 1022–1024.
3. ^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.
4. ^An application of games to the completeness problem for formalized theories, A. Ehrenfeucht, Fundamenta Mathematicae 49 (1961), 129–141.
5. ^Stanford Encyclopedia of Philosophy, entry on Logic and Games.
6. ^A Course in Model Theory, Bruno Poizat, tr. Moses Klein, New York: Springer-Verlag, 2000.
7. ^Linear Orderings, Joseph G. Rosenstein, New York: Academic Press, 1982.
8. ^[https://www.sciencenews.org/article/subtle-logic-winning-game Example of the Ehrenfeucht-Fraïssé game.]
9. ^Course on combinatorial games in finite model theory by Phokion Kolaitis (.ps file)
10. ^{{cite book|author=Neil Immerman|title=Descriptive complexity|url=https://books.google.com/books?id=kWSZ0OWnupkC&pg=PA91|year=1999|publisher=Springer|isbn=978-0-387-98600-5}}, chapter 6
- {{cite book | last1=Grädel | first1=Erich | last2=Kolaitis | first2=Phokion G. | last3=Libkin | first3=Leonid | last4=Maarten | first4=Marx | last5=Spencer | first5=Joel | author5-link=Joel Spencer | last6=Vardi | first6=Moshe Y. | author6-link=Moshe Y. Vardi | last7=Venema | first7=Yde | last8=Weinstein | first8=Scott | title=Finite model theory and its applications | zbl=1133.03001 | series=Texts in Theoretical Computer Science. An EATCS Series | location=Berlin | publisher=Springer-Verlag | isbn=978-3-540-00428-8 | year=2007 }}
External links
- Six Lectures Ehrenfeucht-Fraïssé games at MATH EXPLORERS' CLUB, Cornell Department of Mathematics.
- [https://www.jstor.org/stable/1998979?seq=1#metadata_info_tab_contents Modeloids] I, Miroslav Benda, Transactions of the American Mathematical Society, Vol. 250 (Jun 1979), pp. 47 – 90 (44 pages)