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The Ehrenfeucht-Fraïssé game (EF-game, for short) is played by two players, usually called the spoiler and the duplicator (in the literature, the two players are sometimes also called Samson and Delilah or, simply, player I and player II). The board of the game consists of two structures and of the same vocabulary. The spoiler’s intention is to show a difference between the two structures, while the duplicator tries to make them look alike.
The rules of the classical EF-game are as follows: The players play a certain number r of rounds. Each round i consists of two steps. First, the spoiler chooses either an element a i in the universe of or an element b i in the universe of . Afterwards, the duplicator chooses an element in the other structure, i.e., she chooses an element b i in the universe of if the spoiler’s move was in , respectively, an element a i in the universe of if the spoiler’s move was in .
After rrounds, the game finishes...
Recommended Reading
Ehrenfeucht A. An application of games to the completeness problem for formalized theories. Fundam Math. 1961;49:129–41.
Fagin R. Easier ways to win logical games. In: Immerman N, Kolaitis PG, editors. Descriptive complexity and finite models, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 31. Providence: American Mathematical Society; 1997. p. 1–32.
Fraïssé R. Sur quelques classifications des systèmes de relations. Université d’Alger, Publications Scientifiques; 1954, SérieA(1):35–182.
Libkin L. Elements of finite model theory. Berlin: Springer; 2004.
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Schweikardt, N. (2016). Ehrenfeucht-Fraïssé Games. In: Liu, L., Özsu, M. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4899-7993-3_1271-2
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DOI: https://doi.org/10.1007/978-1-4899-7993-3_1271-2
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