Ehrenfeucht-Fraïssé Games

Synonyms

EF-games; Ehrenfeucht games

Definition

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 𝒜 and ℬ 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 𝒜 or an element b_i in the universe of ℬ. Afterwards, the duplicator chooses an element in the other structure, i.e., she chooses an element b_i in the universe of ℬ if the spoiler’s move was in 𝒜, respectively, an element a_i in the universe of 𝒜 if the spoiler’s move was in ℬ.

After rrounds, the...