Finite Model Theory/FO EFG

Logic and Games Connection between Games and logical Quantifiers... Simple Games S chooses first all n from one/both side(s), then D chooses all one match as in 1. for every i = 1 ... n where the i'th choice must include the i-1'th. Definition Ehrenfeucht-Fraisse-Game (EFG) Let and be finite relational structures, with the same set of relation symbols, and n be a fixed natural number. A game is said to be an Ehrenfeucht-Fraisse-game iff it is constituted the following way 2 Players: Spoiler S and Duplicator D Game tree: Root: a pair of empty sequences Branch: Move of S: S chooses exactly one element of either A or B that has not been chosen already (latter is not relevant in the first move) Move of D: D chooses exactly one element of the remaining set that has not been chosen already, i.e. if S has chosen from A D has to choose from B and if S chose one from B D has to choose from A. Terminator: The game is finished iff α and β are of size n. Turn: Moves are carried out alternatingly, S is to move first. Payoff: D wins iff the final position defines a partial isomorphism by mapping the i-th elements of α and β, for all i. Else S wins Information: the players are assumed to have perfect information