# Finite Model Theory/FO EFG

## Logic and Games

Connection between Games and logical Quantifiers...

## Simple Games

1. S chooses first all n from one/both side(s), then D chooses all
2. 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 ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$ 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