Weak Ehrenfeucht-Fraïssé games

Abstract

In this paper we define a game which is played between two players I \textbf {I} and II \textbf {II} and two mathematical structures A \mathcal {A} and B \mathcal {B} . The players choose elements from both structures in α \alpha moves, and at the end of the game player II \textbf {II} wins if the chosen structures are isomorphic. Thus the difference between this and the ordinary Ehrenfeucht-Fraïssé game is that the isomorphism can be arbitrary, whereas in the ordinary EF-game it is determined by the moves of the players. We investigate determinacy of the weak EF-game for different α \alpha (the length of the game) and its relation to the ordinary EF-game.