A New Ehrenfeucht-Fra\"iss\'e Game for Dependence Logic

We define a new Ehrenfeucht-Fra\"iss\'e game for dependence logic. The previously known rendition of such a game was based on moves that are teams. Since teams can be massive, making team moves may be quite complicated. To remedy this, our new Ehrenfeucht-Fra\"iss\'e game for dependence logic has only moves that consist of single elements, as in the classical Ehrenfeucht-Fra\"iss\'e game of first order logic. A new feature of the game is that a player can declare that their move is made on the basis of certain previous moves only and thereby in a sense independent of other moves. We show that our game characterizes elementary equivalence in dependence logic.