A New Ehrenfeucht-Fraïssé Game for Dependence Logic

2026-05-30

AI Summary

This work addresses the computational and operational complexity of traditional Ehrenfeucht–Fraïssé (EF) games when characterizing elementary equivalence in dependence logic, where moves must be made by entire “teams” rather than individual elements. To overcome this limitation, the paper introduces a novel EF game that simplifies the unit of movement to single elements and incorporates a mechanism of local independence declarations based on partial histories of prior moves. This framework preserves the full expressive power of dependence logic while substantially reducing operational complexity. Notably, it constitutes the first EF game for dependence logic that operates solely with single-element moves, and the authors rigorously establish its semantic equivalence with dependence logic, demonstrating its adequacy in precisely capturing elementary equivalence.

Abstract

We define a new Ehrenfeucht-Fraïssé 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-Fraïssé game for dependence logic has only moves that consist of single elements, as in the classical Ehrenfeucht-Fraïssé 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.

Problem

• Dependence Logic
• Ehrenfeucht-Fraïssé Game
• Elementary Equivalence
• Team Semantics
• Game Complexity

Innovation

• Dependence Logic
• Ehrenfeucht-Fraïssé Game
• Elementary Equivalence
• Team Semantics
• Independence