Ehrenfeucht-Fraïssé Games on Random Structures

• Authors: Benjamin Rossman
• Venue: WoLLIC '09 Proceedings of the 16th International Workshop on Logic, Language, Information and Computation
• Year: 2009

Certain results in circuit complexity (e.g., the theorem that AC0 functions have low average sensitivity) [5, 17] imply the existence of winning strategies in Ehrenfeucht-Fraïssé games on pairs of random structures (e.g., ordered random graphs G = G (n ,1/2) and G + = G *** {random edge }). Standard probabilistic methods in circuit complexity (e.g., the Switching Lemma [11] or Razborov-Smolensky Method [19, 21]), however, give no information about how a winning strategy might look. In this paper, we attempt to identify specific winning strategies in these games (as explicitly as possible). For random structures G and G + , we prove that the composition of minimal strategies in r -round Ehrenfeucht-Fraïssé games $\Game_r(G,G)$ and $\Game_r(G^{{+}},G^{{+}})$ is almost surely a winning strategy in the game $\Game_r(G,G^{{+}})$. We also examine a result of [20] that ordered random graphs H = G (n ,p ) and H + = H *** {random k -clique} with p (n ) *** n *** 2/(k *** 1) (below the k -clique threshold) are almost surely indistinguishable by $\lfloor k/4 \rfloor$-variable first-order sentences of any fixed quantifier-rank r . We describe a winning strategy in the corresponding r -round $\lfloor k/4 \rfloor$-pebble game using a technique that combines strategies from several auxiliary games.