Ehrenfeucht-Fraïssé Games on Random Structures

Certain results in circuit complexity (e.g., the theorem that AC 0 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 and is almost surely a winning strategy in the game . We also examine a result of [20] that ordered random graphs H∈=∈G(n,p) and H ∈+∈∈=∈H∈ ∪∈{random k-clique} with p(n)∈ ≪-∈2/(k∈-∈1) (below the k-clique threshold) are almost surely indistinguishable by -variable first-order sentences of any fixed quantifier-rank r. We describe a winning strategy in the corresponding r-round -pebble game using a technique that combines strategies from several auxiliary games. © 2009 Springer Berlin Heidelberg.

Duke Scholars

Author Benjamin Rossman Computer Science