Ehrenfeucht-Fraïssé Games on Random Structures

Abstract

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.