Ehrenfeucht-Fraïssé Games on Random Structures

Abstract. 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 �r(G, G) and�r(G +,G +)isalmostsurely a winning strategy in the game �r(G, G +). We also examine a result of [20] that ordered random graphs H = G(n, p) andH + = H ∪{random k-clique} with p(n) ≪ n −2/(k−1) (below the k-clique threshold) are almost surely indistinguishable by ⌊k/4⌋-variable first-order sentences of any fixed quantifier-rank r. Wedescribeawinningstrategyinthecorresponding r-round ⌊k/4⌋-pebble game using a technique that combines strategies from several auxiliary games.