The complexity of random ordered structures

We show that for random bit strings, U p ( n ) , with probability, p = 1 2 , the first order quantifier depth D ( U p ( n ) ) needed to distinguish non-isomorphic structures is Θ ( lg lg n ) , with high probability. Further, we show that, with high probability, for random ordered graphs, G ≤ , p ( n ) with edge probability p = 1 2 , D ( G ≤ , p ( n ) ) = Θ ( log ∗ n ) , contrasting with the results for random (non-ordered) graphs, G p ( n ) , given by Kim et al. [J.H. Kim, O. Pikhurko, J. Spencer, O. Verbitsky, How complex are random graphs in first order logic? Random Structures and Algorithms 26 (2005) 119–145] of D ( G p ( n ) ) = log 1 / p n + O ( lg lg n ) .