How complex are random graphs in first order logic?
Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
A zero-one law for a random subset
Random Structures & Algorithms
The Strange Logic of Random Graphs
The Strange Logic of Random Graphs
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We show that for random bit strings, U"p(n), with probability, p=12, the first-order quantifier depth D(U"p(n)) needed to distinguish non-isomorphic structures is @Q(lglgn), with high probability. Further, we show that, with high probability, for random ordered graphs, G"@?","p(n) with edge probabiltiy p=12, D(G"@?","p(n))=@Q(log^*n), contrasting with the results of random (non-ordered) graphs, G"p(n), by Kim et al. [J.H. Kim, O. Pikhurko, J. Spencer, O. Verbitsky, How complex are random graphs in first order logic? (2005), to appear in Random Structures and Algorithms] of D(G"p(n))=log"1"/"pn+O(lglgn).