The Tenacity of Zero-One Laws

The Ehrenfeucht-Fraisse game is a two-person game of perfect information which is connected to the Zero-One Laws of first order logic. We give bounds for roughly how quickly the Zero-One Laws converge for random bit strings and random circular bit sequences. We measure the tenaciousness of the second player (“Duplicator”) in playing the EhrenfeuchtFraisse game, by bounding the numbers of moves Duplicator can play and win with probability 1− . We show that for random bit strings and random circular sequences of length n generated with a low probability (p n−1), the number of moves, T (n), is Θ(log2 n). For random bit strings and circular sequences with isolated ones (n−1 p n−1/2), T (n) = O(min(log2 np,− log2 np)). For n−1/2 p and (1 − p) n−1/2, we show that T (n) = O(log ∗ n) for random circular sequences, where log∗ n has the usual definition– the least number of times you iteratively apply the logarithm to get a value less than one.