The Tenacity of Zero-One Laws

Abstract

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 Ehrenfeucht-Fraisse game, by bounding the numbers of moves Duplicator can play and win with probability $1-\epsilon$. We show that for random bit strings and random circular sequences of length $n$ generated with a low probability ($p\ll n^{-1}$), the number of moves, $T_{\epsilon}(n)$, is $\Theta(\log_2 n)$. For random bit strings and circular sequences with isolated ones ($n^{-1}\ll p \ll n^{-1/2}$), $T_{\epsilon}(n) = O(\min(\log_2 np, -\log_2 np^2))$. For $n^{-1/2}\ll p$ and $(1-p) \ll n^{-1/2}$, we show that $T_{\epsilon}(n) = O(\log^* n)$ 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.