### 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-∈. 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(log
_{2} n). For random bit strings and circular sequences with isolated ones (n
^{-1} ≪ p ≪ n
^{-1/2}), T
_{∈}(n) = O(min(log
_{2} np,-log
_{2} np
^{2})). 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-14 |

Number of pages | 14 |

Journal | Electronic Journal of Combinatorics |

Volume | 8 |

Issue number | 2 R |

State | Published - 2001 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Electronic Journal of Combinatorics*,

*8*(2 R), 1-14.

**The tenacity of zero-one laws.** / Spencer, Joel H.; St. John, Katherine.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 8, no. 2 R, pp. 1-14.

}

TY - JOUR

T1 - The tenacity of zero-one laws

AU - Spencer, Joel H.

AU - St. John, Katherine

PY - 2001

Y1 - 2001

N2 - 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-∈. 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(log 2 n). For random bit strings and circular sequences with isolated ones (n -1 ≪ p ≪ n -1/2), T ∈(n) = O(min(log 2 np,-log 2 np 2)). 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.

AB - 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-∈. 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(log 2 n). For random bit strings and circular sequences with isolated ones (n -1 ≪ p ≪ n -1/2), T ∈(n) = O(min(log 2 np,-log 2 np 2)). 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.

UR - http://www.scopus.com/inward/record.url?scp=4043067728&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4043067728&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4043067728

VL - 8

SP - 1

EP - 14

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2 R

ER -