### Abstract

We give a complete characterization for the limit probabilities of first order sentences over sparse random bit strings at the threshold of adjacency. For strings of length n, we let the probability that a bit is "on" be c/√n, for a real positive number c. For every first order sentence φ, we show that the limit probability function: f
_{φ}(c) = lim
_{n→∞} Pr[U
_{n}, c/√n has the property φ] (where U
_{n}, c/√n is the random bit string of length n) is infinitely differentiable. Our methodology for showing this is in itself interesting. We begin with finite models, go to the infinite (via the almost sure theories) and then characterize f
_{φ}(c) as an infinite sum of products of polynomials and exponentials. We further show that if a sentence φ has limiting probability 1 for some c, then φ has limiting probability identically 1 for every c. This gives the surprising result that the almost sure theories are identical for every c.

Original language | English (US) |
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Title of host publication | STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings |

Pages | 94-104 |

Number of pages | 11 |

Volume | 1373 LNCS |

DOIs | |

State | Published - 1998 |

Event | 15th Annual Symposium on Theoretical Aspects of Computer Science, STACS 98 - Paris, France Duration: Feb 25 1998 → Feb 27 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 1373 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 15th Annual Symposium on Theoretical Aspects of Computer Science, STACS 98 |
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Country | France |

City | Paris |

Period | 2/25/98 → 2/27/98 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings*(Vol. 1373 LNCS, pp. 94-104). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1373 LNCS). https://doi.org/10.1007/BFb0028552

**Random sparse bit strings at the threshold of adjacency.** / Spencer, Joel H.; St. John, Katherine.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings.*vol. 1373 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1373 LNCS, pp. 94-104, 15th Annual Symposium on Theoretical Aspects of Computer Science, STACS 98, Paris, France, 2/25/98. https://doi.org/10.1007/BFb0028552

}

TY - GEN

T1 - Random sparse bit strings at the threshold of adjacency

AU - Spencer, Joel H.

AU - St. John, Katherine

PY - 1998

Y1 - 1998

N2 - We give a complete characterization for the limit probabilities of first order sentences over sparse random bit strings at the threshold of adjacency. For strings of length n, we let the probability that a bit is "on" be c/√n, for a real positive number c. For every first order sentence φ, we show that the limit probability function: f φ(c) = lim n→∞ Pr[U n, c/√n has the property φ] (where U n, c/√n is the random bit string of length n) is infinitely differentiable. Our methodology for showing this is in itself interesting. We begin with finite models, go to the infinite (via the almost sure theories) and then characterize f φ(c) as an infinite sum of products of polynomials and exponentials. We further show that if a sentence φ has limiting probability 1 for some c, then φ has limiting probability identically 1 for every c. This gives the surprising result that the almost sure theories are identical for every c.

AB - We give a complete characterization for the limit probabilities of first order sentences over sparse random bit strings at the threshold of adjacency. For strings of length n, we let the probability that a bit is "on" be c/√n, for a real positive number c. For every first order sentence φ, we show that the limit probability function: f φ(c) = lim n→∞ Pr[U n, c/√n has the property φ] (where U n, c/√n is the random bit string of length n) is infinitely differentiable. Our methodology for showing this is in itself interesting. We begin with finite models, go to the infinite (via the almost sure theories) and then characterize f φ(c) as an infinite sum of products of polynomials and exponentials. We further show that if a sentence φ has limiting probability 1 for some c, then φ has limiting probability identically 1 for every c. This gives the surprising result that the almost sure theories are identical for every c.

UR - http://www.scopus.com/inward/record.url?scp=78649823425&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649823425&partnerID=8YFLogxK

U2 - 10.1007/BFb0028552

DO - 10.1007/BFb0028552

M3 - Conference contribution

AN - SCOPUS:78649823425

SN - 3540642307

SN - 9783540642305

VL - 1373 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 94

EP - 104

BT - STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings

ER -