### 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 |

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) |
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Volume | 1373 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### 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 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

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