Random sparse bit strings at the threshold of adjacency

Joel H. Spencer, Katherine St. John

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationSTACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings
Pages94-104
Number of pages11
Volume1373 LNCS
DOIs
StatePublished - 1998
Event15th Annual Symposium on Theoretical Aspects of Computer Science, STACS 98 - Paris, France
Duration: Feb 25 1998Feb 27 1998

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1373 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other15th Annual Symposium on Theoretical Aspects of Computer Science, STACS 98
CountryFrance
CityParis
Period2/25/982/27/98

Fingerprint

Adjacency
Strings
Limiting
First-order
Infinite sum
Finite Models
Probability function
Differentiable
Polynomial
Methodology
Polynomials

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Spencer, J. H., & St. John, K. (1998). Random sparse bit strings at the threshold of adjacency. In 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.

STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Vol. 1373 LNCS 1998. p. 94-104 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1373 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Spencer, JH & St. John, K 1998, Random sparse bit strings at the threshold of adjacency. in 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
Spencer JH, St. John K. Random sparse bit strings at the threshold of adjacency. In STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Vol. 1373 LNCS. 1998. p. 94-104. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/BFb0028552
Spencer, Joel H. ; St. John, Katherine. / Random sparse bit strings at the threshold of adjacency. STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Vol. 1373 LNCS 1998. pp. 94-104 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{af0365e33c424f398e07a35f40aacda4,
title = "Random sparse bit strings at the threshold of adjacency",
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.",
author = "Spencer, {Joel H.} and {St. John}, Katherine",
year = "1998",
doi = "10.1007/BFb0028552",
language = "English (US)",
isbn = "3540642307",
volume = "1373 LNCS",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "94--104",
booktitle = "STACS 98 - 15th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings",

}

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

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 -