### Abstract

It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in such a formula can serve as a measure for the "first order complexity" of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log* n, the inverse of the TOWER-function. The general picture, however, is still a mystery.

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

Pages (from-to) | 119-145 |

Number of pages | 27 |

Journal | Random Structures and Algorithms |

Volume | 26 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2005 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*26*(1-2), 119-145. https://doi.org/10.1002/rsa.20049

**How complex are random graphs in first order logic?** / Kim, Jeong Han; Pikhurko, Oleg; Spencer, Joel H.; Verbitsky, Oleg.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 26, no. 1-2, pp. 119-145. https://doi.org/10.1002/rsa.20049

}

TY - JOUR

T1 - How complex are random graphs in first order logic?

AU - Kim, Jeong Han

AU - Pikhurko, Oleg

AU - Spencer, Joel H.

AU - Verbitsky, Oleg

PY - 2005/1

Y1 - 2005/1

N2 - It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in such a formula can serve as a measure for the "first order complexity" of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log* n, the inverse of the TOWER-function. The general picture, however, is still a mystery.

AB - It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in such a formula can serve as a measure for the "first order complexity" of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log* n, the inverse of the TOWER-function. The general picture, however, is still a mystery.

UR - http://www.scopus.com/inward/record.url?scp=22144460478&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22144460478&partnerID=8YFLogxK

U2 - 10.1002/rsa.20049

DO - 10.1002/rsa.20049

M3 - Article

AN - SCOPUS:22144460478

VL - 26

SP - 119

EP - 145

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1-2

ER -