How complex are random graphs in first order logic?

  • Authors:

  • Jeong Han Kim;Oleg Pikhurko;Joel H. Spencer;Oleg Verbitsky

  • Affiliations:

  • Microsoft Research, One Microsoft Way, Redmond, Washington 98052;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890;Courant Institute, New York University, New York, New York 10012;Department of Mechanics and Mathematics, Kyiv University, Ukraine

  • Venue:
  • Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
  • Year:
  • 2005

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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 &THgr;(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. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005