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 Probl. Peredachi Inf., 2017, Volume 53, Issue 4, Pages 95–108 (Mi ppi2255)

Large Systems

Quantifier alternation in first-order formulas with infinite spectra

M. E. Zhukovskii

Derzhavin Tambov State University, Tambov, Russia

Abstract: The spectrum of a first-order formula is the set of numbers $\alpha$ such that for a random graph in a binomial model where the edge probability is a power function of the number of graph vertices with exponent $-\alpha$ the truth probability of this formula does not tend to either zero or one. In 1990 J. Spenser proved that there exists a first-order formula with an infinite spectrum. We have proved that the minimum quantifier depth of a first-order formula with an infinite spectrum is either 4 or 5. In the present paper we find a wide class of first-order formulas of depth 4 with finite spectra and also prove that the minimum quantifier alternation number for a first-order formula with an infinite spectrum is 3.

 Funding Agency Grant Number Russian Science Foundation 15-11-10021 The research was carried out at the expense of the Russian Science Foundation, project no. 15-11-10021.

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English version:
Problems of Information Transmission, 2017, 53:4, 391–403

Bibliographic databases:

UDC: 621.391.1+519.1
Revised: 15.04.2017

Citation: M. E. Zhukovskii, “Quantifier alternation in first-order formulas with infinite spectra”, Probl. Peredachi Inf., 53:4 (2017), 95–108; Problems Inform. Transmission, 53:4 (2017), 391–403

Citation in format AMSBIB
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