Abstract
The spectrum of a first-order formula is the set of numbers α 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 −α 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.
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Original Russian Text © M.E. Zhukovskii, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 4, pp. 95–108.
The research was carried out at the expense of the Russian Science Foundation, project no. 5-11-10021.
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Zhukovskii, M.E. Quantifier Alternation in First-Order Formulas with Infinite Spectra. Probl Inf Transm 53, 391–403 (2017). https://doi.org/10.1134/S003294601704007X
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DOI: https://doi.org/10.1134/S003294601704007X