Abstract
Let Sp(d∀) denote the class of spectra of prenex first-order sentences (with equality) withd universal quantifiers. Let NRAM(T(n)) denote the class of sets (of positive integers) accepted by Nondeterministic Random Access Machines, NRAM, (with successor as the only arithmetical operation) in timeO(T(n)) wheren is the input integer. We prove Sp(d∀) = NRAM(n d) ford ≥ 2. Moreover, each spectrum of Sp(d∀) is the spectrum of ad-universal-quantifier sentence with relation and function symbols of arityd only. Similar results hold for generalized spectra and alternating spectra.
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Grandjean, E. Universal quantifiers and time complexity of random access machines. Math. Systems Theory 18, 171–187 (1985). https://doi.org/10.1007/BF01699468
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DOI: https://doi.org/10.1007/BF01699468