Limiting Cases for Spectrum Closure Results

Abstract

The spectrum of a first-order sentence is the set of cardinalities of its finite models. Given a spectrum S and a function f, it is not always clear whether or not the image of S under f is also a spectrum. In this paper, we consider questions of this form for functions that increase very quickly and for functions that increase very slowly. Roughly speaking, we prove that the class of all spectra is closed under functions that increase arbitrarily quickly, but it is not closed under some natural slowly increasing functions.

Fifty years of the spectrum problem: survey and new results

“…Proof of (ii). In [73], a result similar to (ii) is proved about the number of binary predicates instead of the number of unary functions. It is not hard to see that his result extends to the case of (ii).…”

“…Class C is closed under f if for any spectrum S in C, f (S) is in C. In the spirit of Theorems 6.11 and 6.13 and Proposition 6.12 one can relate the collapse of hierarchies to the possible closure of class of spectra under some function. In [73], such problems are studied and several related results are given. (ii) The unary hierarchy collapses to f-spec 2 1 if and only if f-spec 2 1 is closed under function f : n → ⌈ n 2 ⌉.…”

“…(i) (Hunter,[73]) The arity hierarchy collapses to spec 1 2 if and only if spec 1 2 is closed under function f : n → ⌈ √ n⌉.…”