On the Spectra of Universal Relational Sentences

Abstract

Ramsey's theorem in both the finite and infinite forms are very well known, but it is less well known that in his original paper (Ramsey, 1930), these fundamental results were introduced as lemmas for establishing various decidability results. Implicit or explicit in his paper are the following two decidability results: (1) it is decidable whether a universal relational sentence (defined below) has an infinite model and (2) there is an algorithm for determining the spectrum (defined below) of a universal relational sentence. As a consequence of (2), we have: (3) it is decidable whether two universal relational sentences have the same spectrum. Theorems 1-3 below are essentially from (Ramsey, 1930) and establish an algorithm for (3) whose run time is bounded by a linear stack of 2s. The purpose of this note is to establish that there is no elementary recursive algorithm for (3); i.e., none whose run time is bounded by a fixed finite stack of 2s. It can be seen from (Lewis, 1978) and Theorem 1 below (a if and only if c) that (1) above is nondeterministic exponential time complete with respect to polynomial time reductions. A universal relational sentence is a sentence of predicate calculus with identity and no constant or function symbols (thus only relation symbols), which is the universal closure of a quantifier free formula. Let R(k,m,n) be the least q such that q~(n)~. (See Graham, Rothschild, and Spencer, 1980, p. 7 for notation.) The spectrum of a sentence is the set of all cardinalities of finite models of the sentence. Let M be a relational structure. An atomic SOI (sequence of indiscernibtes) for M is a sequence bn, finite or infinite, such that for all atomic for-

DOI: 10.1016/S0019-9958(84)80034-0

Cite this paper

@article{Friedman1984OnTS, title={On the Spectra of Universal Relational Sentences}, author={Harvey M. Friedman}, journal={Information and Control}, year={1984}, volume={62}, pages={205-209} }