The spectrum of a first-order sentence is the set of the cardinalities of its finite models. In this paper, we consider the spectra of sentences over binary relations that use at least three variables. We show that for every such sentence $\Phi$, there is a sentence $\Phi'$ that uses the same number of variables, but only one symmetric binary relation, such that its spectrum is linearly proportional to the spectrum of $\Phi$. Moreover, the models of $\Phi'$ are all bipartite graphs. As a corollary, we obtain that to settle Asser's conjecture, i.e., whether the class of spectra is closed under complement, it is sufficient to consider only sentences using only three variables whose models are restricted to undirected bipartite graphs.

Source : oai:arXiv.org:1706.08691

DOI : 10.23638/LMCS-14(2:4)2018

Volume: Volume 14, Issue 2

Published on: April 25, 2018

Submitted on: June 30, 2017

Keywords: Computer Science - Logic in Computer Science,F.4.1,F.1.3

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