5 research outputs found
Bounded degree and planar spectra
The finite spectrum of a first-order sentence is the set of positive integers
that are the sizes of its models. The class of finite spectra is known to be
the same as the complexity class NE. We consider the spectra obtained by
limiting models to be either planar (in the graph-theoretic sense) or by
bounding the degree of elements. We show that the class of such spectra is
still surprisingly rich by establishing that significant fragments of NE are
included among them. At the same time, we establish non-trivial upper bounds
showing that not all sets in NE are obtained as planar or bounded-degree
spectra
Bounded degree and planar spectra
The finite spectrum of a first-order sentence is the set of positive integers
that are the sizes of its models. The class of finite spectra is known to be
the same as the complexity class NE. We consider the spectra obtained by
limiting models to be either planar (in the graph-theoretic sense) or by
bounding the degree of elements. We show that the class of such spectra is
still surprisingly rich by establishing that significant fragments of NE are
included among them. At the same time, we establish non-trivial upper bounds
showing that not all sets in NE are obtained as planar or bounded-degree
spectra