Bounded degree and planar spectra
Seminar Room 1, Newton Institute
There are many problems in finite model theory about which we know a lot in the unrestricted classes, but which are still not thoroughly researched in the case where we restrict the class of considered models (for example, in terms of properties of their Gaifman graphs). In this talk we consider the problem of spectra of formulae. A set of integers S is a spectrum of phi if n \in S iff phi has a model of size n. It is well known that S is a spectrum of some formula iff the problem of deciding whether n \in S is in NP when n is given in unary (equivalently, in NE when n is given in binary).
Restricting the class of models we can get, for example, bounded degree spectra (S is a bounded degree spectrum of phi iff phi has a bounded degree model of size n), weak planar spectra (S is a bounded degree spectrum of phi iff phi has a planar model of size n), and forced planar spectra (S is a spectrum of phi which admits only planar models).
We provide the complexity theoretic characterizations for these cases, similar to the one above. In case of bounded degree spectra, there is a very small
(polylogarythmic) gap between our lower and upper bound. In case of weak planar spectra the gap is polynomial. We also provide a weaker complexity theoretic characterization of forced planar spectra.