Investigation of binary spectra by explicit polynomial transformations of graphs

Abstract.We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle.Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.

show abstract

“…(ii) The FOL-sentence axiomatizing fields of characteristic p has an f-spectrum with /?"+'. In [54] it is shown: (\ufi\uf\,...,\uf\).…”

Section: (I) Let Sq{*) Be With Relation Symbols X ~ {U S} U Unary Amentioning

confidence: 99%

On spectra of sentences of monadic second order logic with counting

Fischer

Makowsky

2004

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…Explicit transformations on structures have been employed more recently to prove some new closure results. In particular, More proves that F 2 is closed under every polynomial with rational coefficients that is asympotically greater than the identity [7]. Given this result, it is natural to ask what happens if we look at subdiagonal functions.…”

Section: B  Mmentioning

confidence: 99%

“…In this paper, we have studied general closure properties for SPEC using modeltheoretic techniques. The use of model-theoretic techniques in the study of spectra has been explicitly advocated in [5] and [7]. Unfortunately, the transformations used in those papers appear to be somewhat limited in their potential.…”

Section: Cmentioning

confidence: 99%

Limiting Cases for Spectrum Closure Results

Hunter

2004

AJL

View full text Add to dashboard Cite

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.

show abstract

First-order spectra with one binary predicate

Durand¹,

Ranaivoson²

1996

Theoretical Computer Science

View full text Add to dashboard Cite

Investigation of binary spectra by explicit polynomial transformations of graphs

Cited by 4 publications

References 8 publications

On spectra of sentences of monadic second order logic with counting

On spectra of sentences of monadic second order logic with counting

Limiting Cases for Spectrum Closure Results

First-order spectra with one binary predicate

Contact Info

Product

Resources

About