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On spectra of sentences of monadic second order logic with counting

Published online by Cambridge University Press:  12 March 2014

E. Fischer  and
J. A. Makowsky
E. Fischer
Affiliation:
Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel, E-mail: eldar@cs.technion.ac.il
J. A. Makowsky
Affiliation:
Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel, E-mail: janos@cs.technion.ac.il
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004 , pp. 617 - 640
Copyright
Copyright © Association for Symbolic Logic 2004

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