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AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

Published online by Cambridge University Press:  23 October 2018

OLIVIER CARTON ,
THOMAS COLCOMBET  and
GABRIELE PUPPIS
OLIVIER CARTON
Affiliation:
INSTITUT DE RECHERCHE EN INFORMATIQUE FONDAMENTALE (IRIF) UNIVERSITÉ PARIS DIDEROT PARIS, FRANCE E-mail: olivier.carton@irif.fr
THOMAS COLCOMBET
Affiliation:
CNRS / INSTITUT DE RECHERCHE EN INFORMATIQUE FONDAMENTALE (IRIF) UNIVERSITÉ PARIS DIDEROT PARIS, FRANCE E-mail: thomas.colcombet@irif.fr
GABRIELE PUPPIS
Affiliation:
CNRS / LABORATOIRE BORDELAIS DE RECHERCHE EN INFORMATIQUE (LABRI) UNIVERSITÉ BORDEAUX BORDEAUX, FRANCE E-mail: gabriele.puppis@labri.fr
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Abstract

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

Keywords

03-XXgeneralized wordsrecognizabilitymonoidslinear orderingsmondaic second-order logic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 1147 - 1189
Copyright
Copyright © The Association for Symbolic Logic 2018 

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