AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

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 col...

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Bibliographic Details
Published in:The Journal of symbolic logic 2018-09, Vol.83 (3), p.1147-1189
Main Authors: CARTON, OLIVIER, COLCOMBET, THOMAS, PUPPIS, GABRIELE
Format: Article
Language:English
Subjects:
Computer Science
Formal Languages and Automata Theory
Logic
Logic in Computer Science
Mathematics
Philosophy
Realism
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Summary: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.
ISSN:0022-4812
1943-5886
DOI:10.1017/jsl.2018.7