Topological complexity of locally finite omega -languages

Locally finite omega languages were introduced by Ressayre in [Formal Languages defined by the Underlying Structure of their Words, Journal of Symbolic Logic, 53, number 4, December 1988, p. 1009-1026]. These languages are defined by local sentences and extend ω-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ω-languages are analytic sets, the class LOCω of locally finite ω-languages meets all finite levels of the Borel hierarchy and there exist some locally finite ω-languages which are Borel sets of infinite rank or even analytic but non-Borel sets. This gives partial answers to questions of Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992] and of Duparc, Finkel, and Ressayre [Computer Science and the Fine Structure of Borel Sets, Theoretical Computer Science, Volume 257 (1-2), 2001, p.85-105].