Radix cross-sections for length morphisms

Authors:
Sylvain Lombardy;Jacques Sakarovitch
Affiliations:
LIGM, Université Paris-Est;LTCI, CNRS / ENST, Paris
Venue:
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Year:
2010

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Abstract

We prove that the radix cross-section of a rational set for a length morphism, and more generally for a rational function from a free monoid into ℕ, is rational. This property no longer holds if the image of the function is a subset of a free monoid with two or more generators. The proof is based on several results on finite automata, such as the lexicographic selection of synchronous relations and the iterative decomposition of unary rational series with coefficients in the tropical semiring. It also makes use of a structural construction on weighted transducers that we call the length difference unfolding.