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Automata, Borel functions and real numbers in Pisot base

Published online by Cambridge University Press:  24 April 2007

Benoit Cagnard
Affiliation:
UMR CNRS 6134, université de Corse, BP 52, 20250 CORTE, France; cagnard@univ-corse.fr; simonnet@univ-corse.fr
Pierre Simonnet
Affiliation:
UMR CNRS 6134, université de Corse, BP 52, 20250 CORTE, France; cagnard@univ-corse.fr; simonnet@univ-corse.fr
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Abstract

This note is about functions ƒ : Aω → Bω whose graph is recognized by a Büchi finite automaton on the product alphabet A x B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such function are continuous or not. In 1920 W. Sierpinski showed that a function $f : \mathbb{ R} \rightarrow \mathbb{R} $ is Baire class 1 if and only if both the overgraph and the undergraph of f are Fσ. We show that such characterization is also true for functions on infinite words if we replace the real ordering by the lexicographical ordering on Bω. From this we deduce that it is decidable whether such function are of Baire class 1 or not. We extend this result to real functions definable by automata in Pisot base.

Type
Research Article
Copyright
© EDP Sciences, 2007

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