Automata, Borel functions and real numbers in Pisot base
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 27-44.

This note is about functions $f:{A}^{\omega }\to {B}^{\omega }$ whose graph is recognized by a Büchi finite automaton on the product alphabet $A×B$. 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 $f:ℝ\to ℝ$ is Baire class 1 if and only if both the overgraph and the undergraph of $f$ are ${F}_{\sigma }$. We show that such characterization is also true for functions on infinite words if we replace the real ordering by the lexicographical ordering on ${B}^{\omega }$. 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.

DOI : https://doi.org/10.1051/ita:2007007
Classification : 03D05,  68Q45,  68R15,  54H05
Mots clés : Borel set, Borel function, automata, sequential machine
@article{ITA_2007__41_1_27_0,
author = {Cagnard, Benoit and Simonnet, Pierre},
title = {Automata, Borel functions and real numbers in Pisot base},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {27--44},
publisher = {EDP-Sciences},
volume = {41},
number = {1},
year = {2007},
doi = {10.1051/ita:2007007},
zbl = {1156.03036},
mrnumber = {2330041},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ita:2007007/}
}
Cagnard, Benoit; Simonnet, Pierre. Automata, Borel functions and real numbers in Pisot base. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 27-44. doi : 10.1051/ita:2007007. http://archive.numdam.org/articles/10.1051/ita:2007007/

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