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Cagnard, Benoit; Simonnet, Pierre
Automata, Borel functions and real numbers in Pisot base. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, 41 no. 1 (2007), p. 27-44
Texte intégral djvu | pdf | Analyses MR 2330041 | Zbl pre05238552
Class. Math.: 03D05, 68Q45, 68R15, 54H05
Mots clés: Borel set, Borel function, automata, sequential machine

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This note is about functions $ f: A^\omega \rightarrow B^\omega $ whose graph is recognized by a Büchi finite automaton on the product alphabet $A \times B$. 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 $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.


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