Automata, Borel functions and real numbers in Pisot base
This note is about functions f:A(omega)-> B-omega whose graph is recognized by a Buchi 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:R -> 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.
automata borel function borel set sequential machine