Solution of Church’s Problem: A Tutorial

Church’s Problem (1957) asks for the construction of a finite-state procedure that transforms any input sequence α letter by letter into an output sequence β such that the pair (α, β) satisfies a given specification. Even after the solution by Büchi and Landweber in 1969 (for specifications in monadic second-order logic over the structure (N,+1)), the problem has stimulated research in automata theory for decades, in recent years mainly in the algorithmic study of infinite games. We present a modern solution which proceeds in several stages (each of them of moderate difficulty) and provides additional insight into the structure of the synthesized finite-state transducers.