New types of automata to solve fixed point problems
Résumé
The aim of this article is to find the supremum of {X|AX subset of or equal to X union B and X subset of or equal to K}, where A, B and K are three fixed, rational languages. This problem and its solution are formally inspired by retroaction in control theory. To get the solution, we introduce a new type of automata whose acceptance condition is a positive boolean formula. The time complexity of our algorithm depends linearly on the size of A, quadratically on the size of K and exponentially on the size of B, where the size is measured by the number of states of the minimal automaton.