Minimizing local automata

Abstract

We design an algorithm that minimizes irreducible deterministic local automata by a sequence of state mergings. Two states can be merged if they have exactly the same outputs. The running time of the algorithm is O(min(m(n-r+1), m log n)), where m is the number of edges, n the number of states of the automaton, and r the number of states of the minimized automaton. In particular, the algorithm is linear when the automaton is already minimal and contrary to Hopcroft's minimization algorithm that has a O (kn log n) running time in this case, where k is the size of the alphabet, and that applies only to complete automata. (Note that kn ges m.) While Hopcroft's algorithm relies on a negative strategy, starting from a partition with a single class of all states, and partitioning classes when it is discovered that two states cannot belong to the same class, our algorithm relies on a positive strategy, starting from the trivial partition for which each class is a singleton. Two classes are then merged when their leaders have the same outputs. The algorithm applies to irreducible deterministic local automata, where all states are considered both initial and final. These automata, also called covers, recognize symbolic dynamical shifts of finite type. They serve to present a large class of constrained channels, the class of finite memory systems, used for channel coding purposes. The algorithm also applies to irreducible deterministic automata that are left-closing and have a synchronizing word. These automata present shifts that are called almost of finite type. Almost-of-finite-type shifts make a meaningful class of shifts, intermediate between finite type shifts and sofic shifts.