Minimization of Automata

This chapter is concerned with the design and analysis of algorithms for minimizingfinite automata. Getting a minimal automaton is a fundamental issue in the use and implementationof finite automata tools in frameworks like text processing, image analysis, linguistic computerscience, and many other applications.Although minimization algorithms are rather old, there were recently some new developmentswhich are explained or sketched in this chapter.There are two main families of minimization algorithms. The first are by a sequence of refine-ments of a partition of the set of states, the second by a sequence of fusions or merges of states.Hopcroft’s and Moore’s algorithms belong to the first family. The linear-time minimization ofacyclic automata of Revuz belongs to the second family.We describe, prove and analyze Moore’s and Hopcroft’s algorithms. One of our studies isthe comparison of these algorithms. It appears that they quite different both in behavior and incomplexity. In particular, we show that it is not possible to simulate the computations of one ofthe algorithm by the other. We also state the results known about average complexity of bothalgorithms.We describe the minimization algorithm by fusion for acyclic automata, and for a more gen-eral class of automata, namely those recognizing regular languages having polynomially boundednumber of words.Finally, we consider briefly incremental algorithms for building minimal automata for finite setsof words. We consider the updating of a minimal automaton when a word is added or removed.