  This chapter is concerned with the design and analysis of algorithms for
minimizing finite automata. Getting a minimal automaton is a fundamental issue
in the use and implementation of finite automata tools in frameworks like text
processing, image analysis, linguistic computer science, and many other
applications. There are two main families of minimization algorithms. The first
by a sequence of refinements 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 of acyclic automata of
Revuz belongs to the second family.
  One of our studies is upon the comparison of the nature of Moore's and
Hopcroft's algorithms. This gives some new insight in both algorithms. As we
shall see, these algorithms are quite different both in behavior and in
complexity. In particular, we show that it is not possible to simulate the
computations of one of the algorithm by the other. We describe the minimization
algorithm by fusion for so-called local automata. A special case of
minimization is the construction o minimal automata for finite sets. We
consider briefly this case, and in particular describe incremental algorithms.
Finally, we consider the case of updating a minimal automaton when a word is
added or removed from the set it recognizes.
