Finite automats (FA's) are of fundamental importance in theory and in
applications. The following basic minimization problem is studied: Giv
en a DFA (deterministic FA), find a minimum equivalent nondeterministi
c FA (NFA). This paper shows that the natural decision problem associa
ted with it is PSPACE-complete. More generally, let A --t B denote the
problem of converting a given FA of type A to a minimum FA of type B.
This paper also shows that most of these problems are computationally
hard. Motivated by the question of how much nondeterminism suffices t
o make the decision problem involving an NFA computationally hard, the
authors study the complexity decision problems for FA's and present s
everal intractability results, even for cases in which the input is de
terministic or nondeterministic with a very limited nondeterminism. Fo
r example, it is shown that it is PSPACE-complete to decide if L(M(1))
.L(M(2))=L(M(3)), where M(1), M(2), and M(3) are DFAs. These problems
are related to some classical problems in automata theory (such as dec
iding whether an FA has the finite power property), as well as recent
ones (such as determining the diversity of a given FA).