On the Hardness of Determining Small NFA’s and of Proving Lower Bounds on Their Sizes

Abstract

In contrast to the minimization of deterministic finite automata (DFA's), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. This fact motivates the following computational problems: (i) Find a minimal NFA for a regular language L, if Lis given by another suitable formal description, resp. come up with a small NFA. (ii) Estimate the size of minimal NFA's or find at least a good approximation of their sizes. Here, we survey the known results striving to solve the problems formulated above and show that also for restricted versions of minimization of NFA's there are no efficient algorithms. Since one is unable to efficiently estimate the size of a minimal NFA in an algorithmic way, one can ask at least for developing mathematical proof methods that help in proving good lower bounds on the size of a minimal NFA for a given regular language. We show here that even the best known methods for this purpose fail for some concrete regular languages. Finally, we give an overview of the results about the influence of the degree of ambiguity on the size of NFA's and discuss the relation between the descriptional complexity of NFA's and NFA's with i¾?-transitions.