On the limits of the communication complexity technique for proving lower bounds on the size of minimal NFA’s

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. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s. Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages. “To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages. The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2 O ( n 1 / 4 ) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head.