Minimizing nfa's and regular expressions

Abstract

We show inapproximability results concerning minimization of nondeterministic finite automata (nfa's) as well as of regular expressions relative to given nfa's, regular expressions or deterministic finite automata (dfa's). We show that it is impossible to efficiently minimize a given nfa or regular expression with n states, transitions, respectively symbols within the factor o ( n ) , unless P = PSPACE . For the unary case, we show that for any δ > 0 it is impossible to efficiently construct an approximately minimal nfa or regular expression within the factor n 1 − δ , unless P = NP . Our inapproximability results for a given dfa with n states are based on cryptographic assumptions and we show that any efficient algorithm will have an approximation factor of at least n poly ( log n ) . Our setup also allows us to analyze the minimum consistent dfa problem.