Optimal lower bounds on regular expression size using communication complexity

The problem of converting deterministic finite automata into (short) regular expressions is considered. It is known that the required expression size is 2(Theta(n)) in the worst case for infinite languages, and for finite languages it is n(Omega(log log n)) and n(O(log n)), if the alphabet size grows with the number of states n of the given automaton. A new lower bound method based on communication complexity for regular expression size is developed to show that the required size is indeed n(Theta(logn)). For constant alphabet size the best lower bound known to date is Omega(n(2)), even when allowing infinite languages and nondeterministic finite automata. As the technique developed here works equally well for deterministic finite automata over binary alphabets, the lower bound is improved to n(Omega(log n)).