Descriptional Complexity of Determinization and Complementation for Finite Automata.

In this paper we study the subset construction that transforms nondeterministic finite automata (NFA) to deterministic finite automata (DFA). It is well known that given a n -state NFA, the subset construction algorithm produces a 2 n -state DFA in the worst case. It has been shown that given n, m ( n < m ≤ 2 n ), there is a n -state NFA N such that the minimal DFA recognizing L ( N ) has m states. However this construction requires O ( n 2 ) number of transitions in the worst case. We give an alternative solution to this problem that requires asymptotically fewer transitions. We also investigate the question of the complementation of NFA. In this case also, it known that given n, m ( n < m ≤ 2 n ), there exists a n -state NFA N such that the minimal NFA recognizing the complement of L ( N ) needs m states. We provide regular languages such that given n, k ( k > 1 and n > k ), the NFA recognizing these languages need n states and the NFA recognizing their complement needs ( k + 1) n − ( k + 1) 2 + 2 states. Finally we show that for given n, k > 1, there exists a O ( n )-state NFA A such that the minimal NFA recognizing the complement of L ( A ) has between O ( n k −1 ) and O ( n 2 k ) states. Importantly however, the constructed NFA's have a small number of transitions, typically in the order of O ( n ) or O ( n 2 / log 2( n )). These are better than the comparable results in the literature.