On the State Complexity of Operations on Two-Way Finite Automata

Abstract

The number of states in two-way deterministic finite automata (2DFAs) is considered. It is shown that the state complexity of basic operations is: at least m + n − o(m + n) and at most 4m + n + 1 for union; at least m + n − o(m + n) and at most m + n + 1 for intersection; at least n and at most 4n for complementation; at least $\Omega(\frac{m}{n}) + \frac{2^{\Omega(n)}}{\log m}$ and at most $2m^{m+1}\cdot 2^{n^{n+1}}$ for concatenation; at least $\frac{1}{n} 2^{\frac{n}{2}-1}$ and at most $2^{O(n^{n+1})}$ for both star and square; between n and n + 2 for reversal; exactly 2n for inverse homomorphism. In each case m and n denote the number of states in 2DFAs for the arguments.

Supported by VEGA grant 2/6089/26 and by Academy of Finland grant 118540.