State Complexity of Operations on Two-Way Deterministic Finite Automata over a Unary Alphabet

Abstract

The paper determines the number of states in a two-way deterministic finite automaton (2DFA) over a one-letter alphabet sufficient and in the worst case necessary to represent the results of the following operations: (i) intersection of an m-state 2DFA and an n-state 2DFA requires between m + n and m + n + 1 states; (ii) union of an m-state 2DFA and an n-state 2DFA, between m + n and 2m + n + 4 states; (iii) Kleene star of an n-state 2DFA, (g(n) + O(n))2 states, where \(g(n)=e^{\sqrt{n \ln n}(1+o(1))}\) is the maximum value of lcm(p 1, …, p k ) for \(\sum p_i \leqslant n\), known as Landau’s function; (iv) k-th power of an n-state 2DFA, between (k − 1)g(n) − k and k(g(n) + n) states; (v) concatenation of an m-state and an n-state 2DFAs, \(e^{(1+o(1)) \sqrt{(m+n)\ln(m+n)}}\) states.