State complexity of operations on two-way finite automata over a unary alphabet

Abstract

The paper determines the number of states in two-way deterministic finite automata (2DFA) over a one-letter alphabet sufficient and in the worst case necessary to represent the results of basic language-theoretic operations on 2DFAs with a certain number of states. It is proved that (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(1+o(1))nlnn is the maximum value of lcm(p1,…,pk) for ∑pi⩽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 2DFA and an n-state 2DFA, e(1+o(1))(m+n)ln(m+n) states. It is furthermore demonstrated that the Kleene star of a two-way nondeterministic automaton (2NFA) with n states requires Θ(g(n)) states in the worst case, its k-th power requires (k⋅g(n))Θ(1) states, and the concatenation of an m-state 2NFA and an n-state 2NFA, eΘ((m+n)ln(m+n)) states.