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

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(p ₁,⋯, p _k) for ∑p _i ≤ 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, e ^{Θ(√ m+n)ln(m+n))} states.