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((l+0(1))root nlnn) is the maximum value of lcm(p(1), ..., P-k) for Sigma 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+0(1))root(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 (-)(g(n)) states in the worst case, its k-th power requires (k.g(n))((-)(1)) states, and the concatenation of an m-state 2NFA and an n-state 2NFA, e((-)(root(m+n) In(m+n))) states. (C) 2012 Elsevier B.V. All rights reserved.