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

Michal Kunc, Alexander Okhotin
2012 Theoretical Computer Science  
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
more » ... te 2DFA, (g(n) + O(n)) 2 states, where g(n) = e (1+o(1)) √ n ln n 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) ) 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 ( √ (m+n) ln(m+n)) states.
doi:10.1016/j.tcs.2012.04.010 fatcat:2ubboafp2jchtndocer5qfkxe4