State complexity of operations on two-way finite automata over a unary alphabet
| Authors | |
|---|---|
| Year of publication | 2012 |
| Type | Article in Periodical |
| Magazine / Source | Theoretical Computer Science |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.1016/j.tcs.2012.04.010 |
| Field | General mathematics |
| Keywords | Finite automata; Two-way automata; Regular languages; Unary languages; State complexity; Landau's function |
| Description | 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 is 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, exp((1+O(1))sqrt((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 Theta(g(n)) states in the worst case, its k-th power requires (k g(n))^(Theta(1)) states, and the concatenation of an m-state 2NFA and an n-state 2NFA, exp(Theta(sqrt((m+n)ln(m+n)))) states. |
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