Size of Unary One-Way Multi-head Finite Automata

Abstract

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case an upper bound of O(n·F(t·n)^k − 1) and a lower bound of n·F(n)^k − 1 states is shown, where t is a constant and F denotes Landau’s function. Since both bounds are of order $e^{\Theta(\sqrt{n \cdot \ln(n)})}$, the trade-off for the simulation is tight in the order of magnitude. For the latter case we obtain an upper bound of O(n ^2k) and a lower bound of Ω(n ^k) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.