On the Succinctness of Deterministic, Nondeterministic, Probabilistic and Quantum Finite Automata

We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm} of cyclic languages, where Lm = {a | k ∈ N}. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language Lm with isolated cut point on one-way probabilistic finite automata is p1 1 +p α2 2 + · · ·+ps s , with p α1 1 p α2 2 · · · ps s being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept Lm with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms. Mathematics Subject Classification. 68Q10, 68Q19, 68Q45.