State Complexity of Reversals of Deterministic Finite Automata with Output

We investigate the worst-case state complexity of reversals of deterministic finite automata with output (DFAOs). In these automata, each state is assigned some output value, rather than simply being labelled final or non-final. This directly generalizes the well-studied problem of determining the worst-case state complexity of reversals of ordinary deterministic finite automata. If a DFAO has n states and k possible output values, there is a known upper bound of kn for the state complexity of reversal. We show this bound can be reached with a ternary input alphabet. We conjecture it cannot be reached with a binary input alphabet except when k = 2, and give a lower bound for the case 3 ≤ k < n. We prove that the state complexity of reversal depends solely on the transition monoid of the DFAO and the mapping that assigns output values to states.