State Complexity of Projection on Languages Recognized by Permutation Automata and Commuting Letters

The projected language of a general deterministic automaton with $n$ states is recognizable by a deterministic automaton with $2^{n-1} + 2^{n-m} - 1$ states, where $m$ denotes the number of states incident to unobservable non-loop transitions, and this bound is best possible. Here, we derive the tight bound $2^{n - \lceil \frac{m}{2} \rceil} - 1$ for permutation automata. For a state-partition automaton with $n$ states (also called automata with the observer property) the projected language is recognizable with $n$ states. Up to now, these, and finite languages projected onto unary languages, were the only classes of automata known to possess this property. We show that this is also true for commutative automata and we find commutative automata that are not state-partition automata.