State Complexity of Prefix-Free Regular Languages

We investigate the state complexities of basic operations for prefix-free regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worst-case for the minimal deterministic finite-state automaton (DFA) that accepts the language obtained from the operation. We know that a regular language is prefix-free if and only if its minimal DFA has only one final state and the final state has no out-transitions whose target state is not a sink state. Based on this observation, we reduce the state complexities for prefix-free regular languages compared with the state complexities for (general) regular languages. For both catenation and Kleene star operations of (general) regular languages, the state complexities are exponential in the size of given minimal DFAs. On the other hand, if both regular languages are prefix-free, then the state complexities are at most linear. We also demonstrate that we can reduce the state complexities of intersection and union operations based on the structural properties of prefix-free minimal DFAs.