Ordering Regular Languages and Automata: Complexity

Given an order of the underlying alphabet we can lift it to the states of a finite deterministic automaton: to compare states we use the order of the strings reaching them. When the order on strings is the co-lexicographic one \emph{and} this order turns out to be total, the DFA is called Wheeler. This recently introduced class of automata -- the \emph{Wheeler automata} -- constitute an important data-structure for languages, since it allows the design and implementation of a very efficient tool-set of storage mechanisms for the transition function, supporting a large variety of substring queries. In this context it is natural to consider the class of regular languages accepted by Wheeler automata, i.e. the Wheeler languages. An inspiring result in this area is the following: it has been shown that, as opposed to the general case, the classic determinization by powerset construction is \emph{polynomial} on Wheeler automata. As a consequence, most classical problems, when considered on this class of automata, turn out to be "easy" -- that is, solvable in polynomial time. In this paper we consider computational problems related to Wheelerness, but starting from non-deterministic automata. We also consider the case of \emph{reduced} non-deterministic ones -- a class of NFA where recognizing Wheelerness is still polynomial, as for DFA's. Our collection of results shows that moving towards non-determinism is, in most cases, a dangerous path leading quickly to intractability. Moreover, we start a study of "state complexity" related to Wheeler DFA and languages, proving that the classic construction for the intersection of languages turns out to be computationally simpler on Wheeler DFA than in the general case. We also provide a construction for the minimum Wheeler DFA recognizing a given Wheeler language.