State-Complexity Hierarchies of Uniform Languages of Alphabet-Size Length

We study the state complexity of a class of simple languages. If A is an alphabet of k letters, a k-language is a nonempty set of words of length k, that is, a uniform language of length k. By a new construction, we show that the maximal state complexity of a k-language is (k − 1)/(k − 1) + 2 + 1, and every k-language of this complexity is also a uniform language of length k of the maximal state complexity previously known. We then prove that, for every i between minimal and maximal complexities, there is a language of complexity i: for each i we exhibit such a language. We introduce “pi automata” accepting languages whose words are permutations of the alphabet; the complexities of these languages form a complete hierarchy between k − k + 3 and 2 + 1. We start with an automaton with k−k+3 states and show that states can be added one at a time, until the automaton has 2+1 states. We construct another class of automata, based on k-ary trees, whose languages define a complete hierarchy of complexities between 2 + 1 and the maximal complexity. Here, we start with an automaton with the maximal number of states and remove states one at a time, until an automaton with 2 + 1 states is reached. ∗This research was supported by the Natural Sciences and Engineering Research Council of Canada under grants no. OGP000871 and R220259.