State-Complexity Hierarchies of Uniform Languages of Alphabet-Size Length.
Janusz Brzozowski, Stavros Konstantinidis · DCFS · 2008
We study the state complexity of certain simple languages. If A is an alphabet of k letters, then a k-language is a nonempty set of words of length k, that is, a uniform language of length k. We show that the minimal state complexity of a k-language is k+2, and the maximal, (k^k^-^1-1)/(k-1)+2^k+1. We prove constructively that, for every i between the minimal and maximal bounds, there is a language of state complexity i. We introduce a class of automata accepting sets of words that are permutations of A; these languages define a complete hierarchy of complexities between k^2-k+3 and 2^k+1. The languages of another class of automata, based on k-ary trees, define a complete hierarchy of complexities between 2^k+1 and (k^k^-^1-1)/(k-1)+2^k+1. This provides new examples of uniform languages of maximal complexity.