State-complexity hierarchies of uniform languages of alphabet-size length

  • Authors:

  • Janusz Brzozowski;Stavros Konstantinidis

  • Affiliations:

  • David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1;Department of Mathematics and Computing Science, Saint Marys University, Halifax, NS, Canada B3H 3C3

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

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.