Operational union-complexity

Abstract

Union-free languages are described by regular expressions using only concatenation and Kleene-star. Every regular language can be given as a union of finitely many union-free languages. By the minimal number of union-free languages needed in such union-free decompositions of a regular language, its union-complexity is defined. In this paper, the union-complexity of the languages obtained by various operations is studied, e.g., having two languages with union-complexities n and m, respectively, what could be the union-complexity of their union/concatenation/shuffle. Particularly, it is shown that the Kleene-star of any regular language has union-complexity 1. In some cases, e.g., at union and concatenation, the resulting language has a bounded union-complexity. In some other cases, e.g., at complement, the resulted language can have arbitrarily large union-complexity. At (k-th) power of a language, the case of the unary alphabet and the general case (alphabet with at least two symbols) have different upper bounds. While, in case of shuffle, there is an unbounded growth in the general case, while for languages over the unary alphabet the growth is bounded. Tight upper bounds are shown for all of the above mentioned cases (whenever the growth is bounded). At intersection we also show an unbounded growth in the general case, especially, over a binary alphabet.