New results on the generalized star-height problem

Abstract

We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, complementation is considered as a basic operator. We rst show that the class of languages of star-height n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective star-free substitutions). It is known that languages recognized by a commutative group are of star-height 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by (Z=2Z) n . In the same direction, we show that one of the languages that was conjectured to be of star height 2 during the past ten years, is in fact of star height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form M (G N), where M and N are aperiodic monoids, and G is a commutative group, then L is of star-height 1. Finally we show that every rational language is the inverse image, under some morphism between free monoids, of a language of (restricted) star-height 1. The determination of the star-height of a rational language is an old problem of formal language theory (see Brzozowski [1], for an historical survey). The restricted star-height problem has been recently solved by Hashiguchi [4], but here we are interested in that aspect of the problem concerning generalized starheight, in which complementation is considered as a basic operator. Thus, in the rest of this paper, the word "star-height" will always refer to generalized star-height. The aim of this paper is to present some new results related to the starheight problem : \Is there an algorithm to compute the star-height of a given rational language" (this language can be given, for instance, by a rational expression). The star-height problem seems to be extremely dicult, and very little is known on the subject. For instance, it is not yet known whether there