Some Results on the Generalized Star-Height Problem

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 first 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 difficult, and very little is known on the subject. For instance, it is not yet known whether there Research supported by the PRC Mathématiques et Informatique. Research supported by National Science Foundation Grant CCR-8700700. Research supported by NSERC Grant A4546 and FCAR Grant 86-EQ-2933