The generalized star-height problem in formal language theory is the open question whether all regular languages can be expressed using generalized regular expressions with a limited nesting depth of Kleene stars. Here, generalized regular expressions are defined like regular expressions, but they have a built-in complement operator. For a regular language, its generalized star height is defined as the minimum nesting depth of Kleene stars needed in order to describe the language by means of a generalized regular expression, hence the name of the problem.
More specifically, it is an open question whether a nesting depth of more than 1 is required, and if so, whether there is an algorithm to determine the minimum required star height.
Regular languages of star-height 0 are also known as star-free languages. The theorem of Schützenberger provides an algebraic characterization of star-free languages by means of aperiodic syntactic monoids. In particular star-free languages are a proper decidable subclass of regular languages.
Famous quotes containing the words generalized, star, height and/or problem:
“One is conscious of no brave and noble earnestness in it, of no generalized passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in manner—in brief, what is there is the feeble, uninspiring quality of German painting and English music.”
—H.L. (Henry Lewis)
“You know when there’s a star, like in show business, the star has her name in lights on the marquee! Right? And the star gets the money because the people come to see the star, right? Well, I’m the star, and all of you are in the chorus.”
—Babe Didrikson Zaharias (1911–1956)
“The lotus’ stem is as long as the depth of water,
So men’s height is just as great as their inner strength.”
—Tiruvalluvar (c. 5th century A.D.)
“The government is huge, stupid, greedy and makes nosy, officious and dangerous intrusions into the smallest corners of life—this much we can stand. But the real problem is that government is boring. We could cure or mitigate the other ills Washington visits on us if we could only bring ourselves to pay attention to Washington itself. But we cannot.”
—P.J. (Patrick Jake)