Computing The Star Height of Regular Languages
In contrast, the second question turned out to be much more difficult, and the question became a famous open problem in formal language theory for over two decades (Brzozowski 1980). For years, there was only little progress. The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be decidable (McNaughton 1967). But the general problem remained open for more than 25 years until it was settled by Hashiguchi, who in 1988 published an algorithm to determine the star height of any regular language. The algorithm wasn't at all practical, being of non-elementary complexity. To illustrate the immense resource consumptions of that algorithm, Lombardy and Sakarovitch (2002) give some actual numbers:
| “ |
leads to computations that are by far impossible, even for very small examples. For instance, if L is accepted by a 4 state automaton of loop complexity 3 (and with a small 10 element transition monoid), then a very low minorant of the number of languages to be tested with L for equality is: |
” |
|
—S. Lombardy and J. Sakarovitch, Star Height of Reversible Languages and Universal Automata, LATIN 2002 |
||
Notice that alone the number has 10 billion zeros when written down in decimal notation, and is already by far larger than the number of atoms in the observable universe.
A much more efficient algorithm than Hashiguchi's procedure was devised by Kirsten in 2005. This algorithm runs, for a given nondeterministic finite automaton as input, within double-exponential space. Yet the resource requirements of this algorithm still greatly exceed the margins of what is considered practically feasible.
Read more about this topic: Star Height Problem
Famous quotes containing the words star, height, regular and/or languages:
“For rigorous teachers seized my youth,
And purged its faith, and trimm’d its fire,
Show’d me the high, white star of Truth,
There bade me gaze, and there aspire.
Even now their whispers pierce the gloom:
What dost thou in this living tomb?”
—Matthew Arnold (1822–1888)
“The Woodrovian style, at the height of the Wilson hallucination, was much praised by cornfed connoisseurs.”
—H.L. (Henry Lewis)
“This is the frost coming out of the ground; this is Spring. It precedes the green and flowery spring, as mythology precedes regular poetry. I know of nothing more purgative of winter fumes and indigestions. It convinces me that Earth is still in her swaddling-clothes, and stretches forth baby fingers on every side.”
—Henry David Thoreau (1817–1862)
“It is time for dead languages to be quiet.”
—Natalie Clifford Barney (1876–1972)