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:
“One year
They sent a million here:
Here men were drunk like water, burnt like wood.
The fat of good
And evil, the breast’s star of hope
Were rendered into soap.”
—Randall Jarrell (1914–1965)
“It will be seen that we contemplate a time when man’s will shall be law to the physical world, and he shall no longer be deterred by such abstractions as time and space, height and depth, weight and hardness, but shall indeed be the lord of creation.”
—Henry David Thoreau (1817–1862)
“A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,—the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white man’s treatment of them, and probably have reason to be so.”
—Henry David Thoreau (1817–1862)
“Science and technology multiply around us. To an increasing extent they dictate the languages in which we speak and think. Either we use those languages, or we remain mute.”
—J.G. (James Graham)