Oracles and Halting Problems
It is possible to posit the existence of an oracle which computes a non-computable function, such as the answer to the halting problem or some equivalent. A machine with an oracle of this sort is a hypercomputer.
Interestingly, the halting paradox still applies to such machines; although they determine whether particular Turing machines will halt on particular inputs, they cannot determine, in general, if machines equivalent to themselves will halt. This fact creates a hierarchy of machines, called the arithmetical hierarchy, each with a more powerful halting oracle and an even harder halting problem.
Read more about this topic: Oracle Machine
Famous quotes containing the words oracles, halting and/or problems:
“Tried by a New England eye, or the more practical wisdom of modern times, they are the oracles of a race already in its dotage; but held up to the sky, which is the only impartial and incorruptible ordeal, they are of a piece with its depth and serenity, and I am assured that they will have a place and significance as long as there is a sky to test them by.”
—Henry David Thoreau (1817–1862)
“Of the two
who feign anger,
sulk in mock sleep,
and give ear
to the other’s halting sighs,
who’s the winner?”
—Hla Stavhana (c. 50 A.D.)
“Imagination is a valuable asset in business and she has a sister, Understanding, who also serves. Together they make a splendid team and business problems dissolve and the impossible is accomplished by their ministrations.... Imagination concerning the world’s wants and the individual’s needs should be the Alpha and Omega of self-education.”
—Alice Foote MacDougall (1867–1945)