Relation To Turing Machines
The Turing computable sets of natural numbers are exactly the sets at level of the arithmetical hierarchy. The recursively enumerable sets are exactly the sets at level .
No oracle machine is capable of solving its own halting problem (a variation of Turing's proof applies). The halting problem for a oracle in fact sits in .
Post's theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees. In particular, it establishes the following facts for all n ≥ 1:
- The set (the nth Turing jump of the empty set) is many-one complete in .
- The set is many-one complete in .
- The set is Turing complete in .
The polynomial hierarchy is a "feasible resource-bounded" version of the arithmetical hierarchy in which polynomial length bounds are placed on the numbers involved (or, equivalently, polynomial time bounds are placed on the Turing machines involved). It gives a finer classification of some sets of natural numbers that are at level of the arithmetical hierarchy.
Read more about this topic: Arithmetical Hierarchy
Famous quotes containing the words relation to, relation and/or machines:
“You must realize that I was suffering from love and I knew him as intimately as I knew my own image in a mirror. In other words, I knew him only in relation to myself.”
—Angela Carter (1940–1992)
“There is a relation between the hours of our life and the centuries of time. As the air I breathe is drawn from the great repositories of nature, as the light on my book is yielded by a star a hundred millions of miles distant, as the poise of my body depends on the equilibrium of centrifugal and centripetal forces, so the hours should be instructed by the ages and the ages explained by the hours.”
—Ralph Waldo Emerson (1803–1882)
“There are bills to be paid, machines to keep in repair,
Irregular verbs to learn, the Time Being to redeem
From insignificance.”
—W.H. (Wystan Hugh)