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:
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)
“As machines become more and more efficient and perfect, so it will become clear that imperfection is the greatness of man.”
—Ernst Fischer (1899–1972)