Properties
- Every set is Turing equivalent to its complement
- Every computable set is Turing reducible to every other computable set. Because these sets can be computed with no oracle, they can be computed by an oracle machine that ignores the oracle it is given.
- The relation is transitive: if and then . Moreover holds for every set A, and thus the relation is a preorder (it is not a partial order because and does not necessarily imply ).
- There are pairs of sets such that A is not Turing reducible to B and B is not Turing reducible to A. Thus is not a linear order.
- There are infinite decreasing sequences of sets under . Thus this relation is not well-founded.
- Every set is Turing reducible to its own Turing jump, but the Turing jump of a set is never Turing reducible to the original set.
Read more about this topic: Turing Reduction
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)