Low (computability)

In computability theory, a Turing degree [X] is low if the Turing jump [X&prime;] is 0&prime;. A set is low if it has low degree. Since every set is computable from its jump, any low set is computable in 0&prime;, but the jump of sets computable in 0&prime; can bound any degree recursively enumerable in 0&prime; (Schoenfield Jump Inversion). X being low says that its jump X&prime; has the least possible degree in terms of Turing reducibility for the jump of a set.

There are various related properties to low degrees:
 * A degree is lown if its n'th jump is the n'th jump of 0.
 * A set X is generalized low if it satisfies X&prime; ≡T X + 0&prime;, that is: if its jump has the lowest degree possible.
 * A degree d is generalized low n if its n'th jump is the (n-1)'st jump of the join of d with 0&prime;.

More generally, properties of sets which describe their being computationally weak (when used as a Turing oracle) are referred to under the umbrella term lowness properties.

By the Low basis theorem of Jockusch and Soare, any nonempty $$\Pi^0_1$$ class in $$2^\omega$$ contains a set of low degree. This implies that, although low sets are computationally weak, they can still accomplish such feats as computing a completion of Peano Arithmetic. In practice, this allows a restriction on the computational power of objects needed for recursion theoretic constructions: for example, those used in the analyzing the proof-theoretic strength of Ramsey's theorem.