Basis theorem (computability)

Effectively closed sets are a topic of study in classical computability theory. An effectively closed set is the set of all paths through some computable subtree of the binary tree ${\displaystyle 2^{<\omega }}$. These sets are closed, in the topological sense, as subsets of the Cantor space ${\displaystyle 2^{\omega }}$, and the complement of an effective closed set is an effective open set in the sense of effective Polish spaces. Stephen Cole Kleene proved in 1952 that there is a nonempty, effectively closed set with no computable point (Cooper 1999, p. 134). There are basis theorems that show there must be points that are not "too far" from being computable, in an informal sense.

A class ${\displaystyle X\subseteq 2^{\omega }}$ is a basis for effectively closed sets if every nonempty effectively closed set includes a member of X (Cooper 2003, p. 329). Basis theorems show that particular classes are bases in this sense. These theorems include (Cooper 1999, p. 134):

• The low basis theorem: each nonempty ${\displaystyle \Pi _{1}^{0}}$ set has a member that is of low degree.
• The hyperimmune-free basis theorem: each nonempty ${\displaystyle \Pi _{1}^{0}}$ set has a member that is of hyperimmune-free degree.
• The r.e. basis theorem: each nonempty ${\displaystyle \Pi _{1}^{0}}$ set has a member that is of r.e. degree.

In the second bullet, a set X has hyperimmune-free degree if every total function from the natural numbers to the natural numbers is dominated by a total computable function.

There are also basis theorems for lightface ${\displaystyle \Sigma _{1}^{1}}$ sets. These basis theorems are studied as part of hyperarithmetical theory. One theorem is the Gandy basis theorem, which is analogous to the low basis theorem. The Gandy basis theorem shows that each nonempty ${\displaystyle \Sigma _{1}^{1}}$. set has an element that is hyperarithmetically low, that is, which has the same hyperdegree as Kleene's set ${\displaystyle {\mathcal {O}}}$.

• Cooper, S. B. (1999). "Local degree theory", in Handbook of Computability Theory, E.R. Griffor (ed.), Elsevier, pp. 121–153. ISBN 978-0-444-89882-1
• — (2003), Computability Theory, Chapman-Hall. ISBN 1-584-88237-9

• Simpson, S. "A survey of basis theorems", slides from Computability Theory and Foundations of Mathematics, Tokyo Institute of Technology, February 18–20, 2013.