Alpha recursion theory

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In recursion theory, the mathematical theory of computability, alpha recursion (often written α recursion) is a generalisation of recursion theory to subsets of admissible ordinals α. An admissible ordinal is closed under Σ1(Lα) functions. Admissible ordinals are models of Kripke–Platek set theory. In what follows α is considered to be fixed.

The objects of study in α recursion are subsets of α. A is said to be α recursively enumerable if it it is Σ1 definable over Lα. A is recursive if both A and α / A (its complement in α) are recursively enumerable.

Members of Lα are called α finite and play a similar role to the finite numbers in classical recursion theory.

We say R is a reduction procedure if it is recursively enumerable and every member of R is of the form \langle H,J,K\rangle where H, J, K are all α-finite.

A is said to be α-recusive in B if there exist R0,R1 reduction procedures such that:

K \subseteq A \leftrightarrow \exists H: \exists J:[<H,J,K> \in R_0 \wedge H \subseteq B \wedge J \subseteq \alpha / B ],
K \subseteq \alpha / A \leftrightarrow \exists H: \exists J:[<H,J,K> \in R_1 \wedge H \subseteq B \wedge J \subseteq \alpha / B ].

If A is recursive in B this is written \scriptstyle A \le_\alpha B. By this definition A is recursive in \scriptstyle\varnothing (the empty set) if and only if A is recursive. However it should be noted that A being recursive in B is not equivalent to A being Σ1(Lα[B]).

We say A is regular if \forall \beta \in \alpha: A \cap \beta \in L_\alpha or in other words if every initial portion of A is α-finite.

[edit] Results in α recursion

Shore's splitting theorem: Let A be α recursively enumerable and regular. There exist α recursively enumerable B0,B1 such that A=B_0 \cup B_1 \wedge B_0 \cap B_1 = \varnothing \wedge A \not\le_\alpha B_i (i<2).

Shore's density theorem: Let A, C be α-regular recursively enumerable sets such that \scriptstyle A <_\alpha C then there exists a regular α-recursively enumerable set B such that \scriptstyle A <_\alpha B <_\alpha C.

[edit] References

  • Gerald Sacks, Higher recursion theory, Springer Verlag, 1990
  • Robert Soare, Recursively Enumerable Sets and Degrees, Springer Verlag, 1987