Xi function

The xi function is a googological function defined by Adam P. Goucher, based on. It is by far the fastest-growing function yet defined &mdash; in the fast-growing hierarchy, its corresponding ordinal is the fixed point of \(\alpha \rightarrow \omega_\alpha^\text{CK}\). Thus it is an uncomputable function.

SKI calculus uses three symbols called combinators: S, K, I. A process called beta-reduction takes the leftmost operator and performs one of the following operations:


 * I(x) = x
 * K(x, y) = x
 * S(x, y, z) = x(z, y(z))

For example, repeated beta-reduction on S(K, S, K(I, S)) produces K(K(I, S), S(K(I, S))) = K(I, S) = I. Some SKI expressions beta-reduce to I, some reduce to another small expression, while others keep on growing forever. If a SKI expression beta-reduces to a string consisting of n combinators, we say that it has output of size n.

Suppose we add an additional symbol Ω, the oracle combinator:


 * Ω(x, y, z) = y if β-reduction of x becomes I, and z otherwise.

If we start with a string of n symbols and we beta-reduce it, the largest possible output is called Ξ(n). It is possible for a construction in SKIΩ calculus to define completely paradoxical and inconsistent statements, and such statements are ignored in the computation of Ξ. Goucher gave Ξ(1000000) as an example of a number much larger than Rayo's number.

We could add another oracle combinator Ω’ which works like Ω, expect that it checks if SKIΩ formula is well founded. Using this new combinator w can make a faster growth rate variant of Ξ, called Ξ2.