Xi function

The xi function is a googological function defined by Adam P. Goucher, based on a variant of. It is by far the fastest-growing function yet defined &mdash; in the fast-growing hierarchy, its corresponding ordinal is the first fixed point of \(\alpha \rightarrow \omega_\alpha^\text{CK}\) (informally, this can be pictured as \(\omega^\text{CK}_{\omega^\text{CK}_{\omega^\text{CK}_{._{._.}}}}\)). It surpasses Rado's sigma function and Rayo's function, and 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.

SKI calculus alone is no more powerful than a Turing machine. But we can greatly increase its strength by adding 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 \(\Xi(n)\). 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, except that it can check if a SKIΩ formula is well-founded (i.e. does not create a paradox). Using this new combinator, we can make a variant of Ξ, called Ξ2, which grows even faster than the ordinary xi function.