Xi function

The xi function is an uncomputable googological function defined by Adam P. Goucher, based on a variant of. In the fast-growing hierarchy, its corresponding ordinal is the \(\omega_\omega^\text{CK}\). The xi function surpasses Rado's sigma function. Goucher mistakenly claimed that Ξ outgrew Rayo(n) and that it took the place as the fastest-growing function yet defined.

Definition
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 finite output is called Ξ(n). Thanks to Gödel incompleteness, it is possible for a SKIΩ calculus statement to be inconsistent or paradoxical, and such statements are ignored in the computation of Ξ.

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.

Values
Some exact values and bounds are shown below:

\begin{eqnarray} \Xi(1) &=& 1 \\ \Xi(2) &=& 2 \\ \Xi(3) &=& 3 \\ \Xi(4) &=& 4 \\ \Xi(5) &=& 6 \\ \Xi(6) &=& 17 \\ \Xi(7) &=& 51 \\ \Xi(8) &\geq& 98 \\ \Xi(9) &\geq& 167 \\ \Xi(10) &\geq& 296 \\ \Xi(11) &\geq& 513 \\ \Xi(12) &\geq& 846 \\ \Xi(n) &>& 7F_{n-2} \text{ (for } n\geq 7) \\ \end{eqnarray}

Where \(F_{n-2}\) denotes n-2th member of the.

A top down approach to finding bounds, by constructed the Fast-growing hierarchy in SKI calculus has yielded the following lower bounds to a weaker version of the function, which lacks the \(Ω\) combinator:

\begin{eqnarray} \Xi(37) &\geq& 5f_3(3)+1 \\ \Xi(42) &\geq& 5f_3(4)+1 \\ \Xi(43) &\geq& 5f_\omega+1(2)+1 \\ \Xi(48) &\geq& 5f_\omega+1(3)+1 \\ \Xi(50) &\geq& 5f_\omega+2(2)+1 &\geq& \text{Graham's Number} \\ \Xi(55) &\geq& 5f_\omega+2(3)+1 \\ \Xi(57) &\geq& 5f_\omega+3(2)+1 \\ \Xi(62) &\geq& 5f_\omega+3(3)+1 \\ \Xi(64) &\geq& 5f_\omega+4(2)+1 \\ \Xi(68) &\geq& 5f_\omega.2+1(2)+1 \\     \end{eqnarray}

Where f refers to the Fast-growing hierarchy.

Winning sequences
Below is the list of beta-reductions of the expressions that will have maximal length. For \(\Xi(1),\Xi(2),\Xi(3)\) and \(\Xi(4)\) the process is trivial: they are S, S(S), S(SS) and S(SSS) respectively. From \(5 \leq n \leq 7\), it goes as follows:

\(\Xi(5)\)
SSS(SI) S(SI)S(SI)

\(\Xi(6)\)
SSS(SI)S S(SI)(S(SI))S SIS(S(SI)S) I(S(SI)S)(S(S(SI)S)) S(SI)S(S(S(SI)S)) SI(S(S(SI)S))(S(S(S(SI)S))) I(S(S(S(SI)S)))(S(S(SI)S)(S(S(S(SI)S)))) S(S(S(SI)S))(S(S(SI)S)(S(S(S(SI)S))))

\(\Xi(7)\)
\(\Xi(7)\) marks the first time the oracle combinator gives the xi function an advantage over standard SKI calculus.

SSS(SI)SΩ S(SI)(S(SI))SΩ SIS(S(SI)S)Ω I(S(SI)S)(S(S(SI)S))Ω S(SI)S(S(S(SI)S))Ω SI(S(S(SI)S))(S(S(S(SI)S)))Ω I(S(S(S(SI)S)))(S(S(SI)S)(S(S(S(SI)S))))Ω S(S(S(SI)S))(S(S(SI)S)(S(S(S(SI)S))))Ω S(S(SI)S)Ω((S(S(SI)S)(S(S(S(SI)S))))Ω) S(SI)S((S(S(SI)S)(S(S(S(SI)S))))Ω)(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω)) SI((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω)) I(S((S(S(SI)S)(S(S(S(SI)S))))Ω))(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω)) S((S(S(SI)S)(S(S(S(SI)S))))Ω)(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω)) S(S(SI)S)(S(S(S(SI)S)))Ω(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))) S(SI)SΩ((S(S(S(SI)S)))Ω)(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))) SIΩ(S(Ω))((S(S(S(SI)S)))Ω)(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))) I(S(Ω))(Ω(S(Ω)))((S(S(S(SI)S)))Ω)(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))) SΩ(Ω(S(Ω)))((S(S(S(SI)S)))Ω)(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))(((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))) Ω((S(S(S(SI)S)))Ω)((Ω(S(Ω)))((S(S(S(SI)S)))Ω))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))((((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω))) Ω((S(S(SI)S)(S(S(S(SI)S))))Ω)((((S(S(SI)S)(S(S(S(SI)S))))Ω)(S((S(S(SI)S)(S(S(S(SI)S))))Ω)))(Ω((S(S(SI)S)(S(S(S(SI)S))))Ω)))

\(\Xi(8)\) and higher
The best known combinator for \(\Xi(8)\) is SSS(SI)S(SΩ)

The best known combinator for \(\Xi(9)\) is SSS(SI)S(S(SΩ))

The best known combinator for \(\Xi(10)\) is SSS(SI)S(S(S(SΩ)))

The best known combinator for \(\Xi(11)\) is SSS(SI)S(S(S(S(SΩ))))

The best known combinator for \(\Xi(12)\) is SSS(SI)S(S(S(S(S(SΩ)))))

Trees
The combinators can be written as trees. The function is on the left and the arguments are on the right.