Busy beaver function

Rado's sigma function (or busy beaver function) \(\Sigma(n)\) is equal to the maximum number of 1s that can be written (in the finished tape) with an n-state, 2-color Turing machine before halting. Turing machines that produce these numbers are called busy beavers.

Rado showed that it the function grows faster than any computable function, and thus it is uncomputable &mdash; determining whether a given Turing machine is a busy beaver takes an infinite number of steps. Despite this, it is possible to evaluate \(\Sigma(n)\) for some small values of \(n\); it has been proven that \(\Sigma(1) = 1\), \(\Sigma(2) = 4\), \(\Sigma(3) = 6\), and \(\Sigma(4) = 13\). No other values are exactly known, but some lower bounds are \(\Sigma(5) \geq 4098\) and \(\Sigma(6) \geq 3.514 \cdot 10^{18276}\).

Milton Green proved that \(\Sigma(2n) \gg 3 \uparrow^{k-2} 3\), yielding the following lower bounds:

\[\Sigma(8) > 3 \uparrow\uparrow 3 = 7625597484987\]

\[\Sigma(10) > 3 \uparrow\uparrow\uparrow 3\]

\[\Sigma(12) > 3 \uparrow\uparrow\uparrow\uparrow 3\]

\(\Sigma(6)\) is already known to be much greater than 7625597484987, so these lower bounds are very weak.

The function's growth rate is comparable to \(f_{\omega^\text{CK}_1}(n)\) in the fast-growing hierarchy, where \(\omega^\text{CK}_1\) is the, the set of all recursive ordinals.

\(\Sigma(64)\) has been proved to be larger than Graham's number. It is the smallest known value of \(\Sigma(n)\) with this property.

Higher order busy beaver functions and Turing machines
Define standard busy beaver function as 1st order and standard Turing machine as 1st order.

Then, 2nd order busy beaver function, \(\Sigma_2(n)\) is equal to the maximum number of 1s that can be written (in the finished tape) with an n-state, 2-color Oracle Turing machine (2nd order Turing machine) before halting, where Oracle Turing machine is a special Turing machine that contains some black box, "oracle" that predicts when the usual Turing machine halts. Thus, such machine has solve and compute arbitrary 1st order busy beaver number. However, even the Oracle Turing machine can't determine when it halts itself, and thus \(\Sigma_2(n)\) is uncomputable even for this machine.

In general, X-th order busy beaver function can be computed using X+1-th order Turing machine, but is uncomputable for all turing machines of N-th order, where \(N \leq X\).

X-th order busy beaver function has growth rate comparable to \(f_{\omega^\text{CK}_1 \times X}(n)\) in the fast-growing hierarchy.