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For example, the busy beaver game can also be generalized to two dimensions using Turing machines on two-dimensional tapes, or to Turing machines that are allowed to stay in the same place as well as move to the left and right. [10] Alternatively a "busy beaver function" for diverse models of computation can be defined with Kolmogorov ...
This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem, a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot ...
Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant. Similarly, most subsets of the natural numbers are not computable. The halting problem was the first such set to be constructed.
As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers, the sequence of which grows faster than any computable sequence. Though too large to ever be computed in full, the sequence of digits of Graham's number can be computed explicitly via simple algorithms; the last 13 digits are ...7262464195387.
The connection is made through the Busy Beaver function, where BB(n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 27-state Turing machine that halts if and only if Goldbach's conjecture is false. [37]
[36] [37] The connection is made through the Busy Beaver function, where BB(n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 15 state Turing machine that halts if and only if a conjecture by Paul Erdős (closely related to the Collatz conjecture) is false. Hence if BB(15) was known, and this machine ...
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The Busy Beaver function Σ(n) grows faster than any computable function. Hence, it is not computable; [ 2 ] only a few values are known. Computability theory originated in the 1930s, with the work of Kurt Gödel , Alonzo Church , Rózsa Péter , Alan Turing , Stephen Kleene , and Emil Post .