Busy Beaver Functions

The Busy Beaver Game is the search for Turing machines which maximize various Busy Beaver Functions. All Busy Beaver functions are non-computable. There are several, related functions with different authors referring to to one or the other as "the Busy Beaver function". Therefore, it is recommended that you use a more specific designation when referring to one specific Busy Beaver function.

The two most commonly used Busy Beaver functions are:

• The Maximum Shift function $S(n,m)$ which is the most commonly used Busy Beaver function by bbchallenge and is generally called $BB(n,m)$ here.
• The Maximum Score function $\mathrm{\Sigma }(n,m)$ which is Tibor Radó's original Busy Beaver function.

Max Shift Function S(n, m)

The Maximum Shift or Maximum Step function is the largest number of steps (or shifts) that any Turing machine (of a certain size) takes before halting. It was introduced by Tibor Radó in his seminal Busy Beaver paper. He used the notation $S(n)$ to define it for Turing machines with $n$ states and 2 symbols. This was later extended to $S(n,m)$ for $n$ states and $m$ symbols. Notably, the halting transition counts as a step, so the TM with rule A0 -> 1RZ halts in 1 step.

Ben-Amram calls this the $time(n)$ function. Harland calls it the "frantic frog" function $ff(n)$.

In his 2020 Survey, Scott Aaronson used the notation $BB(n,m)$ for the Max Shift function and refers to it as "the" Busy Beaver function.

Max Score Function Σ(n, m)

The Maximum Score function is the largest number of ones (or non-zero symbols in general) left on the tape by any halting Turing machine (of a certain size) at the moment it halts. It was also introduced by Tibor Radó in his seminal paper. He called it the "score" of the Turing machine. He used the notation $\mathrm{\Sigma }(n)$ to define it for Turing machines with $n$ states and 2 symbols. This was later extended to $\mathrm{\Sigma }(n,m)$ for $n$ states and $m$ symbols.

Ben-Amram calls this the $ones(n)$ function. Harland calls it the "busy beaver" function $bb(n)$. Before Aaronson's survey, this was the function that most people called "the" Busy Beaver function.

Other Busy Beaver functions

In addition to the above functions, there are a couple others that have appeared in the literature:

• Maximum space: Ben-Amram call this $space(n)$, bbchallenge calls it BB_SPACE(n) or $B{B}_{space}(n)$. This is the total number of tape cells read before halting. According to Ben-Amram, it includes the starting cell, but not necessarily the cell the halting transition moves to.
• Maximum consecutive ones: Ben-Amram call this $num(n)$. A TMs only qualifies if it halts with all ones consecutive on tape, largest number of consecutive ones on tape at halt wins.