Busy beaver function

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Not to be confused with Maximum shifts function.

Busy beaver function
Growth rate	>* all computable functions
Alternate names	BB, Radó's sigma function
Notations	\(\Sigma(n)\), \(\text{BB}(n)\)
Author(s)	Tibor Radó
Year	1961

The busy beaver function (a.k.a. BB function or Radó's sigma function) is a distinctive fast-growing function from computability theory. The classic definition, originally denoted as \(\Sigma(n)\), is defined as the maximum number of ones that can be written in the finished tape with an n-state, 2-color halting Turing machine starting from a blank tape before halting. These Turing machines are called busy beavers.

Radó showed that this function eventually dominates all computable functions, and thus it is uncomputable. That is, no algorithm that terminates after a finite number of steps can compute \(\Sigma(n)\) for arbitrary \(n\). However, it is computable by an oracle Turing machine with an oracle for the halting problem.

A related function, also defined by Radó, is defined as the maximum number of steps a n-state halting Turing machine takes starting from a blank tape before halting. This function, denoted \(S(n)\), is also referred to as the busy beaver function. This may lead to confusion, so the alternative function is sometimes referred to as the frantic frog function. Recent efforts in researching busy beavers consider \(\text{BB}(n)\) to be defined in terms of the maximum shifts.

It is the most well-known of the uncomputable functions, and is one of the fastest-growing functions ever arising out of professional mathematics. Only a handful of significant googological functions surpass it — for example, the xi function, the ITTM busy beaver function and the Rayo function.

	A	B	C	D	E
0	1RB	1RC	1RD	1LA	H
1	1LC	1RB	0LE	1LD	0LA

Informal explanation

Generally, \(\Sigma(n,m)\) indicates the maximum number of non-blank symbols that can be written (in the finished tape) with an n-state, m-color halting Turing machine starting from a blank tape before halting.

\(\Sigma(n)\) also can be defined as the largest number in set \(T = \{n_1,n_2,\cdots,n_k\}\), which contains outputs of all Turing machines with \(n\) states and 2-colors. The size of \(T\) is not larger than \((4n+4)^{2n}\) (total number of Turing machines with \(n\) states), which shows how small portion of numbers \(\Sigma(n)\) can define.

Busy Beaver Turing Machines - Computerphile

The fifth and sixth busy beaver

Implementation of the fifth and sixth busy beavers

It is possible for multiple different Turing machines of a given size to be busy beavers. For example, the following 3-state, 2-symbol machines both produce the maximum of six 1's:

The Robot of Eternity Inn

Imagine an endless row of hotel rooms, and each room contains a lightbulb and a switch that controls it. Initially, all the rooms are dark. A robot starts at one of the rooms and has the ability to operate switches and move to adjacent rooms.

The robot has several states that it can be in, and each state determines what it should do based on whether the current room is light or dark. For example, a robot's rules could include these states:

• The "scared" state:
  1. If the room is dark, turn on the light and move to the room to the left.
  2. If the room is light, do nothing and go to the "normal" state.
• The "normal" state:
  1. If the room is light, turn off the light and move to the room on the right.
  2. Otherwise, go to the "scared" state.

One special state is the "stop" state. When the robot finds itself in this state, the process is complete.

Suppose a robot has n states (not including the "stop" state), and it stops. What is the maximum number of light rooms at this point?

This system is in direct allegory to Turing machines. The hotel is the tape, the robot is the Turing machine, and dark rooms and light rooms are 0 and 1 cells.

Example

Suppose n = 3. It turns out that the following ruleset wins:

• State "normal" (initial state):
  1. If the room is dark, turn it on and move to the right. Then go to state "beware".
  2. If the room is light, move left and go to state "frivolous".
• State "beware":
  1. If the room is dark, turn it on and move to the left. Then go to state "normal".
  2. If the room is light, move right. (Stay in state "beware".)
• State "frivolous":
  1. If the room is dark, turn it on and move to the left. Then go to state "beware".
  2. If the room is light, stop.

Uncomputability

With the right rules, Turing machines (TMs) can perform any computable operation despite their apparent simplicity. If a computer can calculate it with finite time and space (even in theory), a TM can also do it with finite (though probably longer) time and space. From a computability perspective, TMs and Intel processors are one and the same, although the former is much simpler. This important fact is known as the Church–Turing thesis. (Some equivalent simple devices include tag systems, certain cellular automata, SKI combinator calculus, and the Brainf**k programming language.)

It is easy to simulate a TM, but it is much harder to predict the behavior of a TM. This is because some TMs never halt, and the only way to test for that is to A) simulate them infinitely, or B) find and prove a pattern. Option A is physically impossible, and option B is difficult — how does a computer recognize arbitrary patterns? — and also problematic because many TMs exhibit chaotic behavior and do not have simple patterns.

The underlying issue here is that computers are no more powerful than TMs. To oversee arbitrary TMs, we need something more powerful than a TM. But, by the Church–Turing thesis, the computers we have today are computationally equivalent to TMs! Overseeing them, and thus computing \(\Sigma(n)\), is impossible.

The Deity of Eternity Inn

The set of all TM rule sets is countably infinite — TMs can be mapped one-to-one onto the natural numbers. In fact, let's label them TM #1, TM #2, TM #3, ... The exact mapping we use is unimportant, as long as it's possible for some TM to recover the ruleset given machine's number.

Note that each of the TMs in this sequence can have any finite number of states. There are \((4n + 4)^{2n}\) Turing machines for \(n\) states, so TM #1-64 might be the one-state machines, TMs #65-20801 all the two-state machines, TMs #20802-16798018 all the three-state machines, etc.

We augment the Robot of Eternity Inn by adding in a god. Three new special states are introduced: "ask", "yes", and "no". If the robot enters the "ask" state, it contacts Triakula, the Goddess of Large Numbers. Triakula then does the following steps:

• She counts all the light rooms and calls the result x.
• She simulates TM #x (which is impossible for anyone but a deity).
• If TM #x halts, she puts the robot in the "yes" state. Otherwise, she puts it in the "no" state.

We ask the same question as before. Given access to the god, what is the maximum number of light rooms possible the robot halts? This is a function of n, the number of states the robot has (excluding "ask", "yes", "no", and "halt").

This new extended situation is an allegory for oracle Turing machines. There are several equivalent definitions for them; some use an additional tape for the oracle, and some use different states than "ask", "yes", and "no".

Higher Deities of Eternity Inn

In the true spirit of googology, you can always go a step further. We can once again transcend oracle Turing machines by enumerating them TM_2, TM₂ #1, TM₂ #2, TM₂ #3, ..., where TM₂ stands for "level-2 Turing machine," or an oracle Turing machine. Transcending those we can get higher successive tiers of Turing machines: TM₃, then TM₄, then TM₅, ...

What do we do after all these levels? We can create a new "level-ω Turing machine", or TM_ω, that has access to all these levels of Turing machines. Take this table:

TM1#1	TM1#2	TM1#3	TM1#4	TM1#5	⋯
TM2#1	TM2#2	TM2#3	TM2#4	TM2#5	⋯
TM3#1	TM3#2	TM3#3	TM3#4	TM3#5	⋯
TM4#1	TM4#2	TM4#3	TM4#4	TM4#5	⋯
TM5#1	TM5#2	TM5#3	TM5#4	TM5#5	⋯
⋮	⋮	⋮	⋮	⋮	⋱

Although this table extends infinitely in two dimensions instead of just one, complete enumeration using the naturals is entirely possible using this winding pattern (although alternative schemes are okay provided they enumerate all cell entries):

TMω#1	TMω#2	TMω#9	TMω#10	TMω#25	⋯
TMω#4	TMω#3	TMω#8	TMω#11	TMω#24	⋯
TMω#5	TMω#6	TMω#7	TMω#12	TMω#23	⋯
TMω#16	TMω#15	TMω#14	TMω#13	TMω#22	⋯
TMω#17	TMω#18	TMω#19	TMω#20	TMω#21	⋯
⋮	⋮	⋮	⋮	⋮	⋱

We now have the definition of TM_ω #n, so we have successfully transcended all order-n Turing machines.

By the way, the maximum output of a TM_a with n states is Σ_a(n).

Beyond

By adding an oracle for TM_ω we can continue with what could be called "level-ω+1 Turing machine", or TM_ω+1, then adding another level of oracle we can get TM_ω+2, and so on. After we define TM_ω+a for all positive integers a we can use a trick similar to the one used in the previous section to construct "level-ω+ω Turing machine", which would be able to solve halting problem for all TM_ω+a.

With the help of ordinal numbers, we can continue this method further, and we could define TM_α for any countable ordinal number α. However, due to the increasing complexity of encodings of high order oracles, this becomes very impractical to work with, which is why mathematicians have been considering different ways of extending the notion of computability. One of them worth mentioning is an infinite time Turing machine devised by J. D. Hamkins and A. Lewis.

Computing values

Background

The behavior of 3-state busy beaver.

It is very hard to prove whether specific TM is a busy beaver or not. There exist \((4n+4)^{2n}\) TM's with n states to verify; this is Radó's \(N\) function. By exploiting symmetry and machines with inaccessible states, the count can be reduced to \((2n-1)(4n)^{2n-2}\), immediately eliminating the most trivial TMs.

It is exceedingly difficult to compute exact values of \(\Sigma(n)\), but some lower bounds can be found. There are two general approaches to the problem:

Bottom-up Technique (for small \(n\))

• Run through some set of possible Turing machines with \(n\) states.
• If a machine runs for more than X steps, for some large limit X, it is assumed to be running indefinitely and is ignored.
• Out of the machines we didn't ignore, find the machine that writes the most ones.

Top-down Technique (for large \(n\))

• Take the some quickly growing function \(f\).
• Simulate the computation of \(f(n)\) using a Turing machine.

For example, if we can compute the Goodstein function on a machine with 100 states, then it is very likely that \(\Sigma(100) > G(n)\) for relatively large n.

First of the methods could be potentially used to evaluate \(\Sigma(n)\) exactly for small \(n\), but the method quickly becomes infeasible. For this reason, the top-down technique is generally used.

Note that if \(f(n)\) is any computable function, \(\Sigma(n)\) will eventually outgrow it. The tricky question is when does the sigma function outgrow it, and this is difficult to answer without actually evaluating \(\Sigma(n)\).

The function's growth rate is believed to be comparable to \(f_{\omega^\text{CK}_1}(n)\) in the fast-growing hierarchy associated to Kleene's \(\mathcal{O}\) with respect to a system of fundamental sequences, where \(\omega^\text{CK}_1\) is the Church-Kleene ordinal, the set of all recursive ordinals. But it is unknown which function is faster if the enumeration of Turing machines in the definition of Kleene's \(\mathcal{O}\) is chosen to be generic.

Small values

Using the bottom-up method, the following values and bounds have been found:

\begin{eqnarray} \Sigma(1) &=& 1 \\ \Sigma(2) &=& 4 \\ \Sigma(3) &=& 6 \\ \Sigma(4) &=& 13 \\ \Sigma(5) &=& 4,098 \\ \Sigma\left(6\right) &\geq& \frac{-11 + 3^\frac{13 + 3^\frac{23 + 3^\frac{7 + 3^\frac{21 + 3^\frac{7 + 3^\frac{23 + 3^\frac{7 + 3^\frac{23 + 3^\frac{7 + 3^\frac{21 + 3^\frac{7 + 3^\frac{23 + 3^\frac{7 + 3^\frac{21 + 3^\frac{7 + 3^4}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{8}}{2} &> 10 \uparrow\uparrow 15\ \end{eqnarray}

For \(\Sigma(1)\), the proof is trivial. Single-state Turing machines either halt after the first step or continue moving out in one direction infinitely. For 2, 3, and 4, behavior becomes more complicated, but it is possible to prove the values. \(\Sigma(5) = 4,098\) follows from the fact that this amount of ones is written by the 5-state Turing machine which takes the longest to halt. The online collaborative project The Busy Beaver Challenge is dedicated to deciding the halting of every 5-state machine, and any machine which takes less time to halt also writes less ones. Beyond \(\Sigma(5)\), no exact values are known.

Using the 6-state Turing machine for the estimation \(\Sigma(6) \geq 3.514 \cdot 10^{18,267}\), Wythagoras and Cloudy176 proved that \(\Sigma(7) > 10^{10^{10^{10^{18,705,352}}}}\). Recent work by Shawn Ligocki and Pavel Kropitz in 2022, increased the lower bound of \(\Sigma(6)\) above the previous bound of \(\Sigma(7)\) to a 14-fold exponential expression (but within a 15-fold exponential) using a Collatz-like six-state Turing machine.

Green's machines

Milton Green proved that \(\Sigma(2n) \gg 3 \uparrow^{n-2} 3\), so e.g.:

\begin{eqnarray} \Sigma(6) \gg 3 \uparrow 3 = 27 \\ \Sigma(8) \gg 3 \uparrow\uparrow 3 = 7,625,597,484,987 \\ \Sigma(10) \gg 3 \uparrow\uparrow\uparrow 3 \\ \Sigma(12) \gg 3 \uparrow\uparrow\uparrow\uparrow 3 \\ \end{eqnarray}

\(\Sigma(6)\) is already known to be much, much greater than 27, so these lower bounds are very weak.

Turing machines discovered by Milton Green are known as Class M Turing machines, and k-th Class M Turing machine has 2k states. To construct the first Class M Turing machine (that simply increments the original number), we use the following rules:

0 _ 1 r 1
0 1 1 l 0
1 _ _ l halt
1 1 1 r 1

Next, every nth Class M Turing machine has 2n states and are constructed as follows:

0 _ _ l 2
0 1 1 l 0
T
2n-1 _ _ l halt
2n-1 1 _ l 2

Here T indicates the table for the (n-1)th Class M Turing machine, except that every state numbers in rules increased by one, and the halting rule replaced with

2n-2 _ 1 r 2n-1

Using Green's machines, Shawn Ligocki showed that there are better bounds \(\Sigma(9) > 10 \uparrow\uparrow 28\) and \(\Sigma(11) > 3 \uparrow\uparrow\uparrow 720618962331271\).

Graham's number vs. Σ(n)

Graham's number \(G\) is a famously large number, but even it is surpassed by relatively small values of the Busy Beaver function. Lower bounds for Graham's number include:

• 2010: \(\Sigma(64) > G\) (previously \(\Sigma(91)\) and \(\Sigma(72)\)) by "res001".
• 2013: \(\Sigma(25) > G\) by Deedlit11.
• 2016: \(\Sigma(18) > G\) by Wythagoras.
• 2021: \(\Sigma(16) > G\) by Daniel Nagaj.
• 2024: \(S(14) > G\) by Racheline.

Larger values

Wythagoras proved \(\Sigma(38) > f_{\omega2}(167)\), \(\Sigma(64) > f_{\omega^2}(4,098)\) and \(\Sigma(85) > f_{\varepsilon_0}(1,907)\).

Wythagoras also showed that \(\Sigma(61, 6) \gg H(3)\) and LittlePeng9 proved \(\Sigma(134, 7) \gg U(3)\) where H and U are Chris Bird's functions. Wythagoras has also shown that \(\Sigma(38,3) \gg f_{\varepsilon_0}(374,676,379)\).

A list of mostly unverified lower bounds for S(n) is available here (wiki page, may be unreliable).

Tape changes

Below are pictures illustrating tape changes for two, three, four, and five-state busy beavers.

The sixth busy beaver -test run-

6 state record holder (compressed to speed up and later exclude seven state-color combos)

5 state busy beaver and frantic frog

4 state busy beaver (also a frantic frog)

3 state busy beaver (not a frantic frog)

2 state busy beaver (also a frantic frog)

Related functions

Output tape of one or more busy beavers in Międzychód, Poland.

Tibor Radó introduced three uncomputable functions in his 1962 paper On Non-Computable Functions, which two of them grow uncomputably fast and are both referred to as the busy beaver.

S function (frantic frog)

Main article: Maximum shifts function

Tibor Radó introduced another uncomputably fast-growing function, \(S(n)\), defined in terms of the number of steps the champion machine took before halting. The busy beaver function may also refer to \(S(n)\), or even the pair \(\langle\Sigma(n), S(n)\rangle\). To avoid ambiguity, \(S(n)\) is referred to as the frantic frog function.

\(S(n)\) is defined as the maximum finite number of steps that an n-card, 2-color halting Turing machine can perform. Some authors refer to this function as the busy beaver function. For all \(n\), \(S(n) \geq \Sigma(n)\) because a machine that prints \(\Sigma(n)\) ones must undergo at least as many state transitions. It has been proven that the frantic frog function has a comparable growth rate as the busy beaver function, in the sense that \(S(n) \in \Sigma(O(n))\).

Analogously, \(S(n,m)\) is the maximum finite number of steps that an n-card, m-color Turing machine can perform.

A machine that produces \(\Sigma(n)\) is not necessarily the same machine that halts in exactly \(S(n)\) steps. The three-state busy beaver is never the three-card frantic frog, but it is also known that the four-card and five-card frantic frog is also a busy beaver. The following 3-state, 2-symbol busy beaver produces the maximum of six 1's but performs only 14 moves:

	A	B	C
0	1RB	0RC	1LC
1	1RH	1RB	1LA

In comparison, the following frantic frog runs for the maximum of 21 moves but produces only five 1's:

	A	B	C
0	1RB	1LB	1LC
1	1RH	0RC	1LA

\(S(643)\) is the smallest known value of the maximum shifts function that is undecidable in ZFC set theory as long as it is consistent. In other words, \(S(643)\) is the smallest known number undecidable in ZFC generated by maximum shifts function in the sense that the equality \(\underbrace{1+ \cdots +1}_m = S(643)\) is unprovable in ZFC set theory for any meta-theoretic natural number \(m\) as long as it is consistent.

The first such TM constructed had 7,918 states. Stefan O'Rear soon improved the bound to 1,919 states, and later to 748 states.

In 2023, Johannes Riebel slightly improved the bound to 745 states.

The bound was significantly improved to 643 and then 636 states in 2024 by Rohan Ridenour.

N_e function (number of stoppers)

Radó formulated the number of halting machines in terms of the set of all valid busy beaver entries \(M, s\), where \(M\) is the Turing machine, and \(s\) is the exact number of steps the halting Turing machine took.

The set of all halting n-card machines is denoted \(E_n\), and the cardinality of this set is denoted \(N_e(n)\). The number of n-card Turing machines that halt after exactly or not more than s steps are respectively denoted \(N(s,n)\) and \(G(s,n)\).

It is straightforward to show that for \(N_e(n) < N(n)\) for all \(n\). Radó used \(N\), \(G\), and \(S\) to show that \(N_e\) is also uncomputable.

Trivial extensions

One can naturally define function \(f\) so that \(f(n) = \Sigma^n(n)\), and then even construct a recursive hierarchy around it. However, to go sufficiently beyond \(\Sigma(n)\), we need more sophisticated paradigm.

Higher-order busy beaver functions

It is impossible for a Turing machine (or anything less powerful) to compute \(\Sigma(n)\) for arbitrary \(n\). However, we can create a second-order Turing machine, or oracle Turing machine, that has access to a black box ("oracle") that can determine when an ordinary Turing machine halts. The maximum number of ones that can be written with an n-state, two-color oracle Turing machine is denoted \(\Sigma_2(n)\) — the second-order busy beaver function. Unlike the simple variations above, this new function has a sizeable advantage over the original function.

Although a second-order Turing machine can solve the halting problem for first-order Turing machines, it cannot predict when it halts itself. Thus \(\Sigma_2(n)\) is uncomputable even with respect to an oracle Turing machine.

In general, if one defines the notion of an \(x\)-th order Turing machine in a suitable way for a recursive ordinal \(x\), \(\Sigma_x(n)\) can be computed using an order-\((x + 1)\) Turing machine, but not for any lower-order machines.

Unfortunately, there is no single, standard definition for these oracle Turing machines, so the higher-order sigma functions do not have any standard definition. At least, under any reasonable formulation of the notion of a higher-order Turing machine, well-orderings of ordinal type \(\omega_1^{\textrm{CK}}\) are uncomputable even with respect to an \(x\)-th oracle Turing machine for any recursive ordinal \(x\). In particular, the fast-growing hierarchy associated to Kleene's \(\mathcal{O}\) with respect to a system of fundamental sequences is uncomputable even with respect to such a higher-order Turing machine. Therefore it is not obvious at all whether higher-order busy beaver functions surpass the fast-growing hierarchy associated to Kleene's \(\mathcal{O}\) or not, although googologists tend to believe that the first-order busy beaver function is comparable with it and hence the second-order busy beaver function significantly surpasses it without reasoning.

Functional busy beaver

A lambda calculus based busy beaver BBλ is defined at OEIS entry A333479. Since it measures program size in bits rather than states, it allows many more values to be determined (37 so far versus only 6 for TMs), and the gap between the largest known value and values beyond Graham’s Number is a mere 13 program bits. A closely related variant defined at entry A361211 can be directly expressed in terms of Kolmogorov complexity, which Mikhail Andreev argues in https://arxiv.org/pdf/1703.05170 is crucial for applications in Information Theory.

Pseudocode

This is an "escape time algorithm" for finding lower bounds — if a Turing machine does not halt after t steps, it is assumed to go on infinitely. Setting t to infinity yields the correct value for BB(n), but this obviously cannot be done on a computer.

function BB(n, t):
    max := 0
    for each rule set r:
        simulate a Turing machine with ruleset r for t steps
        if the machine halts:
            if number of 1's printed > max:
                max := number of 1's printed
    return max

This method was used to compute \(\Sigma(3)\) and \(\Sigma(4)\) as well as the bounds for \(\Sigma(5)\) and \(\Sigma(6)\).

The "bottom-up" method of finding bounds by constructing Turing machines with large outputs is a tricky problem. So far it has been done only by humans, and computer-assisted proofs bounding \(\Sigma(n)\) for large \(n\) are yet to be explored. An example of such a program is Skelet's BBprover program.

Monotony

\(\Sigma(n) = m\) is clearly monotonical. At least, if we have \(n\)-state Turing machine that halts on symbol 0, then for \(n+1\)-state Turing machine we can replace the halting rule with the rule that goes to the state \(n+1\). In the state \(n+1\), we go to the halting state and write 1, making \(m+1\) ones and proving that \(\Sigma(n+1) \geq m+1\). If we have \(n\)-state Turing machine that halts on symbol 1, we do the same except that in \(n+1\)-th state head goes to the left or right corner of 1's, writes 1 and halts.

See also

• Computable function
• Fast-growing hierarchy
• loader.c
• marxen.c
• Xi function
• Rayo's number

External links

• Busy Beaver on Wikipedia
• Anthony Morphett's Turing machine simulator
• Anthony Morphett's Turing machine simulator - old version

Uncomputable numbers

Busy beaver numbers (based on Turing theories): \(\Sigma(1919)\) (1919th busy beaver number) · Fish number 4 · \(\Xi(10^6)\) · \(\Sigma_{\infty}(10^9)\)
Rayo's numbers (based on Set theories): Rayo's number (Rayo(10¹⁰⁰)) · Fish number 7 · BIG FOOT (FOOT¹⁰(10¹⁰⁰)) · Little Bigeddon · Sasquatch · Large Number Garden Number
Miscellany: Hollom's number · Oblivion · Utter Oblivion · (Ultimate Oblivion) · (Hyper oblivion) · (Ultra Oblivion)