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Questions tagged [busy-beaver]

The Busy Beaver problem is about finding the sequence of n-state Turing Machines writing the most 1's on a tape.

Tibor Radó defined Busy Beavers as Turing Machines that among all halting Turing Machines with the same number of states write the most 1s on the tape, i.e. the $nth$ Busy Beaver writes the most $1$s among all $n$-state Turing Machines.

The Busy Beaver function $\Sigma(n)$ maps $n$ to the number of $1$s written by the $nth$ Busy Beaver. It is proven to be uncomputable as well as growing faster than any computable function.

Radó also defined the Max Shift function $S(n)$ that maps $n$ to the greatest possible number of state shifts made by halting $n$-state Turing Machines which is equivalent to the greatest number of steps. Like $\Sigma(n)$ this function is uncomputable. Since writing a $1$ on the tape requires a shift, $\Sigma(n) \leq S(n)$.

Discussions about the Busy Beaver problem often involve the Max Shift function, since it has similar properties and is easier to reason about.

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Goldbach Conjecture and Busy Beaver numbers?

Background: I am a complete layman in computer science. I was reading about Busy Beaver numbers here, and I found the following passage: Humanity may never know the value of BB(6) for certain, let alone that of BB(7) or any higher number in the…
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Computation of busy beaver function

The busy beaver max shifts function, $S(n)$, has known values for $n\leq4$. Is there some basic, structural reason why it's inconceivable that we will ever find $S(n)$ for $n>4$? What is so different about $n=4$ than $n=5$? Or $n=6$? Somewhere…
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Are there any functions with Big O (Busy Beaver(n))?

So, I was reading this article by Scott Aaronson on big numbers, and he mentioned that the Busy Beaver sequence increases faster than all sequences computable by Turing Machines. Faster than exponentials, faster than the Aaronson sequence, and…
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Wanted: Concrete Example of Busy Beaver Holdout

I understand from the Wikipedia page on the Busy Beaver problem that the Busy Beaver values for 5-state 2-symbol (quintuple) Turing-machines have not been determined, because there are 'holdout' machines whose halting behavior is as of yet…
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What is the minimum acceleration of a macro machine?

In the context of Turing machines, consider a $k$-sized macro machine (k-MM) which operates on groups of $k$ symbols at once. This is a common optimization in the search for Busy Beavers, explained e.g. by Holkner: Acceleration Techniques for Busy…
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Busy Beaver machines on semi-infinite tape

The Busy Beaver problem is to find the largest number of non-blank characters that are printed by a terminating Turing machine of no more than a given size on the blank input. The usual Busy Beaver machines run on a blank tape that is infinite in…
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Is a secondary TM sufficient to detect all loops?

Procedure: Start a secondary TM in parallel with the first, but have the second perform exactly 1 step each 2 steps the first TM performs (i.e. it runs at half speed). If the second machine ever reaches the same configuration as the first, a loop…
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What is the smallest $n$ such that $BB(n) > $Graham's number?

BB represents the busy beaver function here. Do we even have any idea of what order of magnitude $n$ would have? Is it possibly around 10, or more like 1000?
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Short SK combinator expression with long reduction / Busy Beaver for SK combinators

Question (short and simple version): Can anyone suggest a very short SK combinator expression with a ridiculously long, but still terminating, reduction path (ignoring loops)? Question (longer version): I note that my question above seems closely…
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Uncomputability of Busy Beaver Function

https://en.wikipedia.org/wiki/Busy_beaver#Proof_for_uncomputability_of_S.28n.29_and_.CE.A3.28n.29 So this is wikipedia's proof of why Busy Beaver Function is uncomputable. But I don't get two things. How do you know that < Create_n0 | Double |…
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Understanding proof for Busy Beaver being uncomputable

I found this proof on http://jeremykun.com/2012/02/08/busy-beavers-and-the-quest-for-big-numbers/ and have highlighted the part I don't understand in bold. (BB(n) is defined as the number of steps made by n-state Busy Beavers.) Theorem: BB(n) is…
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Busy Beaver problem - Proof by contradiction

I am trying to understand a proof regarding the Busy Beaver problem that uses a proof by contradiction approach to show $\sum(n)$ is Turing-uncomputable: Find $\sum(n) = max \{\sum(M) | M \in M(n) \}$ Where $n$ : Number of states $M(n)$: Set of…
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Given an n-state TM, can we construct an m-state TM (m>n) to determine if it halts?

BB(n) is roughly the maximum number of new states an n-state TM can run into without halting. So for a particular n, if we know BB(n), then we can find out if an arbitrary n-state TM halts by running it for BB(n) steps. So given some n-state TM, can…
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Busy beaver with Turing Machine

Basic setups: A Turing machine M operates on a doubly infinite tape. The tape is divided into cells. Each cell can hold either a 0 or a 1 (meaning, we will consider only TM’s with two symbols). The machine has a head, which is positioned on top of…
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Is there a difference between extremely slow growing functions and constants with respect to computable functions?

So let's say we have the function $f(n)$ that gives $k$ such that $k$ is the smallest number that gives a busy beaver function $B$ value from input $k$ that is greater than $n$. Or more succinctly the smallest $k$ with $B(k) > n$. Can there be…
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