Is busy beaver the fastest growing function known to man?

Question

I just had this interesting question. What is the fastest growing function known to man? Is it busy beaver?

We know functions such as $x^2$, but this function grows slower than $2^x$, which in turn grows slower than $x!$, which in turn grows slower than $x^x$. We can then combine functions, to have $(x^x)!$ that grows faster than $x^x$, and so on.

Then we arrive at recursive functions such as Ackermann's function $A(x,x)$ that grows much faster than $(x^x)!$. Then people though about busy beaver $B(x)$ function that grows even faster than Ackermann's function.

At this point I haven't heard of any other functions that grow faster than busy beaver. Does it mean that there are no other functions that can possibly grow quicker than busy beaver? (Aside from factorial of $B(x)$ and like $A(B(x), B(x))$, etc.)

there is a way to formalize and order "fast growing" math functions in terms of TCS time & space complexity which unfortunately doesnt appear to have been done (much?) so far. asked a similar/related question on tcs.se which lists some of these functions from math theory by name incl ackermanns fn but unfortunately was closed. but maybe if it gets upvoted enough it could be reopened, wink wink =) ... but seriously, think it could lead to important bridge thms someday. — vzn, Sep 23 '12 at 02:20
what nobody has mentioned much so far is that it doesnt seem to make much sense to study the relative growths of functions that are uncomputable such as BB but there is a notion for "well behaved" computable fns called time and space constructibility and those do have an ordering (but of course not a maximum...) please note that BB and Ackermanns are in a sense incomparable because the former is uncomputable and the later is computable. see also blums speedup theorem and the gap theorem. — vzn, Sep 23 '12 at 15:09
@vzn Why would growth only make sense for computable functions? Asymptotic growth is a mathematical concept unrelated to computability at all. — Raphael, Sep 23 '12 at 16:07
hard to get the details all in a comment. agree with aarons answer below. the point is that BB and ackermanns fn are not at the same "order" in the "undecidability hierarchy". fns higher in the "order" are in a sense "larger" than the lower order fns. ie guess BB might be seen as 1st order & ackermann as 0th order. — vzn, Sep 23 '12 at 16:19
@vzn for BB the growth rate implies uncomputability. but uncomputability does not imply high growth rate. — Sasho Nikolov, Sep 23 '12 at 16:41
SN .. example? the point is, how do you "compare" the growth of a recursive function like ackermann and an uncomputable one like BB? which one "grows faster"? the only plausible answer is that BB "grows faster" than ackermann right? and in some sense doesnt any uncomputable function "grow faster" than a computable one? or are they incomparable? ie to some degree its "apples and oranges" wrt current theory — vzn, Sep 23 '12 at 18:50
Hi @vzn. The function $f$ such that $f(n) = 1$ if the $n$'th Turing machine halts, and $f(n) = 0$ otherwise is uncomputable but grows more slowly than the Ackerman function. On the other hand, it is easy to prove that for some fixed constant $c$, for all $n > c$, BB$(n) > $ Ackerman$(n)$. If this were not the case, you could solve the halting problem by running a turing machine $T$ with description length $n$ for only Ackerman$(n)$ steps and seeing if it halted before then or not. — Aaron, Sep 23 '12 at 21:49
@vzn maybe you have another idea of "grows faster".. what I (and I believe others) mean is the partial order given by $f = \omega(g)$. — Sasho Nikolov, Sep 24 '12 at 02:19
ok in this case there is a way to compare the two fns but there are surely other contrived examples where the intrinsic undecidability of one function messes up the possibility of comparing another without it... true? — vzn, Sep 24 '12 at 02:50
also suspect that actually proving BB(n) > Ackermann(n) is not provable somehow in some theory or model. but dont have ref on that right now. in other words, am suspicious of glitches that can arise in comparing a recursive vs a recursively enumerable sequence (ie computable vs uncomputable) — vzn, Sep 24 '12 at 15:45
"but there are surely other contrived examples where the intrinsic undecidability of one function messes up the possibility of comparing another without it" -- why would it? It is easy to "mess up" comparison of functions in terms of Landau notation (think oscillation) but that works for computable functions, too. — Raphael, Sep 25 '12 at 06:48
this question on growth of non computable fns vs computable ones seems related. mummerts answer about the hyperimmune-free degree seems to sketch out some of the theoretical qualifications for comparisons — vzn, Sep 25 '12 at 17:43
another analogy is the weierstrauss function which is "everywhere continuous but nowhere differentiable." ie just because a fn is defined for all values of x like BB does not mean that all operations on it (such as comparison with other fns) can be defined. it seems that comparing function growth rates might have some similarity to differentiability. the blum axioms also show there exist "pathological" languages that resist analysis by asymptotic analysis. lots of advanced theory on related stuff.... — vzn, Sep 25 '12 at 17:53
@vzn and yet, computability implies next to nothing about growth rates. — Sasho Nikolov, Sep 30 '12 at 23:40
it is not easy to describe but the Lafitte paper cited below seems to show that growth rate and uncomputability of the busy beaver fn are tightly coupled & cannot be fully regarded as independent phenomena.... if anyone has another ref that indicates otherwise plz cite it.... — vzn, Oct 01 '12 at 03:27
Known to man? There may of course be some that only women know about. Unfortunatly most computer scientists are men so these never make it into the literature. — alan2here, Nov 23 '18 at 18:44

score 51 · Answer 1 · answered Sep 22 '12 at 22:27

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The busy beaver function grows faster than any computable function. However, it can be computed by a Turing machine which has been given access to an oracle for solving the halting problem. You can then define a "second order" busy beaver function, that grows faster than any function that can be computed even by any Turing machine with an oracle for the halting problem. You can keep doing this forever, building up a hierarchy of ever faster growing busy beaver functions.

See Scott Aaronson's excellent essay on this topic, Who Can Name the Bigger Number?.

answered Sep 22 '12 at 22:27

Aaron

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Do you have a resource/reasoning as to why an oracle TM for HALT_TM can solve busy beaver? – Ryan Dougherty Feb 22 '15 at 04:44
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Ryan: Solving the halting problem is (computationally) equivalent to knowing Busy Beaver. 1) Does program[length=n] halt? Simulate it for BusyBeaver(n) steps. 2) What is BusyBeaver(n)? For every program of length<n, throw it away if it halts, and take the max score amongst the others. – ninjagecko Dec 13 '15 at 16:01
@ninjagecko do you mean not halts – Christopher King Jun 18 '17 at 02:31

score 36 · Answer 2 · answered Sep 22 '12 at 23:58

There is no such thing as "the fastest growing function". In fact, there is even no sequence of fastest growing functions. This was already shown by Hausdorff. Given two functions $f,g\colon \mathbb{N} \longrightarrow \mathbb{N}$, say that $g$ grows faster than $f$ if $$ \lim_{n\rightarrow\infty} \frac{g(n)}{f(n)} = \infty. $$ Given a function $f$, the following function $g$ grows faster than $f$: $$ g(n) = nf(n). $$ Given a sequence of functions $f_n$, the following function $g$ grows faster than all of them: $$ g(n) = n \max_{m \leq n} f_m(n). $$ A natural question to ask is whether there is a "scale" of fastest growing functions. This is a well-ordered set of functions $g_\alpha$ which is "cofinal", that is, given any function $f$, there is a faster-growing function $g_\alpha$. (Instead of a well-ordered set, we can equivalently talk about a chain, that is, any two functions in the set need to be comparable.) The existence of a scale is independent of ZFC: assuming CH, there is a scale, while in Cohen's model that falsifies CH (adding $\omega_1$ reals), no scale exists.

score 5 · Answer 3 · edited Sep 28 '12 at 16:21

Other answers address the question directly. For more and deeper background, this paper by Lafitte on the subject considers the larger context of busy beaver-like functions. It also has some results and theorems fitting the idea into a more general framework. It shows that (informally) "busy beaver-like functions" have a close connection to Chaitin incompleteness phenomena (Theorem 2.1). It also shows that there are theories that are not "powerful" enough to "comprehend" the busy beaver-like functions, i.e. they are unprovable in those theories due to Godel-related incompleteness. It shows the idea of assuming busy beaver-like results as axioms and a logical progression of theories that results similar to ideas originally envisioned by Turing.

[1] Busy beavers gone wild by Grégory Lafitte. Abstract:

We show some incompleteness results à la Chaitin using the busy beaver functions. Then, with the help of ordinal logics, we show how to obtain a theory in which the values of the busy beaver functions can be provably established and use this to reveal a structure on the provability of the values of these functions.

the other answer is completely different. hmmm, speaking of "emphasis on language", would an example of that be a moderator saying "hell no"? anyway the abbrevs can be seen as a generous gift to people who like to earn +2 for edits =) — vzn, Sep 28 '12 at 16:48
You say yourself that this does not answer directly, so why did you not post as a comment? — Raphael, Sep 29 '12 at 10:09

score 0 · Answer 4 · edited Sep 28 '12 at 16:23

The Hartmanis-Stearns time and space hierarchy theorems prove that there is no "fastest growing" function in terms of time or space because the scale is unbounded. But it does give an ordering such that all "well behaved" computable/recursive functions can be compared. But many "fast growing" math functions do not seem to have been evaluated in terms of time/space complexity so far despite it being a somewhat obvious or even glaring theoretical "gap" to fill. Doing so could lead to important "bridge theorems".

Is busy beaver the fastest growing function known to man?

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