The Insane Ackermann Function 

Ackerman has only one function: to pwn titans

New favourite word unlocked: quinquagintillion

pov: you've been inventing your sorting algorithms and ended up with complexity of A(n) ☠️

zeke won't enjoy studying this function

"Ackermann function" "Colossal number" Aot mentioned

in the 50's scientists once attempted to calculate A[5], but the piece of paper on which the calculations were performed quickly turned into a black hole

You've def earned it for summoning a titanic fandom.

And after tetration (for 4) You get pentation (for 5) Pentation is just a tower of tetration Tetration of tetration Tetration 5 times is pentation of five It is represented by a 5 with a small five on its left side bottom (5 with ⁵ at bottom left)

Is this by Levi or mikasa?!

In other words, the Ackerman function would be useful for describing the weight of your mum.

And SQUARE it is mad

It's actually very simple and beautiful idea, suppose you give me a list of functions f_n(x) with x natural, each growing faster than the one before. Now I could apply the diagonal n -> f_n(n) Which grows faster than each f_n. To get Ackermann you just choose some functions f_n, there are variations, the one I learnt was f_1(x)=2x and we get f_(n+1)(x) by iterating f_n(f_n(...f_n(1))...) x times, so that f_2(x)=2^x, f_3(x)=2^2^...^2 x times etc. Hope you have a wonderful day😊

You got this completely wrong. This has nothing to do with the Ackerman function, except that both grow fast. The Ackerman function is a recursive function with 3 arguments. And it grows even faster and takes much longer to compute.

Its main use was to demonstrate a total computable function that is not primitively recursive. The original has three arguments. You essentially showed A(n, n, n) here, where all arguments are equal.

What is A[pi]? What would the analytic continuation look like?

Ackermann Function with Graham's number as both of the parameters.

I drove past Qinquagintillion on my way to Muskoka

KENNYYYY!

Ah yes, the Ackermann function, the mathematical attempt to scale Levi's power level.

Didn't know Mikasa was this good at math

How would A(0) be? I can't think of any operations below addition, and A(0) = 0 seems like an incomplete answer...?

I'm glad we know how many atoms there are in the universe, I would have thought that would be impossible to know.

''This escalated quickly''

Good. Now extend it to the rationals

I had never seen Ackerman's function actual meaning besides being purely recursive. This content is good stuff!!

Actually, A(4) is 4^1.34e+154, the number can't even be written, there's not enough of anmything to write it

Well, that escalated quickly

levi going crazy w this one

Going from 27 to roughly 1 quinguagintillion in one go is kinda crazy

This is why mathematicians ought to be licensed.

You can also write tetration with only 2 symbols. Instead of putting the "exponent" on the top right of the base, you write it on the top left. Quintation being the next step and you write it either with three arrows as you showed or on the bottom left. Pentation on the bottom right

it´s used to slay titans and protect eren

I was not expecting this ackerman

This is not the full ackermann function. The full process is described recursively, as this version relies on knuth arrows, which ackermann does not have credit to. Ackermann's function relies on the "Ackermann's Worm" which is a weaker version of the Beklemishev Worm. You begin with a number, and a "row" number. eg 2 [1] 1. If list empty, return row 2. Else, increment row, 3. If last term = 0, delete it 4. Else, replace the last term with "row" copies of last term - 1 Process for A(2,1) is as follows: 2[1] 1,1[2] 1,0,0,0[3] 1,0,0[4] 1,0[5] 1[6] 0,0,0...[7] [14] A(2,1) = 14

Its threatening to watch that two pop up on the final image with no volume

even cooler is the disjoint set union runtime can be optimized to inverse ackermann, which unlike ackermann grows extremely slowly, becoming near O(1)

Can it take negative or non-integer inputs?

It's really dope hearing someone talk about tetration! In my opinion, it's a very under-discussed operation!

GREAT VIDEO 📹 👍 👏 👌

For something really fast-growing, check out Goodstein sequences.

Benannt nach einem gewissen Sparkassenvorstand und beschreibt die Geschwindigkeit mit der Gelder in schwarze Kassen verschwinden?

Thats a crazy sequence

Ackerman function leads to a colossal number....Hmmmmmmmm. KENNYYY!

The *true limits* of computation are the Busy-Beaver fns. Ack was just the 1st fn to prove "non-primitive" recursive fns exist and are computable

quinquagintillion is a word I’ve never heard before

The reason why it tests the limits of computation it's not because the numbers are large but because it is a small simple testable example of a non-primitive recursive function

Bro summoned the entire aot fandom 😂

"That escalated quickly." meme.

As soon as I heard, "Tower of Powers" I immediately though of Frank Zappa and "Bobby Brown".

Great video. I remember implementing karatsuba and dft integer multiplication for a final project in my advanced algorithms class. I have to say I felt quite smug after your karatsuba performance was equally as shit as mine 😂

Me and the boys about to pull up to the ackerman function

So what is A(inf) or A(-inf) or even A(0.5) would be?

Me typing 10 into a computer and causing a instant black hole

Everyone gangsta till the ackerman function of 1/2

In beaf terms, A(n)={n,n,n-2}

A[5] ?

We can finally measure how heavy caseoh is

i wonder how quickly the equivalent for the commutative hyper-operations is. suppose you have some binary operation + and you want to create a hyperoperation •, then we define the commutative hyperoperations as: ln(a•b)=ln(a)+ln(b) clearly this turns addition into multiplication, but it also turns multiplication into a new operation, which is like a commutative version of exponentiation. this new operation is a^ln(b)=b^ln(a)=exp(ln(a)ln(b)) then just do the ackermann function thing with this. not sure if itd work well because the domains of the commutative hyperoperations are weird

do ackermann numbers pls

Well well well, if ain't that escalated quickly.

What is Quagmire doing here?

Thank you

I'm curious what's before one in that function

I came up with this function in like grade 6 or something after learning about exponentiation.

"and square it" oh you ate that

levi ackermann

What’s the use of this mysterious function? Except being stupidly large ??!

Why stop at 4?

how does this relate to the two parameters version?

I think you calculated my brain cells in the first equation 🥹

when i heard the number of the particles in the universe _squared_ my jaw literally fell off

Ackermann sequence: 2 4 27 10^10^153

Actually the 4 is being raised to 4^256 which is the quinquagintillion. So it's even worse than they stated.

What about smth like A[3.5]? Seems unreasonable to have the 3.5th hyperoperation and do it 3.5 times

What's the next higher function?

But Tree(3) grows MUCH faster. Its insane.

A(0) should be 1 because addition of natural numbers is just repeated addition of 1, so it's 0+1, and A(-1) could be interpreted as -1, using an even simpler operation which just returns its input. Going further, A(-2) could be 0, using the simplest operation which returns the same thing and takes no input. A(-2) = 0 A(-1) = -1 A(1) = 1+1 = 2 A(2) = 2·2 = 4 A(3) = 3³ = 27 A(4) = 4^(4^(4^4)) = 4^(13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,096) From this extension, it's clear that A(-3) and below would be meaningless, but could we define A(½), A(3⁄2), or A(5⁄2)?

Well, it's escalated quickly.... as they say.

So a[5] ... and so on ?? It's a virtual cliffhanger! That's computationally complex!!!

note: this is the single-argument ackermann function the actual function is 3-arguments long

Another way to think about universe squared is an entire universe for ever particle in this universe.

waiting for the release of nigganilion

and if you repeat the tower of powers, you get funky 16ths

“Ah, no one can visualise this thing” Meanwhile this guy:

Is this function defined for decimal numbers

Tower of power mentioned 🎺

What happens at A[5]?

Must be Australian, because that's a Huge Ackermann.

So A(n) = n{n-2}n?

What is A(0) ? Is this Ackermann function continuous?

A new busy beaver value was recently proven BB(5)=47,176,870 The Busy beaver function is the fastest growing function which is still computable and grows even faster than this.

erwin dancho would be proud

My math shower thoughts fr

I mean, the limits of computation as in "if you try to calculate a number with more digits than there are atoms in the observable universe you can't".

And this yet discourages tenacious tricksters

Continue the sequence: 2, 4, ?, ?

Is there an analytic continuetion of that function to Real or complex numbers?

Busy Beaver would like to have a word with mister Ackerman

What about A[G(64)] i.e, Graham's Number??

Well my idea was already found