He's could be the David Attenborough of computer science. "...And here, we observe the C++, in its natural habitat..."

@rykehuss3435

7 жыл бұрын

Nailed it

@xplinux22

7 жыл бұрын

"As you can see, this young male stack frame is prowling through the forest of dense, tangled registers calling for a mate. Alas, with the stack currently empty, his efforts are unfortunately futile." *cue safari music*

@TampaCEO

6 жыл бұрын

That's who he reminds me of!

@douwehuysmans5959

5 жыл бұрын

"but alas, the buffer overflowed and as you see can see the instruction pointer makes a miraculous jump towards the kernel"

@JamesSpeiser

5 жыл бұрын

hahaha

Absolutely fascinating. I can only imagine how much more fascinating it would be if i knew what he was talking about.

@Firestar-rm8df

7 жыл бұрын

basically it's interesting if you like studying theoretical software. It's some interesting concepts, but it's utterly useless for actually writing software. This would do essentially the same thing: #include . . . double time; increase(time, time); sleep( time ); . . . double increase( double time, double temp_time) { time = pow(2, increase(time, temp_time-1); return time; } it's clunky but I'm not going to bother optimizeing and cleaning it up. it basically sleeps for 2^time!, or two raised to time factorial. Edit: at least as far as wasting cycles it would behave simularly

@SteveUrlz

7 жыл бұрын

lol, same

@sidharthcs2110

6 жыл бұрын

Christian Herrera So I'm not alone

@Confucius_76

5 жыл бұрын

yes! loved it though

@Ligma_Male-ub1ud

5 жыл бұрын

Low key just copied and pasted this into an ide I’m in CS1 , but changing my degree plan after this semester

i don't alwys use recursion...but when i use recursion i don't always use recursion.

@RKBock

6 жыл бұрын

to understand recursion, you must first understand recursion

@shadowagent3

5 жыл бұрын

@@RKBock to write a recursive function you must first write a recursive function to write a recursive function

@nandkishorenangre3541

5 жыл бұрын

@@shadowagent3 To write a recursive comment some one else must add up the next recursive comment

@JamesSpeiser

5 жыл бұрын

holds beer

@mohamedb737

4 жыл бұрын

the only way to understand recursive function is to do it recursively

I once tried to compute just a slightly bigger ackerman call like ack(3,3) computed in an instance and I tried ack(3,4) i think, but it exhausted the call stack. So I wrote the expansion to disk, to parse the deferred chain. The whole computation, even though it was just text, created a 10gb textfile while the slightly smaller call ack(3,3) just needed about 15lines of deferred chains. gives you a feeling of how insanely fast the output grows. This is due to the ackerman function actually abstracting hyperoperation. Where ack(3,3) was exponentiation, ack(3,4) was tetration. To put it simply: it is rasing exponentiation itself to a power. Then ack(3,5) would be even more extreme, a pentation - rasing the rasing of exponentation to a power, or just rasing tetration to a power.

@brandenjames2408

8 жыл бұрын

Keist Zenon just to blow your mind even more, pentation isnt raising tetration to a power, its raising tetration to a tetration

@oliviapalmu5585

8 жыл бұрын

So it acts like Graham's number?

@carbrickscity

2 жыл бұрын

Similar, but Graham is kinda one level above Ackermann.

@DenisovichDev

2 жыл бұрын

I wrote 3, 4 in C and it was actually instantaneous. Although 4, 4 caused a buffer overflow

@macchiato_1881

Жыл бұрын

@@DenisovichDev he was parsing the deferred chains. You didn't so that why yours was instant.

Ack(-1, -1); ... The thread running this code has never been seen again.

@lukitree

8 жыл бұрын

Herp Derpington haha, yeah gotta add error checking in there ... #include ... assert (m >= 0 && n >= 0); ...

@321671999

8 жыл бұрын

lukitree don't use tho, use

@OverSoft

8 жыл бұрын

Herp Derpington Define the int's as uint and it'll protect you from this nonsense. :P

@MrMiljaker

8 жыл бұрын

+Herp Derpington Actually, I still think it would terminate, because once m and n get small enough, the sign bit will flip and suddenly we're back at a positive number.

@jakubcislo8029

6 жыл бұрын

Nah, it just produces StackOverflowError.

Sir, thank you for this. From app engineers to web developers, I’ve always felt that it is vitally important for programmers to study the mathematical properties that make our projects work. To do otherwise, to me, is like trying to be a materials engineer without having an understanding of the physical and chemical properties of your raw materials.

I checked the size of 3 * 2 ** 65333 in python the number itself is 280 lines in windows powershell

@EmptyKingdoms

4 жыл бұрын

And it probably just filled the final lines with a bunch of zeroes because it cannot compute that large an integer.

@dramforever

4 жыл бұрын

@@EmptyKingdoms Python can

@EmptyKingdoms

4 жыл бұрын

@@dramforever using what tools?

@dramforever

4 жыл бұрын

​@@EmptyKingdoms For the purpose of explaining, consider this: In principle, you can represent a large number by its digits, and turn the multiplication with pencil and paper method ('schoolbook multiplication') into a program. It can handle anything that your computer can store. (280 lines of characters is like *nothing* compared to GBs of RAM) It won't be fast, but it shouldn't be too slow either. Now you can just start from 3 and keep multiplying by 2, which is again slow but shouldn't be unbearable. (You can even specialize and just write a 'multiply by single digit number' function) There are faster ways to do multiplication (google 'integer multiplication algorithm'), and also for exponentiation, but I'm not really familiar with those, but they are pretty well researched by others so maybe just consult what others already wrote. But actually, big integer implementations, or arbitrary precision integer implementations (look ma, no precision loss for final digits) use binary internally (bytes and words are just chunks of binary bits). 3 * 2**65533 is really neat in binary, so the real deal is converting it to decimal. Again, someone else already figured it out like ages ago. Just google 'radix conversion algorithm'. But why computers sometimes can't handle large numbers? Because really they just can't handle them *fast*. Like *real* fast. Your CPU has dedicated *hardware* circuitry to handle not too large integers and not too wild floats, so that if your program does not need 'weird' computation like computing a very large number it ends up blazingly fast, and most programs actually do just need normal numbers.

@EmptyKingdoms

4 жыл бұрын

@@dramforever thank you.

"and within that VILE second argument.." This is too funny! I like this guy.

this dude's voice is absolutely soothing.

It'd be funny if the program returned -1/12.

@CircusBamse

10 жыл бұрын

unfortunately it has to be an integer and -1/12 is not so something would've gone wrong, but I do like your thinking though.

@son-tchori7085

10 жыл бұрын

Minox sta Some would argue that summing natural integers to infinity would be integer as well... : kzread.info/dash/bejne/qWF9mLqNhr2-p9o.html

@kevinmichelet91

10 жыл бұрын

Minox sta summing positive integers is not supposed to be negative. But 1 + 2 + 3 + 4 + 5 + ... = -1/12 (see numberphile) So if string theory can incorporate that, why not ?

@Falcrist

10 жыл бұрын

You only get -1/12 if you go all the way to infinity. Whatever stupendous number it puts out will be just as far from infinity as 1.

@cemerson

10 жыл бұрын

Minox sta But adding all the natural numbers must be an integer, right?

You know you're a true mathematician when you get scared by how big a number can get. Also makes you appreciate how big infinity actually is. It's bigger than any ack out there.

@leeustadh2735

8 ай бұрын

any ack out there is finite, regardless of how big it is. Infinity on the other hand literally means not finite.

@JordanMetroidManiac

4 ай бұрын

Well, infinity isn’t bigger than any number. Infinity is an undefined value, so it cannot be compared to numbers. For example, you may be tempted to say that 1/0 is infinity because division by a value extremely close to zero results in an extremely huge number, but it’s technically undefined. The reason why it’s crazy to try to comprehend the “hugeness” of infinity is only because that for any number you come up with, you can always add one to it (or do some crazy operation) and it’s even bigger. It goes from incomprehensible to even more incomprehensible. Just wanted to say that infinity isn’t comparable to numbers, but more just an undefined thing that *does not exist in the set of real numbers.* (That’s an obvious fact, but with that in mind, it should also be obvious that saying infinity is bigger than a number is just as senseless as saying infinity is smaller than a number, because they’re apples and oranges.) Moreover, since 1/x has the limit of “positive” and “negative” infinity as x approaches zero from the right and left side respectively, that should clearly indicate that there is only one infinity, and that is the state of being undefined, not existing within the set of real numbers. When 1/x shoots down toward “negative” infinity from the left and comes back down from “positive” infinity after x passes through zero, that’s the function 1/x escaping the set of real numbers to a single place, called undefined, and returning to the set of real numbers in the same direction in which it exited. That said, infinity seems like it exists on either “end” of the real number line, so that would mean it’s just as correct to say that infinity is smaller than every single number you can think of. But if that was true, then it’s a contradiction. Infinity cannot be both bigger than and smaller than every real number, and hence it does not exist on either end of the real number line. The existence of infinity is entirely outside of the set of real numbers. So, finally, infinity cannot be compared to real numbers. It’s not bigger than any ack out there. It’s just that you can construct a number arbitrarily large enough to be bigger than any ack out there, and that’s what is mind-boggling. :)

@charlesmurtaugh3771

Ай бұрын

“It’s longer than you think, dad!”

Ah, nothing like theoretical computer science.

the result will probably be 42

@fuppetti

9 жыл бұрын

Jerónimo Barraco Mármol ack(ack(life, universe),ack(everything, douglasadams))

@truthseeker7497

8 жыл бұрын

+Jerónimo Barraco Mármol A distinct possibility. What was the question?

@nand3kudasai

8 жыл бұрын

+Dan Overlin To know the question we would need a computer and an algorithm far more complicated. It would probably take ten million years to know.

@truthseeker7497

8 жыл бұрын

Another 5 minutes and we would have had the answer . . . damn the Vagons! The best laid plans of mice . . .

@blinky840

7 жыл бұрын

I would give you an upvote but you are already at 42 and I would hate to ruin that.

It's been 15 years I'm working professionally in computer science, writing code and doing something useful with it, thinking that I'm kind of understanding how it all works, from the theory of electronics, to the end user using it, I'm mind blown that some mathematician from the early 20s have already thought of all of this. Plus kudos to mr Brallsford explaining it like it's some basic course by its concise explaination. Love from France

I love listening to Professor Brailsford because it makes me realise, that I'm not that smart and there are so many more interesting things to learn.

It seems that some reductions can be made in the levels of recursion by adding cases for the following: ack(1, n) = n + 2 ack(2, n) = 2*n + 3 ack(3, n) = 1

@KnakuanaRka

3 жыл бұрын

Yeah, that’s one way to improve it.

@Rudxain

2 жыл бұрын

Another optimization is to convert one of the tail-calls into a loop, that leaves us with just 1 recursive call. But thanks to your comment I realized there's a simpler way to optimize the A function, define it in terms of a single Hyper-Operation call: A(m, n) = HyperOP(m, 2, n + 3) - 3 Where the 0th argument of HP is the degree or "order" of the HyperOP we want to use, 1st arg is what we want to hyper-operate, and 2nd arg is the number of times to apply the "sub-HP". To implement HP efficiently, define it like so: HP(0, a, b) = b++ //add 1 HP(1, a, b) = a + b HP(2, a, b) = a × b HP(3, a, b) = a ^ b //a ** b The general case HP(n, a, b) requires recursion, it can be an implicit or explicit stack. It's essentially HP(n - 1, a, HP(n - 1, a, ...)), with b copies of HP calls, IIRC. HP(4, a, b) is tetration, HP(5, a, b) is pentation, etc... Before I realized this optimization, I just used the Wikipedia table with all closed-form expressions that didn't require recursion, and used a memoization hash table when recursion was needed. After implementing the HP optimization I realized Wikipedia already mentioned that in the "Definition" section, but I didn't understand the notation before... *bruh*

@UNKNOWNKINGTHEGAMER

Жыл бұрын

try 4,4

Man I sure love Professor Brailsford, I hang on every word he says. Such a great lecturing voice! Such a wise old wizard.

I've been jobbing business focused programmer for over thirty years. I was at college recursion was kind of worshiped by my tutors 'oh look at the elegant code' they would say. When I got in to the business world the number of times recursion was the best solution was about twice in thirty years!

@BlackMambo

9 жыл бұрын

Brilliant...!

@djdedan

9 жыл бұрын

u guys never sorted anything? i would assume most sort functions are recursive (of course i could be totally wrong - but then again i could be totally right!)? Or do you mean you never had to write any recursive code?

@SteveGouldinSpain

9 жыл бұрын

Strangely no because whenever I've had data it's been inside a database, so you get the database to sort the data before you get the data out. Or if you get it into a webpage then use methods from a library like jQuery datatables to handle the sorting for you. I've never needed to get my hands dirty sorting. Infact I remeber doing all that sorting stuff at college level and pretty much never having to use it like I never used truth tables or Karnough maps!

@djdedan

9 жыл бұрын

Steve Gould haha yeah - it's the curse (?) of software engineering, learn the nuts and bolts but unless you plan to be an expert on, say sorting, you are better off using someone else's sorting routines - so why learn the nuts and bolts in the 1st place. My answer would be: for job interviews! lol! cheers!

@SteveGouldinSpain

9 жыл бұрын

Spot on DjDedan and in fact some the most successful 'programmers' I know hardly write a line of code. They are just very good a cobbling together solutions from other people's work.

Hi Professor Brailsford! I've really enjoyed all your videos, thank you for taking the time to make them. It has been a couple years for me since graduating, and I miss my senior level computer science classes. Your lectures remind me of why I got in to and love the field... and not for a day job

I want Professor Brailsford as my lecturer. He's awesome :D

@hamza3838

6 жыл бұрын

Suave Atore wait is that his name? THATS SUCH A COOL NAME

g64 is graham's number (see one of Brady's other videos). I first saw Ack(g64, g64) in an xkcd comic though. So you're looking at a number created through superexponential means, fed into a function that has a super exponential time complexity.

@MrKastle09

10 жыл бұрын

Graham's number is insane. But the most amazing thing is that the only thing bigger than Graham's Number ... is yo momma XD

@Garbaz

10 жыл бұрын

Thats gonna be fun :D It will stop calculating even before you really startet the ackerman part...

@themennissvids

10 жыл бұрын

Krisna Siv Nope, there's a thread on the xkcd forums called "My Number is Bigger!", they've found numbers that make my mother look negligible.

@hexagonist23

2 жыл бұрын

What about Ack(TREE(g64),TREE(g64))

All the cores in the world won't help you since this C code will run in a single thread. Also the first thing I thought was "wouldn't memoisation help improve performance?" After a quick google search, I discovered that "When a software developer learns about the Ackermann function, he will try to see how much of improvement function memoisation does if any". LOL

@HebaruSan

6 жыл бұрын

How much is it?

@lagduck2209

6 жыл бұрын

HebaruSan probably at least some, if any

@SwissSareth

4 жыл бұрын

Funnily true ...

@sayven

2 жыл бұрын

can confirm lol

Computerphile is such a wonderful treat for me, thank you so much Brady and all of your interviewees for the time and effort putting this together (numberphile as well!). You are doing a service to all mankind, you deserve a trophy!

I'd love to go to any of Professor Brailford's lectures. I love his passion and knowledge!

You could try writing a memoized version of this code, where it remembers the result of a previous computation, and just plops that result in so it won't have to spend time working that out, cuz it seems to me that that's what's slowing it down so much; that it has to go through the same set of computations again and again EDIT: PEOPLE before commenting on why it wouldn't work; please read the other comments! THEY MAY ALREADY HAVE COME TO THE SAME CONCLUSION! Please don't waste precious keystrokes on repeated information, thank you

@Poldovico

9 жыл бұрын

But I wanted to know why they didn't, and the comments you refer to aren't on the first page anymore D:

@greenya84

9 жыл бұрын

This i called "Dynamic programming" :)

@NikolajLepka

9 жыл бұрын

Yuriy Grinevich do your research, memoization has its differences

@MartinFracker

9 жыл бұрын

Yuriy Grinevich Memoization is a technique that can be involved in dynamic programming.

@hdpasd123lol

9 жыл бұрын

Nikolaj Lepka As far as I know your approach is known as " backtracking " , so for the people saying it wont work , it will. It will indeed make it faster.

9:45 I reproduced the function in Python and believe I got it right as it gives me the same results, except for ack(4, 1): _"RecursionError: maximum recursion depth exceeded while calling a Python object"_ lulz at Python

@tombeaudry3143

7 жыл бұрын

Python, please.

@SonnyGhosh

7 жыл бұрын

use like Java, c#, c++, etc

@kraemer-raimund

7 жыл бұрын

You can change the recursion limit in Python.

@bangpaf2328

7 жыл бұрын

Thanks for the replies. I guess I should have known, I'm still pretty wet behind the ears when it comes to programming.

@thomaspappas8946

7 жыл бұрын

+Raimund yes but how?

This is amazing. I have been in computers and programming a long time, (1972) and this is new ground for me. Thanks for sharing it.

I just love that you can look at his face an see how much he loves this topic

Ackerman(4,2) is equal to 42 :)

@TH3MIN3R3000

7 жыл бұрын

Herp Derpingson if only, if only...

ack(tree(3), graham's number)

@ChristopherKing288

7 жыл бұрын

7

@pranamd1

7 жыл бұрын

@abcdefzhij

7 жыл бұрын

The salad is strong in this one. You are already at the magnitude of TREE(3), doing an ackermann function (which is EXTREMELY weak by comparison) won't really do anything.

@zokalyx

6 жыл бұрын

42

@carbrickscity

6 жыл бұрын

The TREE function is way stronger than the Ackermann function. However, the follow-up video of this video was about a function even stronger than the TREE function - the Busy Beaver function.

Just ran into this function while going through SICP, this really helped shed some light on the topic. Thanks!

This is so so interesting... as a computing and business student, I am really amazed and I can actually admire and appreciate all these programs and contributions made!!!! AMAZING!

The thing that makes the Ackermann function irreducible is that the amount of unique (m,n) pairs you need to computer is actually greater than the Ackermann number at (m,n-1). So even if you were trying to compute bottom to top, while it might be more efficient than the naive implementation (no pair calculated twice), it would still require a very large run time, and also require a great deal of memory as well.

The most important question remained unanswered! Why the Ackermann function cannot be put in for loops! A sketch of the proof would be fantastic for Prof. Brailsford to answer given his great teaching skills, maybe for a next Computerphile episode! For people it may look like you only need to write a program that keeps nesting for loops to calculate Ackermann's. Best and great series!

@profdaveb6384

9 жыл бұрын

Essentially, using recursion is a way of dynamically nesting "for" loops by virtue of your chosen programming language creating *dynamically* as many stack frames as are needed..There is in principle no limit except that, as the stack grows huge, you will eventually run out of memory, The problem with static for loops, explicitly written into your program, is that every compiler will put a compile-time limit on how deeply nested they may be.

@paulpinecone2464

2 жыл бұрын

Ok, so to go half way there, make a language structure that takes an array of triples which specify the set of nested loops and a pointer to a function to call inside that is passed a state array of the current values of all the loop variables. Several challenge levels: 1) The array is static, ie, it is just shorthand for writing out all the loops. The compiler or preprocessor can generate them. 2) The array is dynamic but contains constants. Ie, the loop system is known at the start of it and can be allocated as a static data structure. 3) The dynamic array can contain algebraic expressions which need to be evaluated to determine loop criteria. Ie, the structure is known at the start but the loop limited can vary. 4) The array is returned by a function which itself can contain functions to define what the substructure of loops will be at each level. I believe this is called recursion and is no longer a language feature.

@kylek.3689

2 жыл бұрын

Because of one of the definitions of the Ackerman function, where A(m+1,n+1) = A(m, A(m+1, n)) This means the loop depth of the Ackerman function relies on M, and you'd have to give one for loop to case m=1, two loops for case m=2, three for loops for case m=3, and so on. Though, this is without using a heap allocated stack to imitate a function call stack.

The only regret I have in my life is learning this late in my life, from Professor Brailsford's and rightly so, and so lucky I am. What an honor to stand on the shoulders of the masters of the computer coding & math sciences resident in my grandfather's home country. Thank you all at Computerphile. Cheers... Dale Robertson professional student. College of Marin

I love computerphile, wish I found out about them when I was in middle or highschool to explore computer science sooner rather than in college.

The Ackermann function, while useless as a piece of software, it is very useful as a theoretical case study, especially in regards to writing compilers and optimizers. The Ackermann function is, by all standards, a pure function, in that its output is deterministic and relies only on its input arguments and nothing else. What that means is that if you call it with constant actual parameters, say, x = ack(4,1), in your code somewhere, the compiler can pre-compute the result and change it to x = 65533 internally so you don't have to execute it in the compiled program. The case study comes about where you decide at what point the compiler should give up trying to pre-compute it and just leave the function call as it is, otherwise the program will never compile.

@brendawilliams8062

4 ай бұрын

Like 41666….↗️

His computer's a bit slow. I compiled his function pretty much verbatim, and it calculates ack(4,1) in 27 seconds. With optimized compilation, it only takes 5.7 seconds. I don't think I'll bother calculating ack(4,2) though...

@Computerphile

8 жыл бұрын

***** suggest you watch the follow-on video :) >Sean

@philronan6929

8 жыл бұрын

***** Ah. That explains it!

@idanguttsait

8 жыл бұрын

+DaKnOb That is true. I had a StackOverflow on 4,1

@moveaxebx

8 жыл бұрын

+Giora Guttsait Try with any functional PL (supports tail call optimization)

@drobilla

8 жыл бұрын

+moveaxebx The Ackermann function is not tail recursive. That is essentially what the professor is explaining. Tail call optimization is, in a sense, converting a recursion into a for loop (making it iterative), and you can not do this for functions like ack.

example of enumerable recursion with occasional output: "Display all files in drive C" example of undescidable recursion: "write a text file that contains the hash value of the file when the file is hashed."

i wish i had professor like that in morocco... the passion you talk about the subject give motivation and make me love more computer science :)

@repairtech9717

Жыл бұрын

Most of us in the UK where he teaches wish we had a professor like this as well. He comes from a very elite university that most dont have the money to pay for. A quote from the wikipedia for the UON says "Nottingham has about 45,500 students and 7,000 staff, and had an income of £694 million in 2021"

God i wish we had teachers like you guys back when i was at school, i might have actually learned something more than whether tom crosses the road with sally or jill. I wanna go back and absorb all this glorious input, but since creating a time machine isn't on my to do list i think i'll just watch more of these videos. Imho this channel, your numberphile channel and the PBS space time channel are the best places for input on youtube, keep it up fellas it's well appreciated.

@PhilipBlignaut

5 жыл бұрын

Great choice of videos subs!!

@mainmast8955

5 жыл бұрын

it may be time for you/us numerical freaks to push forward.

Ermagherd that lack of indentation is driving me crazy!!

@eliatarasoff5872

10 жыл бұрын

Agreed. That code would be sooo much easier to read if it were indented! This is like...Computers 101 stuff! ^^;

@miciduve

2 жыл бұрын

@@eliatarasoff5872 ermagherd someone learned with python didnt they

I have barely the slightest idea of what is going on in the "guts" of this, but i just really enjoy professor Brailsford talking about it. So thanks, professor!

I think I have some vague idea what he's talking about but but probably I'm totally wrong. I love his dramaticness of code going wrong so that keeps me listening.

Counted the number of recursions required to calculate ack(4, 1) and the result was 2, 862, 984, 010. Mind boggling...

@KnakuanaRka

3 жыл бұрын

Y2K38 How do you calculate that? Is that its own recursive function?

@abhineetsingh1291

3 жыл бұрын

In the recursive function create a static variable and increase every time for a recursive call

@KnakuanaRka

3 жыл бұрын

abhineet singh Makes sense.

If anyone remembers Knuth's arrow notation from Numberphile's video on Graham's number, it is worth noting that Ackermann's function can be represented in arrow notation as well. If we define an arrow operation: 2 {↑^m} n, then that is identical to ack(m+2,n-3) + 3 So, inversely, if we have ack(m,n) we can write its arrow notation as: 2 {↑^m-2} [n+3] - 3 So, as in the video, if we have ack(3,1) = 2{↑^1}4 - 3 Which is 2^4 - 3 = 16 - 3 = 13, same as in the video. (Note that this method only easily works, though, for m > 2, otherwise you have no arrow operator.) So really ack(4,2) may be *uncomputable*, but it can be mathematically represented as 2{↑^2}5 - 3 with these hyperexponentiation operators. If you want to know what ack(4,2) is exactly, well, the best estimates put it approximately at: 2*10^19728 OR 10^10^4.295 So, yes, far bigger than universal timescales or particle numbers. ^_^ en.wikipedia.org/wiki/Knuth's_up-arrow_notation has some more information. Love the videos!

Such a nice voice to listen to. And extremely interesting topics.

This was mindblowingly amazing, thanks!

This guy has great personality!

I computed the value of the Ackermann function of 4 and 1 and my M1 Macbook took roughly 6 seconds which just shows how far we've come in 7 years of processor performance upgrades.

thanks! your way of explaining Ackermann's function is so clear and enthusiastic. great gift!

'How do you define forever?' -- an under-rated question

Fantastic explanation. The bit about the big crunch didn't age well, but otherwise... just magic.

@amihartz

9 ай бұрын

The reliability of the view that the expansion of the universe is speeding up is exaggerated. The Nobel Prize was awarded prior to their data being made public, and when it finally was made public recently there have been tons of papers criticizing it, both in methodology and by using more data from more modern telescopes.

After breaking various online 'big number calculators' I found one that works, and it turns out 2^65533 * 3 minutes rounds out to, unless I've screwed up somewhere, about 1x10^19712 times the age of the universe. Not terribly helpful. (((((2^65533 * 3) / 60) / 24) / 365) / 13700000000) = 1.0434008320186794 × 10^19712

@KnakuanaRka

5 жыл бұрын

If you want help with big powers like that, try making logarithms of everything; multiplying and dividing stuff like that turns into adding and subtracting more manageable numbers.

That voice is really calming and confortable, ideal for teaching!!! :)

Lovely presentation. I really enjoyed the final part... great humour.

I remember when in school we were making the program that solved the “Hanoi tower” problem and then we made a fractal tree that was moving in the wind - it was so beautiful! Since that day I fell in love with recursions and fractals!

Ooops, I think I just crashed a planetary superbrain in the Perseus arm. by asking it to compute Ack(G, G), where G is Grahams Number.

@Jcarr250

9 жыл бұрын

See the thing is, that's barely bigger than G64 already. It's actually smaller than G65 Ack(G64, G64) looks to be 2 ↑^(G64) (G64 + 3) whereas G65 = 3 ↑^(G64) G64 ↑^(G64) refers to G64 Knuth Arrows

@garethdean6382

9 жыл бұрын

Umbrall Let's feed it an xkcd number.

@coopergates9680

9 жыл бұрын

Smonjirez G65 has the G64 arrows between two 3s. His number, though, is a lot less than G66.

@Jcarr250

9 жыл бұрын

Cooper Gates That's what I was saying. It's not even just smaller than G66 it's smaller than G65

@coopergates9680

9 жыл бұрын

Umbrall No, he put G64 arrows between G64 twice instead of two 3s, so it is a lot larger than G65.

Beautiful Masterpiece

On computerphile, you're never too far from the deep end.

i was searching for the secret behind the ackermann power in aot and got this. Nice

I love this guy

the time for computation blew my mind as same as the hanoi tower, this is astonishing

Recursively Enumerable includes undecidable problems. The halting problem is undecidable and recursively enumerable (just run the turing machine and accept if it halts). What you meant to put at the end is unrecognizable problems. An example of an unrecognizable problem is all turing machine input pairs that do not halt.

Such a simple function, yet amazing how long it takes to compute. But, given infinite memory, I wonder much the runtime could be brought down using dynamic programming (memoization).

You use openSUSE! Me too! ^_^ And I had the Ackermann function in a lecture at ETH Zürich! Keep on doing these videos! :D

Wow, that escalated quickly, Mr. Ackermann!

This guy just makes me so happy

At school while learning VB6 long time ago, I made a horse race game project. It had to have random values on the power of the horse but I felt that this PC random function was not enough random to me. So in the race the speed was randomly changing speed by a random factor for a random time. This is how you could have a 200MHz computer completely freeze. At that moment I understood that my code was not optimized.

Would ack(m-1,ack(m-1,ack(m,n-1))) still be superexponential or would it have an even bigger growth?

@computerphile : Thanks for the awesome videos. I am a huge fan. Can you guys please do a video on Dynamics programming?

This guy is amazing, so fun to watch

@7:05 he claims, "But what I would like to draw your attention to, because this is important, is that every time M and N are altered, they are reduced." This statement is wrong. @9:50, you can see that ackerman(m,n) is always greater than both m and n. Because the code contains, "ans = ack(m-1, ack(m,n-1))" whenever neither m nor n are 0, n in fact increases. It might be better to say m always either stays the same and n decreases, or whenever n increases m decreases. Eventually when m reaches 0, an answer is produced immediately.

@KnakuanaRka

3 жыл бұрын

Terje Oseberg Yeah, that’s a whoopsie.

@pvic6959

3 жыл бұрын

i see an issue in the second line as well if n=0, the answer is ack(m-1,1). The second argument (n) is back to 1 even if it was 0 before. whcih will make it keep going.

@KnakuanaRka

3 жыл бұрын

pvic However, m decreases during the call; basically, as you do recursive calls, n decreases to 0, then m decreases as n increases, then n runs down to 0 again, then m decreases again as n is reset, and so on, so m will become 0 eventually, causing the calls to stop recursing. It will take a VERY long time, since n will increase by a crazy amount each time m decreases, but it will run out eventually.

I am SO PROUD that a native english speaker just pronounced the name of my home town (goettingen) correctly! :D

These videos are such an absolute joy for somebody like me who is trying to get into computer science. From the bottom of my heart, thank you.

This channel is so awesome.

It's time to build a computer I don't mind crashing.

That's not at all what superexponential means in the context of computational complexity. In the context of computational complexity theory, anything superexponential is anything with a time complexity (or space complexity) greater than O(c^n), where c is a constant and n is the problem size. For example, the factorial function is superexponential. There are a lot of problems that have factorial complexity, for example in combinatorial optimisation.

I love listening to this guy .... totally inspiring

Fantastic video. Professor Brailsford is awesome. ❤

This algorithm takes longer to compute than the Ultimate Answer to Life, the Universe, and Everything took.

You've discussed algorithms that are necessarily recursive. Given that multi-cores and threads are now mainstream, I wonder if there are any algorithms that are necessarily parallel.

@jirihavel9766

8 жыл бұрын

+aMulliganStew No, you can always serialize it by interleaving instructions from different threads.

@jonathanpark4619

8 жыл бұрын

+Jiří Havel That's still parallelization.

@jirihavel9766

8 жыл бұрын

Jonathan Park I meant that you can always emulate a parallel machine on a serial one. This means that there are no problems that are solvable by parallel but not by serial one.

@jonathanpark4619

8 жыл бұрын

Unless you're executing the threads in a serial manner, that's still parallelization.

@jirihavel9766

8 жыл бұрын

Jonathan Park You can call it this way. I answered to "I wonder if there are any algorithms that are necessarily parallel." And since you can convert any parallel algorithm to a serial one (even if that means emulating some parallel machine), then parallel algorithms can't solve any problem that a serial algorithm can't.

As some people are wondering what ack(4,2) is like: the number has 19729 decimal digits, begins with 20035.. and ends in ...6733. One can compute it by exploiting the knowledge that the special case ack(3,x) is always 2^(x+3)-3. So, a programming language with unbounded integers like Haskell can do that for you. However, ack(4,3) is hopeless anyway, because to even represent the resulting number as a binary number on a computer you need a computer with a very very large piece of computer memory - exceeding the size of the known universe.

OpenSUSE linux in the background! Love that distro.

I downloaded the program files and ran ack(6,6) on my quad core 3.2 ghz machine running Arch Linux. It used 100% of one of my cores, took 1 minute 45 seconds, and ended in a segmentation fault. ack(4,4) also segfaults. ack(3,3) returns in under a second.

I worked out the fast Ackermann function :) function fAck(m,n){ var ack = [4,5,6,2,3,4,5,6,7,3,5,7,9,11,13,5,13,29,61,125,253,13,65533]; console.log( ack[m] ) if( m

@coopergates9680

9 жыл бұрын

kowdermeister So that will throw the error every time it's called he he (original call)

The smirk while saying "You can prove there is an answer and how to get it, but no one will be around long enough to hear it" :D

The program in the description (link) is missing a semicolon in the printf line... I learned how to detect those the hard way Great video as always prof!

i believe that for loops and recursion are only needed when working with indexes and such. but not when doing math... there is always an equation. that will do it next to instantly.

@Ozzah

9 жыл бұрын

steven johnston No. Elementary functions can be defined recursively and there is no closed for solution to them. Some functions are defined as infinite sums of basis functions. Most differential equations have no closed for solutions and must be solved numerically. In all these cases, loops or recursions are absolutely necessary. And it's definitely not true that it can be computed instantly. I currently have 3 computers working for almost a month on one problem ;)

@jervey123

9 жыл бұрын

wuuut? i disagree math is all about recursion... the simplest of elements in math revolve around recursion like say counting but before we can count we have to define natural numbers; how do you define a number? well a number is either zero or the increment of a number, so you can practically create a function called "number 5" which is just inc^5(zero), see? recursion

I think the answer '42' from Hitchhiker's Guide to The Galaxy may be a result of a call of ackerman. Now the question is what were m and n?

@furbyfubar

4 жыл бұрын

That's an easy but not a very interesting call since: ack(0,41) = 42

Mind boggling! Great video!

I had to use recursion on my sediment loading model. For my model it took a DEM and other layers and need to work up a river to measure sediment loss and sediment delivery. So I worked from the origin cell and looked for neighbor cells that fell upriver. So I had to look recursively at each cell until we were at the highest cells and then work downriver. I guess you could do it with a with loop and several for loops but it would take forever.

Sometime in the mid nineties when the big crunch was swept off the table in cosmology, Professor Brailsford had already reached his personal StackOverflow and couldn't take in new information. It happens to all of us.

Would caching make this function more computable? By remembering old results you do not need to recalculate them. Since it only calls the function with reduced arguments, it doesn't seem that hard to build your way up? I am probably missing something :P

@ernhamDjinn

6 жыл бұрын

ack(4,2) will access ack(3, 65k), value that was never computed before, which itself will access ack(2, insanely_large_value), and so on.

@armoredmind-gr2298

5 жыл бұрын

Nope . Thats memorization for recursion . That's not the problem in this case

@mainmast8955

5 жыл бұрын

ur onto something

Fabulous video. Pretty interesting stuff

These guys remember why we code the way we do. Founding fathers indeed.

Ackermann was clearly a very naughty boy. The man's created a monstrosity. Imagine feeding it into the Krell Machine and saying, "chew on that, motherflocker!" as the core starts to overheat with Robbie in the background chiming, "warning, warning, what you have done is wrong" and the Id becomes furious and is going to have you!

i bet it can worked out in a loop with the use of some kind of list

Mark, thank you! I was thinking "I like this guy, but his cosmology is a couple of decades out of date." Glad I'm not the only one to go "Big Crunch?, srsly?"

One's of the greatest professors if it isn't the great, even if the video topic is kind of hard to be comprehensible but within his abilities of clarifying, things can be well understood. Ps. We still want to know why we cannot use for loops for ackman function.