Infinite Monkey Theorem flaw?

#1
Hello everybody,

I think I have found a flaw in the Infinite monkey theorem. I came to the conclusion that codes (such as a sample of written language) can not be generated randomly.

Please look at the attached file (only 1.5 pages) and let me know what I did wrong.

Thank you very much,

I would greatly appreciate your answers.

Alonso
 

Dason

Ambassador to the humans
#4
Decided to hop onto one of my virtual machines and read the document.

SmoothJohn already covered the basic problems with your 'proof'. Which is the other point I want to make. Your document is in no way a disproof. You're basically just saying "Hey guys, it doesn't look like randomness could generate that." But you're wrong.

Take for example coin flipping. If we flip a perfectly fair coin we would expect that after an infinite number of flips the number of heads and the number of tails should be equal. Now listen closely.... In that infinite sequence it is almost sure that you will see a sequence of heads as long as K. What is K you ask? Who cares, it can be any natural number. Let's say K is 8 for the time being. If we randomly group the infinite sequence into blocks of 8 then each block of 8 has probability 1/2^8 of being all heads. Now if we use the naive approach and only look at the successive blocks of 8 you can ask what the probability is that you won't have one of those blocks be a complete run of heads. The probability that any block won't be a run of 8 heads is 1-(1/2^8). So the probability that the neither of the first two blocks is the a sequence of heads is (1 - (1/2^8)^2. The probability that none of the first m blocks contains that sequence of heads is (1-(1/2^8)^M. You can see that 1-1/2^8 < 1 so if we take it to higher and higher powers it becomes smaller and smaller. (Note that I'm being really conservative in that the sequence of 8 heads could have been somewhere that didn't fit perfectly into one of our blocks but this route makes the calculation easier and gets the point across just as well). So taking M to infinity means that the probability that we don't see this sequence goes to 0. But my block size of 8 was completely arbitrary. This works for any block size. So if we have an infinite sequence the probability that there isn't a sequence of heads at least 23423136572352352345 long goes to 0. I hope you can see how this directly relates back to the problem you "disproved".
 
#5
Dear Dason and SmoothJohn:

Thank you very much for your answers, I really appreciate it.

I understand now that I was completely wrong.

Thank you for your time.

Sincerely,

Alonso
 

Dason

Ambassador to the humans
#6
No worries. Don't let this stop you from exploring math/stats though! They're such beautiful topics and we just wanted to make sure you had the details straight ;)
 
#7
Generative Rendering Theorem​

Those familiar with computers will know that files stored on disk have a size specified in terms such as kilobytes or megabytes. Well, what is a byte? A byte is a unit of storage analogous to a litre. A jug that can store a litre can store any volume of liquid up to and including a litre, any volume between 1-1000 millilitres, or even nothing at all. It is the same with a byte. A byte can store any number in the range 1-255, or it can store nothing at all (the number “0”).

An author editing a text document sees nothing but text displayed on screen, yet all of the letters and other characters on display are actually stored as numbers in the “txt” file being edited. Each character has its corresponding number. Although there are several “numbering” standards used for storing text, one of the most common is ASCII (American Standard Code for Information Interchange). In ASCII, the letter “C” is stored in the “txt” file as the number 67, while the word “CAT” is stored in the “txt” file as the number sequence 67, 65, 84.

While a “txt” file containing only a “C” can most certainly be brought into existence by an author using a word processor, there is another process by which a “txt” file containing only a “C” can be brought into existence. This process – called Generative Rendering – exploits the fact that given one byte can store only the numbers 0 and 1-255 (or 0-255, as we programmers say) a program can be written that does nothing more than write out to disk all possible 1-byte “txt” files. Each of these “txt” files will contain a different number in the range 0-255. Among these “txt” files will be a file containing the number 67. When that “txt” file is loaded into a word processor, what is displayed on screen is “C”. This program – a Generative Text Renderer – is presented here in a “language” the layperson can understand:

10 for x = 0 to 255
20 write x to a “txt” file
30 next x

Bytes can be combined in such a way that working together two bytes can be used to store numbers in the range 0-65535. This is like having two jugs, each of which can store only 1 litre but which – when working together – can store up to 2 litres. The Generative Text Renderer can be modified to write out all possible 2-byte “txt” files. Each of these “txt” files will contain a different number in the range 0-65535. Among these “txt” files will be a file containing the two numbers 67 and 65. When that “txt” file is loaded into a word processor, what is displayed on screen is “CA”.
Similarly, three bytes working together can be used to store numbers in the range 0-16777215. The Generative Text Renderer can be modified to write out all possible 3-byte “txt” files. Each of these “txt” files will contain a different number in the range 0-16777216. Among these “txt” files will be a file containing the three numbers 67, 65 and 84. When that “txt” file is loaded into a word processor, what is displayed on screen is “CAT”.

Ninety-one bytes working together can be used to store numbers in the range 0-SomeHugeNumber. The Generative Text Renderer can be modified to write out all possible 91-byte “txt” files. Each of these “txt” files will contain a different number in the range 0-SomeHugeNumber. Among these “txt” files will be a file containing the ninety-one numbers

083 105 110 099 101 032 116 104 101 121 032 097 114 101 032 110 111 032 108 111 110 103 101 114 032 116 119 111 032 098 117 116 032 111 110 101 044 032 108 101 116 032 110 111 032 111 110 101 032 115 112 108 105 116 032 097 112 097 114 116 032 119 104 097 116 032 071 111 100 032 104 097 115 032 106 111 105 110 101 100 032 116 111 103 101 116 104 101 114 046

When that “txt” file is loaded into a word processor, what is displayed on screen is: Since they are no longer two but one, let no one split apart what God has joined together. That’s Matthew 19:6 by the way (NLT).

Generative Rendering Theorem outdoes Infinite Monkey Theorem. While Infinite Monkey Theorem cannot guarantee that even a tome as short as the complete Works of William Shakespeare can be recreated by random chance even when time is not a limiting factor, Generative Rendering Theorem absolutely does guarantee that each and every Work stored in the British Library can be recreated in very little time by supercomputers doing nothing more than counting sequentially from zero to infinity and beyond!
 
#8
Hi @alonsovener! I think you gave up very fast! ;-)

IMT is a very nice issue. I would put together your post and JudgeDracoAmunRa's. We have important things here:

- an ordered text has no randomness enough to be taken as randomly generated at all (and @alonsovener showed it through simple frequency distribution analysis);

- @Dason's answer has no math issues. But we must remember that a given "must happen" event has prob = 1. But a "prob = 1 event" has no 100% chance to happen. "How come?", you may ask. This happens because (1/2)^8 can be done to ANY 8 rounds sequence! 8 heads, 8 tails or an aleatory 8 rounds event. And 8 could be any number as 10^500.000 or a Shakespeare sonet length!! Now I ask you: is it rational to think about a LONG non-aleatory sequence given that an aleatory sequence has the same chance to happen? is it rational to expect any long sequence to spread apart from unbiased frequency distribution (so many space bar hits for exemple)?

Considering a 50 keys keyboard, if you are to have an aleatory trial, frequency distribution WILL tend to 1/50 for each key! That simple. Isn't it compatible to REALITY?

Why would not it happen? Just to exactly generate a predefined text one chose and that has the same chance of an aleatory string? Seems to me like a biased test.

And I would like to make it clear: It is all about PROBABILITY and real RANDOMNESS!! It is not philosophy.

Hope this may feed this discussion.
 
#9
The maths might have been wrong but the idea was right. A random process can never reproduce a work of literature. The missing element in the infinite monkey theory is meaning - the meaning that the words and sentences "code" for. The misguided attempt to prove this mathematically was perhaps prompted by by an intuitive understanding that there is something seriously wrong with the theory. Maths will never catch up with this - for the same reason the Turing test will never be passed.
 

Dason

Ambassador to the humans
#11
I honestly think anybody that thinks they can disprove the infinite monkey theorem really doesn't understand the word "infinite".
 

hlsmith

Less is more. Stay pure. Stay poor.
#12
Well based on Jack's assessment of my spelling, eventually I should have one without any typographical error. Perhaps it may occur next time!
 

Dason

Ambassador to the humans
#13
The maths might have been wrong but the idea was right. A random process can never reproduce a work of literature. The missing element in the infinite monkey theory is meaning - the meaning that the words and sentences "code" for. The misguided attempt to prove this mathematically was perhaps prompted by by an intuitive understanding that there is something seriously wrong with the theory. Maths will never catch up with this - for the same reason the Turing test will never be passed.
What a beautiful proof you put forth. Your argument boils down to "it's not gonna happen cause I say so".

Let me ask a related but different question. Do you think in the same situation proposed in the original theorem that any three letter word will be written at some point?
 
#15
Perhaps, as a non-mathematician I shouldn't have joined a maths discussion. The original questioner said "I think I have found a flaw in the Infinite monkey theorem. I came to the conclusion that codes (such as a sample of written language) can not be generated randomly."

Meaning is the code. Randomness cannot reproduce meaning.
 

Dason

Ambassador to the humans
#16
Perhaps, as a non-mathematician I shouldn't have joined a maths discussion. The original questioner said "I think I have found a flaw in the Infinite monkey theorem. I came to the conclusion that codes (such as a sample of written language) can not be generated randomly."

Meaning is the code. Randomness cannot reproduce meaning.
I'm not exactly sure what your argument here is. It's certainly possible for randomness to generate character strings that represent words/sentences/etc... and that is what the theorem is about. Nobody said the monkeys would understand if/when they finally produced something of significance - just that eventually in their vast piles of output that any finite piece of literature you could think of would be produced eventually.
 
#19
If I may attempt some amateur statistics, the theorem crashes because it is possible that the target outcome (reproducing or producing a specified meaningful set of words or numbers) won't happen - even in infinity . Once that possibility is accepted, the basis collapses. I think the target outcome is impossible, but failure only has to be possible for the theorem to be fatally flawed. Because failure is a possible outcome, it must be included in the probability maths. Once it's included, the monkey theorem immediately crashes.
 
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