Answers.Body: <p>You would probably have to pack recursive structures into finite-dimensional real vectors and there have been such attempts. The finite precision limits goes as far as the recursion can go.</p>
  
  <p>The limitation of <em>feedforward</em> neural networks is restricted to finite input and output spaces, so <em>recurrent</em> may be more suitable for this task as in theory can process arbitrarily long strings of numbers, but it has much more practical difficulties than feedforward network.</p>
  
  <p>These kind of methods are open to debate.</p>
  
  <p>Source: <a href="ftp://ftp.sas.com/pub/neural/FAQ.html" rel="nofollow">SAS FAQ</a></p>
  
  <p>References:</p>
  
  <ul>
  <li>Blair, 1997; Pollack, 1990; Chalmers, 1990; Chrisman, 1991; Plate, 1994; Hammerton, 1998; Hadley, 1999</li>
  </ul>
  
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Answers.UserDisplayName: kenorb
Answers.Body: <p>Pseudo-random number generators are specifically defined to defeat any form of prediction via 'black box' observation. Certainly, some (e.g. linear congruential) have weaknesses, but you are unlikely to have any success in general in predicting the output of a modern RNG. For devices based on chaotic physical systems (e.g. most national lotteries), there is no realistic possibility of prediction.</p>
  
  <p>"Patterns or statistical association" is a much weaker criterion than 'prediction'. Some very recent work has applied topological data analysis to visualize patterns within the infamous Randu RNG.</p>
  
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Answers.UserDisplayName: NietzscheanAI
Body: <p>Given a list of fixed numbers from a mathematical constant, such as <span class="math-container">$\pi$</span>, is it is possible to train AI to attempt to predict the next numbers of this constant?</p>
  <p>Which AI or neural network would be more suitable for this task?</p>
  <p>Especially, the one which will work without memorizing the entire training set, but the one which will attempt to find some patterns or statistical association.</p>
  
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Comments.CreationDate: 2016-08-03T18:22:27.873Z
Comments.Text: That's going to depend on the PRNG algorithm. A good one would be completely unpredictable if you didn't know its state/seed.
Comments.UserDisplayName: Ben N
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Comments.CreationDate: 2016-08-09T10:03:38.000Z
Comments.Text: I've narrowed down the question to Pi constant, I hope that helps.
Comments.UserDisplayName: kenorb
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Comments.CreationDate: 2016-08-12T10:39:14.763Z
Comments.Text: As a mathematician, I really don't understand what your new question is/means. It is a well known conjecture that every string of (decimal) numbers/digtis appears infinitely many times in pi, so the next numbers could be anything. Also, there is a formula that allows us to calculate the nth (binary) digit of pi without knowing those before that, but finding the location of a string requires more since we can't use that formula backwards.
Comments.UserDisplayName: wythagoras
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Comments.CreationDate: 2016-08-12T10:52:18.777Z
Comments.Text: @wythagoras I mean that maybe ANN can detect some pattern where we humans can't.
Comments.UserDisplayName: kenorb
Tags: <reference-request><recurrent-neural-networks><algorithm-request><sequence-modeling><model-request>
Title: What are the approaches to predict sequence of $\pi$ numbers?
UserDisplayName: kenorb
ThreadText: # What are the approaches to predict sequence of $\pi$ numbers?
  
  (Asked by: username_0 on 2016-08-03)
  
  Given a list of fixed numbers from a mathematical constant, such as $\pi$, is it is possible to train AI to attempt to predict the next numbers of this constant?
  
  Which AI or neural network would be more suitable for this task?
  
  Especially, the one which will work without memorizing the entire training set, but the one which will attempt to find some patterns or statistical association.
  
  **Tags:** reference-request, recurrent-neural-networks, algorithm-request, sequence-modeling, model-request
  > **Comments:**
  > **username_1**: That's going to depend on the PRNG algorithm. A good one would be completely unpredictable if you didn't know its state/seed.  
  > **username_0**: I've narrowed down the question to Pi constant, I hope that helps.  
  > **username_2**: As a mathematician, I really don't understand what your new question is/means. It is a well known conjecture that every string of (decimal) numbers/digtis appears infinitely many times in pi, so the next numbers could be anything. Also, there is a formula that allows us to calculate the nth (binary) digit of pi without knowing those before that, but finding the location of a string requires more since we can't use that formula backwards.  
  > **username_0**: @wythagoras I mean that maybe ANN can detect some pattern where we humans can't.  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_3 on 2016-08-03. Score: 4 | Accepted answer
  
  Pseudo-random number generators are specifically defined to defeat any form of prediction via 'black box' observation. Certainly, some (e.g. linear congruential) have weaknesses, but you are unlikely to have any success in general in predicting the output of a modern RNG. For devices based on chaotic physical systems (e.g. most national lotteries), there is no realistic possibility of prediction.
  
  "Patterns or statistical association" is a much weaker criterion than 'prediction'. Some very recent work has applied topological data analysis to visualize patterns within the infamous Randu RNG.
  
  
  ---
  
  ### Answer by username_0 on 2016-08-03. Score: -1
  
  You would probably have to pack recursive structures into finite-dimensional real vectors and there have been such attempts. The finite precision limits goes as far as the recursion can go.
  
  The limitation of *feedforward* neural networks is restricted to finite input and output spaces, so *recurrent* may be more suitable for this task as in theory can process arbitrarily long strings of numbers, but it has much more practical difficulties than feedforward network.
  
  These kind of methods are open to debate.
  
  Source: [SAS FAQ](ftp://ftp.sas.com/pub/neural/FAQ.html)
  
  References:
  
  * Blair, 1997; Pollack, 1990; Chalmers, 1990; Chrisman, 1991; Plate, 1994; Hammerton, 1998; Hadley, 1999
  
  
  ---
  
  