First of all, almost any program-representation medium (except particularly trivial ones) can be used to represent any program, so that property doesn't tell you anything about the merits and limitations of a particular approach.
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Second, it is not relevant whether a program *can be represented* (that's almost always true), one care whether a program *can be developed* (not just in theory, but in practice) using a particular approach.
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So statements like "NNs are universal approximators (or, 'there are Turing-complete RNN architectures'), therefore DL has no limits" is like saying "you can use a pen & some paper to produce the complete works of Shakespeare". Technically true, but pretty silly at the same time.
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Google X
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Haven't you heard, early MS stated all anyone will ever need is 64k of RAM, or was that megs, maybe it was gigs... maybe terabytes....oh hell gates must have meant 64 googleplexes...
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I don't think any serious CS person would make that straw-man claim. X can be used to represent any function is a pretty interesting fact by itself. It might be trivial for you, it is not for me. (I saw an exchange, I think the other gentleman was making a much broader point)
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I think it goes without saying that any system of interest in what you are talking about has to be Turing complete. But that by itself does not say anything (and that is obvious). It is really a problem for representation (architectural?)/ perhaps in terms of optimization? etc.
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The mistake one level up from this is "X can learn any function given infinite data, therefore X is all we need". It's learning without universe-sized mounds of data that's the trick.
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Same with computation power but I suppose lots of data and lots of computation power are similar enough.
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