Presumably a linear combination of programs means a quantum superposition. But not much makes sense after that. Is the group just S_{2^n}?
Really fun question. One partial idea: you can give the space of programs a finite group structure by using ideas from https://en.wikipedia.org/wiki/Reversible_computing …. And then take a discrete Fourier transform over a finite group. But what does a linear combination of programs mean, initially? Hmm.https://twitter.com/warrenm/status/997326005411897344 …
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I'd rather an interpretation in terms of conventional computers, which have the non-trivial benefit of existing. Yes, the group will be some permutation group, though of course you'd like to relate the idea back to what it means about programs.
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Seems like it would be a highly nonabelian group, and FFTs are most useful in the abelian setting. We're still waiting for something to figure out nonabelian FFT and solve graph isomorphism in quantum polynomial time. :)
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yes, and also a legitimate question is why it is a finite group.
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The convolutional neural network, especially in the context of sequence processing is essentially taking the Fourier transform of a program and is arguably meaningful
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@em_gently
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And that question is related the thought that provoked this: how can one preserve well-formedness (and ideally some sort of useful semantics) across transformations in the program "frequency" domain?
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The Fourier transform over a finite group would imply relationship with the convolution operator over that finite group. Therefore simplifying the convolution calculation and thus providing an alternative way of computing for equivariance. That's my guess. ;-)
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I keep coming back to this half-fleshed-out notion of computation as fundamentally relational, ergo tabular, which seems to fit into this space somehow?https://gist.github.com/mbilokonsky/ebea6bac180f4c3b57eed5b1fc5e170e …
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So a program is a sequence of group operations or programs are acted upon by group operations? If former, "+" can be group composition. But then it cannot be a vector space for programs. But you wouldn't want "+" to be abelian anyhow for programs.
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