Several Comments from Manuel Blum

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Dear Hunter — What a nice idea. And many of your examples below are both new to me and quite interesting. Thanks! And good luck! — manuel

While thinking about your blog, two things occurred to me that might be worth mentioning:

1. If a function f(x) has speedup, then any lower bound on its computation can be improved by a corresponding amount. For example, if every program for computing f(x) can be sped up to run twice as fast (on all but a finite number of integers), then any lower bound G(x) on its run time can be raised from G(x) to 2G(x) (on all but a finite number of integers). For another example, if any program for computing f(x) can be sped up by a sqrt (so that any run time F(x) can be reduced to a runtime of at most sqrt(F(x)), then any lower bound G(x)on its run time can be raised to [G(x)]^2, etc. This is all easy to see.

2. Much harder to see is a curious relation between speedup and inductive inference, which has to do with inferring an algorithm from observation of the sequence of integers that it generates. Theorem: there exists an inductive inference algorithm for inferring all sequences that have optimal algorithms (i.e. have programs that cannot be sped up)! This was quite a surprise (and a breakthrough) for me. Still is. To explain it though, I’d have to explain inductive inference, etc, and this would take me a bit of time. Some day…

Anyway, thanks again for your blog.

Best wishes and warm regards,

manuel (blum)

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