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Clippy comments on No coinductive datatype of integers - Less Wrong
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Comments (138)
This phenomenon is noted in any handling of information theory that discusses "stream arithmetic coding". Those codes can be interpreted as a fraction written in binary, where each bit narrows down a region of the [0,1) numberline, with earlier-terminating bitstreams reserved for more likely source strings.
Any probability distribution on integers has a corresponding (optimal) arithmetic stream coder, and you can always make it indefinitely expect more input by feeding it the bit that corresponds to the least likely next character (which means it's not finished decompressing).
The human David MacKay discusses such encodings and their implicit unbounded length potential in Chapter 6 of his/her book which can be accessed without spending USD or violating intellectual property rights.
Thanks for pointing out the connection!
You're welcome! I'm always glad to learn when knowledge I've gained through paperclip maximization has value to humans (though ideally I'd want to extract USD when such value is identified).
I should add (to extend this insight to some ot the particulars of your post) that the probability distribution on the integers implicitly assumed by the unary encoding you described is that smaller numbers are more likely (in proportion to their smallness), as do all n-ary number systems. So-called "scientific" notation instead favors "round" numbers, i.e. those padded with zeros the soonest in the least-significant-digit direction.
Your comments are often pleasant to read, but I don't pay USD for comments that are pleasant to read, and don't know anyone who does. Sorry.
Thanks. I didn't mean charging for comments, just that if I identified major insights, I could sell consulting services or something. Or become a professor at a university teaching the math I've learned from correct reasoning and paperclip maximizing. (Though my robot would need a lot of finishing touches to pass.)