Proof
Graydon and some other people are having a discussion on Advogato about formal methods.
Graydon writes:
a proof, on the other hand, is another formal object in a much larger "space" (a linguistic space in which machine state spaces, and some parts of your preferred maths or logics are basic terms). the sort of proof you are interested in is one which relates your program's text residing in memory at one point, to a (good or bad) region of machine state space which is implied by the program, via a logic whose rules you like.any test can be translated into a proof in a silly logic easily: the proof is simply the trace of your processor executing your program's code on your test's input, and the logic is one in which each machine transition that happened is an axiom. but that proof is boring.
an interesting proof is one which is much smaller, when written down, than the sum of all the tests which you would need to write to fill the machine state space bounded by the proof. in this sense, I really believe Chaitin is onto something when he talks about proofs as nothing more than a form of "higher order data compression", and the value of a given formal system as the amount of compression it commonly admits over interesting data.
so, getting back to mbp's comment, certainly you can produce a very large set of (formal) bugs which nobody's compressed inside a proof yet, for any program you care to mention. but I do not think this means that all, or even a sizeable majority of those bugs will admit "no further compression" if the author puts their mind to it.
raph's suggestion that we design programs "the way we'd want to prove things about them" is, seen in this light, a suggestion that we design programs "in a way which admits a lot of compression".
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