Re: [Haskell-cafe] type metaphysics

Luke, sorry for being offtopic, but you are more and more intriguing me with topology. I wonder if any stuff from it has, apart from applications in computability/complexity, also computational applications as useful as monoids or rings do (i.e. parallel prefix sums). 2009/2/3 Luke Palmer :

On Mon, Feb 2, 2009 at 4:18 PM, Reid Barton wrote:

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So here's a programming challenge: write a total function (expecting total arguments) toSame :: ((Nat -> Bool) -> Nat) -> (Nat -> Bool,Nat -> Bool) that finds a pair that get mapped to the same Nat.

Ie. f a==f b where (a,b) = toSame f

(Warning: sketchy argument ahead.) Let f :: (Nat -> Bool) -> Nat be a total function and let g0 = const True. The application f g0 can only evaluate g0 at finitely many values, so f g0 = f (< k) for any k larger than all these values. So we can write

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toSame f = (const True, head [ (< k) | k <- [1..], f (const True) == f (< k) ])

and toSame is total on total inputs.

Well done! That's not sketchy at all! There is always such a k (when the result type of f has decidable equality) and it is the "modulus of uniform continuity" of f. This is computable directly, but the implementation you've provided might come up with a smaller one that still works (since you only need to differentiate between const True, not all other streams).

I guess I should hold off on conjecturing the impossibility of things... :-)

Luke

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