Hi Petr.

As Tom hinted, Stream is isomorphic to function-from-Nat (Peano/lazy natural numbers), being the memoized (trie) representation of such functions. The Stream Monad instance corresponds to the function-from-a Monad ("reader") instance, so all such trie Monad instances are useful wherever we want to work monadically with functions but want memoization.

I like Olaf's example use as well: 'fmap not . join' constructs a stream of binary representations of numbers in [0,1] that disagrees with every element of any given enumeration of such numbers. A very terse demonstration of a profoundly important mathematical discovery.

- Conal

On Tue, Jul 12, 2016 at 4:10 AM, Petr Pudlák <petr.mvd@gmail.com> wrote:

Indeed, the question is not wherever it's a monad or not, just to what extent is the monad useful.

Dne po 11. 7. 2016 22:08 uživatel Olaf Klinke <olf@aatal-apotheke.de> napsal:
As someone else on this list whose name I don't recall put it, there is no
choice on whether to make Stream a monad or not. It simply _is_ a monad.
Anyone with some CS education hearing 'diagonal of inifinite list of
infinite lists' should immediately think of Georg Cantor.

import Data.Stream
import Data.Ratio

type Real = Stream Rational
-- approximate a real number by an ascending stream
-- of lower bounds.

supremum :: Stream Real -> Real
supremum = join
-- If a stream of reals is index-wise ascending,
-- the monad instance for Stream computes its supremum.
-- Use this e.g. to compute any mathematical quantity
-- defined as a supremum.

-- Olaf

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