[Haskell-cafe] Why not some subclass of Floating to model NaNs as some handleable bottom?
YueCompl
compl.yue at icloud.com
Sat Aug 7 10:21:32 UTC 2021
Okay, I got it working to some extent:
(and I find it a good showcase for my https://marketplace.visualstudio.com/items?itemName=ComplYue.vscode-ghci <https://marketplace.visualstudio.com/items?itemName=ComplYue.vscode-ghci> extension, improved it a bit to support this very scenario, with the src file at https://github.com/complyue/typing.hs/blob/main/src/PoC/Floating.hs <https://github.com/complyue/typing.hs/blob/main/src/PoC/Floating.hs> )
Obviously my naive implementation `(l, r) / (l', r') = ([x / x' | x <- l, x' <- l'], [y / y' | y <- r, y' <- r'])` is wrong, I think I need to figure out how to represent 1 (the unit number) of this type, even before I can come to a correct definition of the division (/) operation, but so far no clue ...
> On 2021-08-07, at 16:16, YueCompl via Haskell-Cafe <haskell-cafe at haskell.org> wrote:
>
> Another failed attempt:
>
> λ> :set -XTypeSynonymInstances
> λ> :set -XFlexibleInstances
> λ>
> λ> data Q = Q
> λ>
> λ> type NonNegativeNumber = ([Q],[Q])
> λ> :{
> λ| instance Num NonNegativeNumber where
> λ| (l,r) * (l',r') = ([x*x'|x <- l, x' <- l'],[y*y'|y <- r, y' <- r'])
> λ| :}
>
> <interactive>:12:25: error:
> • No instance for (Num Q) arising from a use of ‘*’
> • In the expression: x * x'
> In the expression: [x * x' | x <- l, x' <- l']
> In the expression:
> ([x * x' | x <- l, x' <- l'], [y * y' | y <- r, y' <- r'])
> λ>
> λ> zero = ([],Q)
> λ> infty = (Q,[])
> λ> zero * infty
>
> <interactive>:17:8: error:
> • Couldn't match type ‘Q’ with ‘[a]’
> Expected type: ([a], Q)
> Actual type: (Q, [a0])
> • In the second argument of ‘(*)’, namely ‘infty’
> In the expression: zero * infty
> In an equation for ‘it’: it = zero * infty
> • Relevant bindings include
> it :: ([a], Q) (bound at <interactive>:17:1)
> λ>
>
>
>> On 2021-08-07, at 15:35, YueCompl via Haskell-Cafe <haskell-cafe at haskell.org <mailto:haskell-cafe at haskell.org>> wrote:
>>
>> Great! I'm intrigued by the idea that GHCi can make such math sentences runnable! I've never thought it this way before.
>>
>> But I need some help to get it going:
>>
>> λ> :set -XTypeSynonymInstances
>> λ> :set -XFlexibleInstances
>> λ>
>> λ> import Data.Ratio
>> λ> type Q = Rational -- this is probably wrong ...
>> λ>
>> λ> type NonNegativeNumber = ([Q],[Q])
>> λ> :{
>> λ| instance Num NonNegativeNumber where
>> λ| (l,r) * (l',r') = ([x*x'|x <- l, x' <- l'],[y*y'|y <- r, y' <- r'])
>> λ| :}
>>
>> <interactive>:9:12: warning: [-Wmissing-methods]
>> • No explicit implementation for
>> ‘+’, ‘abs’, ‘signum’, ‘fromInteger’, and (either ‘negate’ or ‘-’)
>> • In the instance declaration for ‘Num NonNegativeNumber’
>> λ>
>> λ> zero = ([],Q)
>>
>> <interactive>:13:13: error: Data constructor not in scope: Q
>> λ> infty = (Q,[])
>>
>> <interactive>:14:10: error: Data constructor not in scope: Q
>> λ>
>> λ> zero * infty -- expect: = ([],[])
>>
>> <interactive>:16:1: error: Variable not in scope: zero
>>
>> <interactive>:16:8: error: Variable not in scope: infty
>> λ>
>>
>> I'd like to do more exercises, but I'm stuck here ...
>>
>>
>>> On 2021-08-07, at 06:16, Olaf Klinke <olf at aatal-apotheke.de <mailto:olf at aatal-apotheke.de>> wrote:
>>>
>>> On Fri, 2021-08-06 at 22:21 +0800, YueCompl wrote:
>>>> Thanks Olaf,
>>>>
>>>> Metaphors to other constructs are quite inspiring to me, though I don't have sufficient theoretical background to fully understand them atm.
>>>>
>>> Pen-and-paper or GHCi experiments suffice here, no fancy theoretical
>>> background needed. Say Q is the type of rationals 0 < q and we express
>>> type NonNegativeNumber = ([Q],[Q])
>>> where the first (infinite) list is the lower approximants and the
>>> second the upper approximants. Multiplication is then defined as
>>> (l,r) * (l',r') = ([x*x'|x <- l, x' <- l'],[y*y'|y <- r, y' <- r'])
>>> The extremes of this type are
>>> 0 = ([],Q)
>>> infty = (Q,[])
>>> It is easily seen that
>>> 0 * infty = ([],[])
>>> a number with no lower and no upper approximants, in other words, NaN.
>>> Excercise: Define division for this type and find out what 1/0 and 0/0
>>> is.
>>>
>>>> Am I understanding it right, that you mean `0/0` is hopeful to get ratified as "a special Float value corresponding to one non-proper Dedekind-cuts", but `NaN` as with IEEE semantics is so broken that it can only live outlaw, even Haskell the language shall decide to bless it?
>>>>
>>> Yes. I think it is vital that we provide a migration path for
>>> programmers coming from other languages. Under the Dedekind
>>> cut/interval interpretation, NaN would behave differently, as I pointed
>>> out. So I'd leave Float as it is, but be more verbose about its
>>> violation of type class laws. To this end, one could have (and now I
>>> might be closer to your initial question) numerical type classes like
>>> HonestEq, HonestOrd, HonestMonoid and HonestRing whose members are only
>>> those types that obey the laws in all elements. Naturally, Float would
>>> not be a member. Who would use these new classes? Probably no one,
>>> because we all like to take the quick and dirty route. But at least it
>>> says clearly: Careful, you can not rely on these laws when using Float.
>>>
>>> Olaf
>>>
>>
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