[Haskell-cafe] Why not some subclass of Floating to model NaNs as some handleable bottom?

[YueCompl]

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> 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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