I'd like to clear the least significant set bit of an arbitrary Bits type. The problem is, that I do not necessarily have the Num instance, so x.&.(x-1) is not an option. The only function I could think of is this:
clearLsb x = x `clearBit` countTrailingZeros x
I benchmarked it against the x&(x-1) version, and it is 1.5 slower on Word32 and Word64 regardless of the level of optimization. I would appreciate if anyone knows some clever hack to do it.
You may overload the function to pick the more efficient implementation at the type level, when it is available. This requires adding a type class, but even with countTrailingZeros implementation you already have to impose some type-class constraint to your function (namely FiniteBits) (1).
In particular, with some language extensions, all Num types can set to use a .&. (a - 1) equation(2):
{-# LANGUAGE FlexibleInstances, UndecidableInstances #-}
import Data.Bits (Bits, (.&.))
class LSB a where
clear :: a -> a
instance (Bits a, Num a) => LSB a where
clear a = a .&. (a - 1) -- more efficient implementation
newtype Foo = ... -- some type not instance of Num
instance LSB Foo where
clear a = ... -- some other approach
1. do also note that with countTrailingZeros you are ruling out Integer type, that is countTrailingZeros (16 :: Integer) will not type check, since it is not an instance of FiniteBits.
2. Though would be better to write explicit instances than use those language extensions
Related
I want to create a Complex type to represent complex numbers.
Following works:
Prelude> data Complex = Complex Int Int
Prelude> :t Complex
Complex :: Int -> Int -> Complex
How can I change this to accept any Num type, instead of just Int.
I tried following:
Prelude> data Complex a = Num a => Complex a a
but got this:
* Data constructor `Complex' has existential type variables, a context, or a specialised result type
Complex :: forall a. Num a => a -> a -> Complex a
(Use ExistentialQuantification or GADTs to allow this)
* In the definition of data constructor `Complex'
In the data type declaration for `Complex'
I'm not really sure what to make of this error. Any help is appreciated.
Traditional data in Haskell is just that: data. It doesn't need to know anything about the properties of its fields, it just needs to be able to store them. Hence there's no real need to constrain the fields at that point; just make it
data Complex a = Complex !a !a
(! because strict fields are better for performance).
Of course when you then implement the Num instance, you will need a constraint:
instance (Num a) => Num (Complex a) where
fromInteger = (`Complex`0) . fromInteger
Complex r i + Complex ρ ι = Complex (r+ρ) (i+ι)
...
...in fact, you need the much stronger constraint RealFloat a to implement abs, at least that's how the standard version does it. (Which means, Complex Int is actually not usable, not with the standard Num hierarchy; you need e.g. Complex Double.)
That said, it is also possible to bake the constraint in to the data type itself. The ExistentialTypes syntax you tried is highly limiting though and not suitable for this; what you want instead is the GADT
data Complex a where
Complex :: Num a => a -> a -> Complex a
With that in place, you could then implement e.g. addition without mentioning any constraint in the signature
cplxAdd :: Complex a -> Complex a -> Complex a
cplxAdd (Complex r i) (Complex ρ ι) = Complex (r+ρ) (i+ι)
You would now need to fulfill Num whenever you try to construct a Complex value though. That means, you'd still need an explicit constraint in the Num instance.
Also, this version is potentially much slower, because the Num dictionary actually needs to be stored in the runtime representation.
Type constructors cannot be constrained in pure Haskell, only functions can. So it is supposed that you declare
data Complex a = Complex a a
and then constrain functions, like
conjugate :: (Num a) => Complex a -> Complex a
conjugate (Complex x y) = Complex x (-y)
In fact, the type and constraint for conjugate can be derived by the compiler, so you can just define the implementation:
conjugate (Complex x y) = Complex x (-y)
However, if you really wish to constrain the type constructor Complex, you can turn on some extensions that enable it, namely ExistentialQuantification or GADTs, as the compiler suggests. To do this, add this line to the very beginning of your file:
{-# LANGUAGE ExistentialQuantification #-}
or
{-# LANGUAGE GADTs #-}
Those are called pragmas.
While you could, as the compiler message instructs, use ExistentialQuantification, you could also define the type like this:
data Complex a = Complex a a deriving (Show, Eq)
It's a completely unconstrained type, so perhaps another name would be more appropriate... This type seems to often be called Pair...
When you write functions, however, you can constrain the values contained in the type:
myFunction :: Num a => Complex a -> a
myFunction (Complex x y) = x + y
I am exposing a function which takes two parameters, one is a minimum bound and the other is a maximum bound. How can I ensure, using types, that for example the minimum bound is not greater than the maximum bound?
I want to avoid creating a smart constructor and returning a Maybe because it would make the whole usage more cumbersome.
Thank you
This doesn't exactly answer your question, but one approach that sometimes works is to change your interpretation of your type. For example, instead of
data Range = {lo :: Integer, hi :: Integer}
you could use
data Range = {lo :: Integer, size :: Natural}
This way, there's no way to represent an invalid range.
This solution uses dependent types (and might be too heavyweight, check if dfeuer's answer is enough for your needs).
The solution makes use of the GHC.TypeLits module from base, and also of the typelits-witnesses package.
Here is a difference function that takes two integer arguments (known statically) and complains at compile-time when the first number is greater than the second:
{-# language TypeFamilies #-}
{-# language TypeOperators #-}
{-# language DataKinds #-}
{-# language ScopedTypeVariables #-}
import GHC.TypeLits
import GHC.TypeLits.Compare
import Data.Type.Equality
import Data.Proxy
import Control.Applicative
difference :: forall n m. (KnownNat n,KnownNat m,n <= m)
=> Proxy n
-> Proxy m
-> Integer
difference pn pm = natVal pm - natVal pn
We can check it from GHCi:
ghci> import Data.Proxy
ghci> :set -XTypeApplications
ghci> :set -XDataKinds
ghci> difference (Proxy #2) (Proxy #7)
5
ghci> difference (Proxy #7) (Proxy #2)
** TYPE ERROR **
But what if we want to use the function with values determined at run time? Say, values that we read from console, or from a file?
main :: IO ()
main = do
case (,) <$> someNatVal 2 <*> someNatVal 7 of
Just (SomeNat proxyn,SomeNat proxym) ->
case isLE proxyn proxym of
Just Refl -> print $ difference proxyn proxym
Nothing -> error "first number not less or equal"
Nothing ->
error "could not bring KnownNat into scope"
In this case, we use functions like someNatVal and isLE. These functions might fail with Nothing. If they succeed, however, they return a value that "witnesses" some constraint. And by pattern-matching on the witness, we bring that constraint into scope (this works because the witness is a GADT).
In the example, the Just (SomeNat proxyn,SomeNat proxym) pattern match brings KnownNat constraints for the two arguments into scope. And the Just Refl pattern match brings the n <= m constraint into scope. Only then we can call our difference function.
So, in a way, we have shifted all the busywork of ensuring that the arguments satisfy the required preconditions out of the function itself.
What you're asking for is dependent types. There is a nice tutorial on it in
https://www.schoolofhaskell.com/user/konn/prove-your-haskell-for-great-safety/dependent-types-in-haskell
Although I don't know how friendly it will be. Do note that dependent typing was improved in GHC 8.0 but I have no experience in that area. I would make sure you're comfortable using template Haskell if you don't want it to be cumbersome.
You needn't invoke the Maybe type to take advantage of 'smart constructors'. If you like, you may accept constructors of either form (min,max) or (max,min) and still create a data type which correctly interprets which is which.
For instance, you could make a little module:
module RangeMinMax (
Range,
makeRange
)
where
data Range = Range (Integer,Integer)
deriving Show
makeRange a b = Range (min a b, max a b)
And now when you create a Range using makeRange, the 2-tuple will automatically be arranged so it's in the form (min,max). Note that the constructor for Range is not exported, so the user of the module is unable to create an invalid Range-- you just need to make sure that you create valid ones in the this module.
I'm trying to create a custom data type. As an example
data Time = Second Int deriving (Show)
However, this is too limiting (we could say later need milliseconds). I would like to instead define something like this:
data Time = Second Num deriving (Show)
This doesn't compile because Num has kind * -> ghc-prim-0.4.0.0:GHC.Prim.Constraint
How do I setup Time such that Second may contain any Num?
One of the best examples of why this might not be so desirable is found here at the Wikibooks section on Classes and Types. They say:
Type constraints in data declarations are less useful than it might seem at first. Consider:
data (Num a) => Foo a = F1 a | F2 a String
Here, Foo is a type with two constructors, both taking an argument of a type a which must be in Num. However, the (Num a) => constraint is only effective for the F1 and F2 constructors, and not for other functions involving Foo. Therefore, in the following example...
fooSquared :: (Num a) => Foo a -> Foo a
fooSquared (F1 x) = F1 (x * x)
fooSquared (F2 x s) = F2 (x * x) s
... even though the constructors ensure a will be some type in Num we can't avoid duplicating the constraint in the signature of fooSquared
This suggests that a reasonable option for you is to just create Time with a generic parameter, and then later ensure that the module functions that operate on Time data always have the necessary constraint for Num.
It won't be so much of a worry that someone goes off and foolishly makes Time String or something -- if they do, then none of the provided module functions are going to be helpful for them, so it doesn't matter so much.
There are also options to look up with GADTs, the {-# LANGUAGE GeneralizedNewtypeDeriving #-} pragma, and the {-# LANGUAGE DatatypeContexts #-} pragma. But usually these start to rope in unnecessary degrees of extra complexity, especially if you're a Haskell novice like me.
There is a deprecated feature called Datatype Contexts that allows you to do that:
{-# LANGUAGE DatatypeContexts #-}
data Num a => Time a = Second a deriving (Show)
t = Second (5 :: Int)
main = print t
This executes on GHC 7.8.3 (sorry, don't have 7.10 to check), but warns you about the deprecation:
t.hs:1:14: Warning:
-XDatatypeContexts is deprecated: It was widely considered a
misfeature, and has been removed from the Haskell language.
Second 5
A non-deprecated way to do it is to use Generalized Algebraic Datatypes (GADTs) (and you'll need standalone deriving as well):
{-# LANGUAGE GADTs, StandaloneDeriving #-}
data Time a where
Second :: Num a => a -> Time a
deriving instance Show a => Show (Time a)
t = Second (5 :: Int)
main = print t
If you try to create a variable with something non-Num, you'll get a compilation error:
t = Second "a"
t.hs:12:5:
No instance for (Num [Char]) arising from a use of ‘Second’
In the expression: Second "a"
In an equation for ‘t’: t = Second "a"
Given the following code:
import Data.Word
data T = T deriving (Eq, Show)
class C a where f :: a -> ()
instance C T where f _ = ()
instance C Word16 where f _ = ()
main = return $ f 0x16
GHC complains that it can't infer what the type for the literal 0x16 should be with the error:
No instance for (Num a0) arising from the literal ‘22’
The type variable ‘a0’ is ambiguous
It is easy to see why this would be -- Haskell allows numeric literals to be of any type which has an instance of Num, and here we can't disambiguate what the type for the literal 0x16 (or 22) should be.
It's also clear as a human reading this what I intended to do -- there is only one available instance of the class C which satisfies the Num constraint, so obviously I intended to use that one so 0x16 should be treated as a Word16.
There are two ways that I know to fix it: Either annotate the literal with its type:
main = return $ f (0x16 :: Word16)
or define a function which essentially does that annotation for you:
w16 x = x :: Word16
main = return $ f (w16 0x16)
I have tried a third way, sticking default (Word16) at the top of the file in the hope that Haskell would pick that as the default type for numeric literals, but I guess I'm misunderstanding what the default keyword is supposed to do because that didn't work.
I understand that typeclasses are open, so just because you can make the assumption in the context quoted above that Word16 is the only numeric instance of C that may not hold in some other module. But my question is: is there some mechanism by which I can assume/enforce that property, so that it is possible to use f and have Haskell resolve the type of its numeric argument to Word16 without explicit annotations at the call site?
The context is that I am implementing an EDSL, and I would rather not have to include manual type hints when I know that my parameters will either be Word16 or some other non-numeric type. I am open to a bit of dirty types/extensions abuse if it makes the EDSL feel more natural! Although if solutions do involve the naughty pragmas I'd definitely appreciate hints on what I should be wary about when using them.
Quick solution with "naughty pragmas" with GHC 7.10:
{-# LANGUAGE TypeFamilies, FlexibleInstances #-}
class C a where f :: a -> ()
instance C T where f _ = ()
instance {-# INCOHERENT #-} (w ~ Word16) => C w where f _ = ()
And with GHC 7.8:
{-# LANGUAGE TypeFamilies, FlexibleInstances, IncoherentInstances #-}
class C a where f :: a -> ()
instance C T where f _ = ()
instance (w ~ Word16) => C w where f _ = ()
Here, GHC essentially picks an arbitrary most specific instance that remains after trying to unify the instances heads and constraints.
You should only use this if
You have a fixed set of instances and don't export the class.
For all use cases of the class method, there is a single possible most specific instance (given the constraints).
Many people advise against ever using IncoherentInstances, but I think it can be quite fun for DSL-s, if we observe the above considerations.
For anybody else wondering about default (I know I was!)
https://www.haskell.org/onlinereport/haskell2010/haskellch4.html#x10-750004.3
Quoting section 4.3.4:
In situations where an ambiguous type is discovered, an ambiguous type variable, v, is defaultable if:
v appears only in constraints of the form C v, where C is a class, and
at least one of these classes is a numeric class, (that is, Num or a subclass of Num), and
all of these classes are defined in the Prelude or a standard library.
So that explains why your default clause is being completely ignored; C is not a standard library type-class.
(As to why this is the rule… can't help you there. Presumably to avoid breaking arbitrary user-defined code.)
The function f below, for a given type 'a', takes a parameter of type 'c'. For different types 'a', 'c' is restricted in different ways. Concretely, when 'a' is any Integral type, 'c' should be allowed to be any 'Real' type. When 'a' is Float, 'c' can ONLY be Float.
One attempt is:
{-# LANGUAGE
MultiParamTypeClasses,
FlexibleInstances,
FunctionalDependencies,
UndecidableInstances #-}
class AllowedParamType a c | a -> c
class Foo a where
f :: (AllowedParamType a c) => c -> a
fIntegral :: (Integral a, Real c) => c -> a
fIntegral = error "implementation elided"
instance (Integral i, AllowedParamType i d, Real d) => Foo i where
f = fIntegral
For some reason, GHC 7.4.1 complains that it "could not deduce (Real c) arising from a use of fIntegral". It seems to me that the functional dependency should allow this deduction. In the instance, a is unified with i, so by the functional dependency, d should be unified with c, which in the instance is declared to be 'Real'. What am I missing here?
Functional dependencies aside, will this approach be expressive enough to enforce the restrictions above, or is there a better way? We are only working with a few different values for 'a', so there will be instances like:
instance (Integral i, Real c) => AllowedParamType i c
instance AllowedParamType Float Float
Thanks
A possibly better way, is to use constraint kinds and type families (GHC extensions, requires GHC 7.4, I think). This allows you to specify the constraint as part of the class instance.
{-# LANGUAGE ConstraintKinds, TypeFamilies, FlexibleInstances, UndecidableInstances #-}
import GHC.Exts (Constraint)
class Foo a where
type ParamConstraint a b :: Constraint
f :: ParamConstraint a b => b -> a
instance Integral i => Foo i where
type ParamConstraint i b = Real b
f = fIntegral
EDIT: Upon further experimentation, there are some subtleties that mean that this doesn't work as expected, specifically, type ParamConstraint i b = Real b is too general. I don't know a solution (or if one exists) right now.
OK, this one's been nagging at me. given the wide variety of instances,
let's go the whole hog and get rid of any relationship between the
source and target type other than the presence of an instance:
{-# LANGUAGE OverlappingInstances, FlexibleInstances,TypeSynonymInstances,MultiParamTypeClasses #-}
class Foo a b where f :: a -> b
Now we can match up pairs of types with an f between them however we like, for example:
instance Foo Int Int where f = (+1)
instance Foo Int Integer where f = toInteger.((7::Int) -)
instance Foo Integer Int where f = fromInteger.(^ (2::Integer))
instance Foo Integer Integer where f = (*100)
instance Foo Char Char where f = id
instance Foo Char String where f = (:[]) -- requires TypeSynonymInstances
instance (Foo a b,Functor f) => Foo (f a) (f b) where f = fmap f -- requires FlexibleInstances
instance Foo Float Int where f = round
instance Foo Integer Char where f n = head $ show n
This does mean a lot of explicit type annotation to avoid No instance for... and Ambiguous type error messages.
For example, you can't do main = print (f 6), but you can do main = print (f (6::Int)::Int)
You could list all of the instances with the standard types that you want,
which could lead to an awful lot of repetition, our you could light the blue touchpaper and do:
instance Integral i => Foo Double i where f = round -- requires FlexibleInstances
instance Real r => Foo Integer r where f = fromInteger -- requires FlexibleInstances
Beware: this does not mean "Hey, if you've got an integral type i,
you can have an instance Foo Double i for free using this handy round function",
it means: "every time you have any type i, it's definitely an instance
Foo Double i. By the way, I'm using round for this, so unless your type i is Integral,
we're going to fall out." That's a big issue for the Foo Integer Char instance, for example.
This can easily break your other instances, so if you now type f (5::Integer) :: Integer you get
Overlapping instances for Foo Integer Integer
arising from a use of `f'
Matching instances:
instance Foo Integer Integer
instance Real r => Foo Integer r
You can change your pragmas to include OverlappingInstances:
{-# LANGUAGE OverlappingInstances, FlexibleInstances,TypeSynonymInstances,MultiParamTypeClasses #-}
So now f (5::Integer) :: Integer returns 500, so clearly it's using the more specific Foo Integer Integer instance.
I think this sort of approach might work for you, defining many instances by hand, carefully considering when to go completely wild
making instances out of standard type classes. (Alternatively, there aren't all that many standard types, and as we all know, notMany choose 2 = notIntractablyMany, so you could just list them all.)
Here's a suggestion to solve a more general problem, not yours specifically (I need more detail yet first - I promise to check later). I'm writing it in case other people are searching for a solution to a similar problem to you, I certainly was in the past, before I discovered SO. SO is especially great when it helps you try a radically new approach.
I used to have the work habit:
Introduce a multi-parameter type class (Types hanging out all over the place, so...)
Introduce functional dependencies (Should tidy it up but then I end up needing...)
Add FlexibleInstances (Alarm bells start ringing. There's a reason the compiler has this off by default...)
Add UndecidableInstances (GHC is telling you you're on your own, because it's not convinced it's up to the challenge you're setting it.)
Everything blows up. Refactor somehow.
Then I discovered the joys of type families (functional programming for types (hooray) - multi-parameter type classes are (a bit like) logic programming for types). My workflow changed to:
Introduce a type class including an associated type, i.e. replace
class MyProblematicClass a b | a -> b where
thing :: a -> b
thang :: b -> a -> b
with
class MyJustWorksClass a where
type Thing a :: * -- Thing a is a type (*), not a type constructor (* -> *)
thing :: a -> Thing a
thang :: Thing a -> a -> Thing a
Nervously add FlexibleInstances. Nothing goes wrong at all.
Sometimes fix things by using constraints like (MyJustWorksClass j,j~a)=> instead of (MyJustWorksClass a)=> or (Show t,t ~ Thing a,...)=> instead of (Show (Thing a),...) => to help ghc out. (~ essentially means 'is the same type as')
Nervously add FlexibleContexts. Nothing goes wrong at all.
Everything works.
The reason "Nothing goes wrong at all" is that ghc calculates the type Thing a using my type function Thang rather than trying to deduce it using a merely a bunch of assertions that there's a function there and it ought to be able to work it out.
Give it a go! Read Fun with Type Functions before reading the manual!