I'm trying to create a data type representing an objects position that can 'GO', 'STOP', move 'FORWARD', and move 'BACKWARD'. I can't figure out how to write the deriving instance of 'Eq' and 'Show' for the functions FORWARD and BACKWARD.
data Moves = GO
| STOP
| FORWARD { f :: Int -> Int -> Int }
| BACKWARD { g :: Int -> Int -> Int }
deriving (Eq, Show)
instance (Eq Moves) where
FORWARD a == FORWARD b = True
FORWARD a == BACKWARD b = True
BACKWARD a == BACKWARD b = True
BACKWARD a == FORWARD b = True
_ == _ = False
The logic for the instance doesn't matter at the moment I just can't figure out how to get it to compile. Thanks
When you automatically derive various standard typeclasses via the deriving clause, this relies on the individual fields or branches of the data structure already having instances for those classes. Since functions have no default instance for either Eq or Show (which is perfectly reasonable, there's no obvious canonical way to either determine if 2 arbitrary functions are equal*, or to print them as strings), and your datatype includes 2 fields whose values are functions, it's impossible for Haskell to derive Eq and Show instances, and this is what the compilation error is about.
The solution is simply to remove the deriving (Eq, Show) part - you don't need this anyway if you're going to define your own custom instances. (The only reason to put a deriving clause is if you need an instance and are happy with the "standard" one - which a lot of the time you would be. Here though, you seem to be wanting to implement your own instances with non-standard logic. That is fine if your use-case demands it.)
*actually in mathematics it's clear what equality of functions means, functions are defined by their graph, which means that for two functions to be equal, they must give the same output for every possible input value. That's still theoretically important in Haskell programming (since the "laws" for various type classes require various functions to be equal), but it's not possible to implement in any reasonable way in general, not least because some functions can run forever without giving a result, so actually computing equality of functions in general is rather stronger than solving the Halting Problem!)
Related
In Haskell id function is defined on type level as id :: a -> a and implemented as just returning its argument without any modification, but if we have some type introspection with TypeApplications we can try to modify values without breaking type signature:
{-# LANGUAGE AllowAmbiguousTypes, ScopedTypeVariables, TypeApplications #-}
module Main where
class TypeOf a where
typeOf :: String
instance TypeOf Bool where
typeOf = "Bool"
instance TypeOf Char where
typeOf = "Char"
instance TypeOf Int where
typeOf = "Int"
tweakId :: forall a. TypeOf a => a -> a
tweakId x
| typeOf #a == "Bool" = not x
| typeOf #a == "Int" = x+1
| typeOf #a == "Char" = x
| otherwise = x
This fail with error:
"Couldn't match expected type ‘a’ with actual type ‘Bool’"
But I don't see any problems here (type signature satisfied):
My question is:
How can we do such a thing in a Haskell?
If we can't, that is theoretical\philosophical etc reasons for this?
If this implementation of tweak_id is not "original id", what are theoretical roots that id function must not to do any modifications on term level. Or can we have many implementations of id :: a -> a function (I see that in practice we can, I can implement such a function in Python for example, but what the theory behind Haskell says to this?)
You need GADTs for that.
{-# LANGUAGE ScopedTypeVariables, TypeApplications, GADTs #-}
import Data.Typeable
import Data.Type.Equality
tweakId :: forall a. Typeable a => a -> a
tweakId x
| Just Refl <- eqT #a #Int = x + 1
-- etc. etc.
| otherwise = x
Here we use eqT #type1 #type2 to check whether the two types are equal. If they are, the result is Just Refl and pattern matching on that Refl is enough to convince the type checker that the two types are indeed equal, so we can use x + 1 since x is now no longer only of type a but also of type Int.
This check requires runtime type information, which we usually do not have due to Haskell's type erasure property. The information is provided by the Typeable type class.
This can also be achieved using a user-defined class like your TypeOf if we make it provide a custom GADT value. This can work well if we want to encode some constraint like "type a is either an Int, a Bool, or a String" where we statically know what types to allow (we can even recursively define a set of allowed types in this way). However, to allow any type, including ones that have not yet been defined, we need something like Typeable. That is also very convenient since any user-defined type is automatically made an instance of Typeable.
This fail with error: "Couldn't match expected type ‘a’ with actual type ‘Bool’"But I don't see any problems here
Well, what if I add this instance:
instance TypeOf Float where
typeOf = "Bool"
Do you see the problem now? Nothing prevents somebody from adding such an instance, no matter how silly it is. And so the compiler can't possibly make the assumption that having checked typeOf #a == "Bool" is sufficient to actually use x as being of type Bool.
You can squelch the error if you are confident that nobody will add malicious instances, by using unsafe coercions.
import Unsafe.Coerce
tweakId :: forall a. TypeOf a => a -> a
tweakId x
| typeOf #a == "Bool" = unsafeCoerce (not $ unsafeCoerce x)
| typeOf #a == "Int" = unsafeCoerce (unsafeCoerce x+1 :: Int)
| typeOf #a == "Char" = unsafeCoerce (unsafeCoerce x :: Char)
| otherwise = x
but I would not recommend this. The correct way is to not use strings as a poor man's type representation, but instead the standard Typeable class which is actually tamper-proof and comes with suitable GADTs so you don't need manual unsafe coercions. See chi's answer.
As an alternative, you could also use type-level strings and a functional dependency to make the unsafe coercions safe:
{-# LANGUAGE DataKinds, FunctionalDependencies
, ScopedTypeVariables, UnicodeSyntax, TypeApplications #-}
import GHC.TypeLits (KnownSymbol, symbolVal)
import Data.Proxy (Proxy(..))
import Unsafe.Coerce
class KnownSymbol t => TypeOf a t | a->t, t->a
instance TypeOf Bool "Bool"
instance TypeOf Int "Int"
tweakId :: ∀ a t . TypeOf a t => a -> a
tweakId x = case symbolVal #t Proxy of
"Bool" -> unsafeCoerce (not $ unsafeCoerce x)
"Int" -> unsafeCoerce (unsafeCoerce x + 1 :: Int)
_ -> x
The trick is that the fundep t->a makes writing another instance like
instance TypeOf Float "Bool"
a compile error right there.
Of course, really the most sensible approach is probably to not bother with any kind of manual type equality at all, but simply use the class right away for the behaviour changes you need:
class Tweakable a where
tweak :: a -> a
instance Tweakable Bool where
tweak = not
instance Tweakable Int where
tweak = (+1)
instance Tweakable Char where
tweak = id
The other answers are both very good for covering the ways you can do something like this in Haskell. But I thought it was worth adding something speaking more to this part of the question:
If we can't, that is theoretical\philosophical etc reasons for this?
Actually Haskellers do generally rely quite strongly on the theory that forbids something like your tweakId from existing with type forall a. a -> a. (Even though there are ways to cheat, using things like unsafeCoerce; this is usually considered bad style if you haven't done something like in leftaroundabout's answer, where a class with functional dependencies ensures the unsafe coerce is always valid)
Haskell uses parametric polymorphism1. That means we can write code that works on multiple types because it will treat them all the same; the code only uses operations that will work regardless of the specific type it is invoked on. This is expressed in Haskell types by using type variables; a function with a variable in its type can be used with any type at all substituted for the variable, because every single operation in the function definition will work regardless of what type is chosen.
About the simplest example is indeed the function id, which might be defined like this:
id :: forall a. a -> a
id x = x
Because it's parametrically polymorphic, we can simply choose any type at all we like and use id as if it was defined on that type. For example as if it were any of the following:
id :: Bool -> Bool
id x = x
id :: Int -> Int
id x = x
id :: Maybe (Int -> [IO Bool]) -> Maybe (Int -> [IO Bool])
id x = x
But to ensure that the definition does work for any type, the compiler has to check a very strong restriction. Our id function can only use operations that don't depend on any property of any specific type at all. We can't call not x because the x might not be a Bool, we can't call x + 1 because the x might not be a number, can't check whether x is equal to anything because it might not be a type that supports equality, etc, etc. In fact there is almost nothing you can do with x in the body of id. We can't even ignore x and return some other value of type a; this would require us to write an expression for a value that can be of any type at all and the only things that can do that are things like undefined that don't evaluate to a value at all (because they throw exceptions). It's often said that in fact there is only one valid function with type forall a. a -> a (and that is id)2.
This restriction on what you can do with values whose type contains variables isn't just a restriction for the sake of being picky, it's actually a huge part of what makes Haskell types useful. It means that just looking at the type of a function can often tell you quite a bit about what it can possibly do, and once you get used to it Haskellers rely on this kind of thinking all the time. For example, consider this function signature:
map :: forall a b. (a -> b) -> [a] -> [b]
Just from this type (and the assumption that the code doesn't do anything dumb like add in extra undefined elements of the list) I can tell:
All of the items in the resulting list come are results of the function input; map will have no other way of producing values of type b to put in the list (except undefined, etc).
All of the items in the resulting list correspond to something in the input list mapped through the function; map will have no way of getting any a values to feed to the function (except undefined, etc)
If any items of the input list are dropped or re-ordered, it will be done in a "blind" way that isn't considering the elements at all, only their position in the list; map ultimately has no way of testing any property of the a and b values to decide which order they should go in. For example it might leave out the third element, or swap the 2nd and 76th elements if there are at least 100 elements, etc. But if it applies rules like that it will have to always apply them, regardless of the actual items in the list. It cannot e.g. drop the 4th element if it is less than the 5th element, or only keep outputs from the function that are "truthy", etc.
None of this would be true if Haskell allowed parametrically polymorphic types to have Python-like definitions that check the type of their arguments and then run different code. Such a definition for map could check if the function is supposed to return integers and if so return [1, 2, 3, 4] regardless of the input list, etc. So the type checker would be enforcing a lot less (and thus catching fewer mistakes) if it worked this way.
This kind of reasoning is formalised in the concept of free theorems; it's literally possible to derive formal proofs about a piece of code from its type (and thus get theorems for free). You can google this if you're interested in further reading, but Haskellers generally use this concept informally rather doing real proofs.
Sometimes we do need non-parametric polymorphism. The main tool Haskell provides for that is type classes. If a type variable has a class constraint, then there will be an interface of class methods provided by that constraint. For example the Eq a constraint allows (==) :: a -> a -> Bool to be used, and your own TypeOf a constraint allows typeOf #a to be used. Type class methods do allow you to run different code for different types, so this breaks parametricity. Even just adding Eq a to the type of map means I can no longer assume property 3 from above.
map :: forall a b. Eq a => (a -> b) -> [a] -> [b]
Now map can tell whether some of the items in the original list are equal to each other, so it can use that to decide whether to include them in the result, and in what order. Likewise Monoid a or Monoid b would allow map to break the first two properties by using mempty :: a to produce new values that weren't in the list originally or didn't come from the function. If I add Typeable constraints I can't assume anything, because the function could do all of the Python-style checking of types to apply special-case logic, make use of existing values it knows about if a or b happen to be those types, etc.
So something like your tweakId cannot be given the type forall a. a -> a, for theoretical reasons that are also extremely practically important. But if you need a function that behaves like your tweakId adding a class constraint was the right thing to do to break out of the constraints of parametricity. However simply being able to get a String for each type isn't enough; typeOf #a == "Int" doesn't tell the type checker that a can be used in operations requiring an Int. It knows that in that branch the equality check returned True, but that's just a Bool; the type checker isn't able to reason backwards to why this particular Bool is True and deduce that it could only have happened if a were the type Int. But there are alternative constructs using GADTs that do give the type checker additional knowledge within certain code branches, allowing you to check types at runtime and use different code for each type. The class Typeable is specifically designed for this, but it's a hammer that completely bypasses parametricity; I think most Haskellers would prefer to keep more type-based reasoning intact where possible.
1 Parametric polymorphism is in contrast to class-based polymorphism you may have seen in OO languages (where each class says how a method is implemented for objects of that specific class), or ad-hoc polymophism (as seen in C++) where you simply define multiple definitions with the same name but different types and the types at each application determine which definition is used. I'm not covering those in detail, but the key distinction is both of them allow the definition to have different code for each supported type, rather than guaranteeing the same code will process all supported types.
2 It's not 100% true that there's only one valid function with type forall a. a -> a unless you hide some caveats in "valid". But if you don't use any unsafe features (like unsafeCoerce or the foreign language interface), then a function with type forall a. a -> a will either always throw an exception or it will return its argument unchanged.
The "always throws an exception" isn't terribly useful so we usually assume an unknown function with that type isn't going to do that, and thus ignore this possibility.
There are multiple ways to implement "returns its argument unchanged", like id x = head . head . head $ [[[x]]], but they can only differ from the normal id in being slower by building up some structure around x and then immediately tearing it down again. A caller that's only worrying about correctness (rather than performance) can treat them all the same.
Thus, ignoring the "always undefined" possibility and treating all of the dumb elaborations of id x = x the same, we come to the perspective where we can say "there's only one function with forall a. a -> a".
I'm writing a parser in Haskell (mostly just to learn). I have a working tokenizer and parser and I want to add line numbers when giving an error message. I have this type:
data Token = Lambda
| Dot
| LParen
| RParen
| Ident String
Back in OO land, I would just create a Metadata object that holds the token's position in the source code. So I could try this:
data Metadata = Pos String Int Int
Then, I could change Token to
data Token = Lambda Metadata
| Dot Metadata
| LParen Metadata
| RParen Metadata
| Ident String Metadata
However, my parser is written using pattern matching on the tokens. So now, all my pattern matching is broken because I need to also account for the Metadata. So that doesn't seem ideal. 99% of the time, I don't care about the Metadata.
So what's the "right" way to do what I want to do?
There’s a wide array of approaches to the design of syntax representations in Haskell, but I can offer some recommendations and reasoning.
It’s advisable to keep metadata annotations out of the Token type, so that it sticks to a single responsibility. If a Token represents just a token, its derived instances for Eq and so on will work as expected without needing to worry about when to ignore the annotation.
Thankfully, the alternatives are simple in this case. One option is to move the annotation info to a separate wrapper type.
-- An #'Anno' a# is a value of type #a# annotated with some 'Metadata'.
data Anno a = Anno { annotation :: Metadata, item :: a }
deriving
( Eq
, Ord
, Show
-- …
)
Now the tokeniser can return a sequence of annotated tokens, i.e. [Annotated Token]. You still need to update the use sites, but the changes are now much simpler. And you can ignore annotations in various ways:
-- Positional matching
f1 (Anno _meta (Ident name)) = …
-- Record matching
f2 Anno { item = Ident name } = …
-- With ‘NamedFieldPuns’
f3 Anno { item } = …
-- 'U'nannotated value; with ‘PatternSynonyms’
pattern U :: a -> Anno a
pattern U x <- Anno _meta x
f4 (U LParen) = …
You can deannotate a sequence of tokens with fmap item to reuse existing code that doesn’t care about location info. And since Anno is a type of kind Type -> Type, GHC can also derive Foldable, Functor, and Traversable for it, making it easy to operate on the annotated item with e.g. fmap and traverse.
This is the preferable approach for Token, but for a parsed AST containing annotations, you may want to make the annotation type a parameter of the AST type, for example:
data Expr a = Add a (Expr a) (Expr a) | Literal a Int
deriving (Eq, Foldable, Functor, Ord, Show, Traversable)
Then you can use Expr Metadata for an annotated term, or Expr () for an unannotated one. To compare terms for equality, such as in unit tests, you can use the Functor instance to strip out the annotations, e.g. void expr1 == void expr2, where void is equivalent to fmap (\ _meta -> ()) here.
In a larger codebase, if there’s a lot of code depending on a data type and you really want to avoid updating it all at once, you can wrap the old type in a module that exports a pattern synonym for each of the old constructors. This lets you gradually update the old code before deleting the adapter module.
Culturally, it’s typical in a self-contained Haskell codebase to simply make breaking changes, and let the compiler tell you everywhere that needs to be updated, since it’s so easy to do extensive refactoring with high assurance that it’s correct. We’re more concerned with backward compatibility when it comes to published library code, since that actually affects other people.
I was reading a Chapter 2 of Purely Functional Data Structures, which talks about unordered sets implemented as binary search trees. The code is written in ML, and ends up showing a signature ORDERED and a functor UnbalancedSet(Element: ORDERED): SET. Coming from more of a C++ background, this makes sense to me; custom comparison function objects form part of the type and can be passed in at construction time, and this seems fairly analogous to the ML functor way of doing things.
When it comes to Haskell, it seems the behavior depends only on the Ord instance, so if I wanted to have a set that had its order reversed, it seems like I'd have to use a newtype instance, e.g.
newtype ReverseInt = ReverseInt Int deriving (Eq, Show)
instance Ord ReverseInt where
compare (ReverseInt a) (ReverseInt b)
| a == b = EQ
| a < b = GT
| a > b = LT
which I could then use in a set:
let x = Set.fromList $ map ReverseInt [1..5]
Is there any better way of doing this sort of thing that doesn't resort to using newtype to create a different Ord instance?
No, this is really the way to go. Yes, having a newtype is sometimes annoying but you get some big benefits:
When you see a Set a and you know a, you immediately know what type of comparison it uses (sort of the same way that purity makes code more readable by not making you have to trace execution). You don't have to know where that Set a comes from.
For many cases, you can coerce your way through multiple newtypes at once. For example, I can turn xs = [1,2,3] :: Int into ys = [ReverseInt 1, ReverseInt 2, ReverseInt 3] :: [ReverseInt] just using ys = coerce xs :: [ReverseInt]. Unfortunately, that isn't the case for Set (and it shouldn't - you'd need the coercion function to be monotonic to not screw up the data structure invariants, and there is not yet a way to express that in the type system).
newtypes end up being more composable than you expect. For example, the ReverseInt type you made already exists in a form that generalizes to reversing any type with an Ord constraint: it is called Down. To be explicit, you could use Down Int instead of ReversedInt, and you get the instance you wrote out for free!
Of course, if you still feel very strongly about this, nothing is stopping you from writing your version of Set which has to have a field which is the comparison function it uses. Something like
data Set a = Set { comparisionKey :: a -> a -> Ordering
, ...
}
Then, every time you make a Set, you would have to pass in the comparison key.
This question already has answers here:
Understanding Haskell's Bool Deriving an Ord
(3 answers)
Closed 7 years ago.
Can someone please explain the following output?
Prelude> compare True False
GT
it :: Ordering
Prelude> compare False True
LT
it :: Ordering
Why are Bool type values ordered in Haskell - especially, since we can demonstrate that values of True and False are not exactly 1 and 0 (unlike many other languages)?
This is how the derived instance of Ord works:
data D = A | B | C deriving Ord
Given that datatype, we get C > B > A. Bool is defined as False | True, and it kind of makes sense when you look at other examples such as:
Maybe a = Nothing | Just a
Either a b = Left a | Right b
In each of the case having "some" ("truthy") value is greater than having no values at all (or having "left" or "bad" or "falsy" value).
While Bool is not Int, it can be converted to the 0,1 fragment of Int since it is an Enum type.
fromEnum False = 0
fromEnum True = 1
Now, the Enum could have been different, reversing 0 and 1, but that would probably be surprising to most programmers thinking about bits.
Since it has an Enum type, everything else being equal, it's better to define an Ord instance which follows the same order, satisfying
compare x y = compare (fromEnum x) (fromEnum y)
In fact, each instance generated from deriving (Eq, Ord, Enum) follows such property.
On a more theoretical note, logicians tend to order propositions from the strongest to the weakest (forming a lattice). In this structure, False (as a proposition) is the bottom, i.e. the least element, while True is the top. While this is only a convention (theory would be just as nice if we picked the opposite ordering), it's a good thing to be consistent.
Minor downside: the implication boolean connective is actually p <= q expressing that p implies q, instead of the converse as the "arrow" seems to indicate.
Let me answer your question with a question: Why is there an Ord instance for ()?
Unlike Bool, () has only one possible value: (). So why the hell would you ever want to compare it? There is only one value possible!
Basically, it's useful if all or most of the standard basic types have instances for common classes. It makes it easier to derive instances for your own types. If Foo doesn't have an Ord instance, and your new type has a single Foo field, then you can't auto-derive an Ord instance.
You might, for example, have some kind of tree type where we can attach several items of information to the leaves. Something like Tree x y z. And you might want to have an Eq instance to compare trees. It would be annoying if Tree () Int String didn't have an Eq instance just because () doesn't. So that's why () has Eq (and Ord and a few others).
Similar remarks apply to Bool. It might not sound particularly useful to compare two bool values, but it would be irritating if your Ord instance vanishes as soon as you put a bool in there.
(One other complicating factor is that sometimes we want Ord because there's a logically meaningful ordering for things, and sometimes we just want some arbitrary order, typically so we can use something as a key for Data.Map or similar. Arguably there ought to be two separate classes for that… but there isn't.)
Basically, it comes from math. In set theory or category theory boolean functions are usually thought of as classifiers of subsets/subobjects. In plain terms, function f :: a -> Bool is identified with filter f :: [a] -> [a]. So, if we change one value from False to True, the resulting filtered list (subset, subobject, whatever) is going to have more elements. Therefore, True is considered "bigger" than False.
This question already has answers here:
Difference between `data` and `newtype` in Haskell
(2 answers)
Closed 8 years ago.
It seems that a newtype definition is just a data definition that obeys some restrictions (e.g., only one constructor), and that due to these restrictions the runtime system can handle newtypes more efficiently. And the handling of pattern matching for undefined values is slightly different.
But suppose Haskell would only knew data definitions, no newtypes: couldn't the compiler find out for itself whether a given data definition obeys these restrictions, and automatically treat it more efficiently?
I'm sure I'm missing out on something, there must be some deeper reason for this.
Both newtype and the single-constructor data introduce a single value constructor, but the value constructor introduced by newtype is strict and the value constructor introduced by data is lazy. So if you have
data D = D Int
newtype N = N Int
Then N undefined is equivalent to undefined and causes an error when evaluated. But D undefined is not equivalent to undefined, and it can be evaluated as long as you don't try to peek inside.
Couldn't the compiler handle this for itself.
No, not really—this is a case where as the programmer you get to decide whether the constructor is strict or lazy. To understand when and how to make constructors strict or lazy, you have to have a much better understanding of lazy evaluation than I do. I stick to the idea in the Report, namely that newtype is there for you to rename an existing type, like having several different incompatible kinds of measurements:
newtype Feet = Feet Double
newtype Cm = Cm Double
both behave exactly like Double at run time, but the compiler promises not to let you confuse them.
According to Learn You a Haskell:
Instead of the data keyword, the newtype keyword is used. Now why is
that? Well for one, newtype is faster. If you use the data keyword to
wrap a type, there's some overhead to all that wrapping and unwrapping
when your program is running. But if you use newtype, Haskell knows
that you're just using it to wrap an existing type into a new type
(hence the name), because you want it to be the same internally but
have a different type. With that in mind, Haskell can get rid of the
wrapping and unwrapping once it resolves which value is of what type.
So why not just use newtype all the time instead of data then? Well,
when you make a new type from an existing type by using the newtype
keyword, you can only have one value constructor and that value
constructor can only have one field. But with data, you can make data
types that have several value constructors and each constructor can
have zero or more fields:
data Profession = Fighter | Archer | Accountant
data Race = Human | Elf | Orc | Goblin
data PlayerCharacter = PlayerCharacter Race Profession
When using newtype, you're restricted to just one constructor with one
field.
Now consider the following type:
data CoolBool = CoolBool { getCoolBool :: Bool }
It's your run-of-the-mill algebraic data type that was defined with
the data keyword. It has one value constructor, which has one field
whose type is Bool. Let's make a function that pattern matches on a
CoolBool and returns the value "hello" regardless of whether the Bool
inside the CoolBool was True or False:
helloMe :: CoolBool -> String
helloMe (CoolBool _) = "hello"
Instead of applying this function to a normal CoolBool, let's throw it a curveball and apply it to undefined!
ghci> helloMe undefined
"*** Exception: Prelude.undefined
Yikes! An exception! Now why did this exception happen? Types defined
with the data keyword can have multiple value constructors (even
though CoolBool only has one). So in order to see if the value given
to our function conforms to the (CoolBool _) pattern, Haskell has to
evaluate the value just enough to see which value constructor was used
when we made the value. And when we try to evaluate an undefined
value, even a little, an exception is thrown.
Instead of using the data keyword for CoolBool, let's try using
newtype:
newtype CoolBool = CoolBool { getCoolBool :: Bool }
We don't have to
change our helloMe function, because the pattern matching syntax is
the same if you use newtype or data to define your type. Let's do the
same thing here and apply helloMe to an undefined value:
ghci> helloMe undefined
"hello"
It worked! Hmmm, why is that? Well, like we've said, when we use
newtype, Haskell can internally represent the values of the new type
in the same way as the original values. It doesn't have to add another
box around them, it just has to be aware of the values being of
different types. And because Haskell knows that types made with the
newtype keyword can only have one constructor, it doesn't have to
evaluate the value passed to the function to make sure that it
conforms to the (CoolBool _) pattern because newtype types can only
have one possible value constructor and one field!
This difference in behavior may seem trivial, but it's actually pretty
important because it helps us realize that even though types defined
with data and newtype behave similarly from the programmer's point of
view because they both have value constructors and fields, they are
actually two different mechanisms. Whereas data can be used to make
your own types from scratch, newtype is for making a completely new
type out of an existing type. Pattern matching on newtype values isn't
like taking something out of a box (like it is with data), it's more
about making a direct conversion from one type to another.
Here's another source. According to this Newtype article:
A newtype declaration creates a new type in much the same way as data.
The syntax and usage of newtypes is virtually identical to that of
data declarations - in fact, you can replace the newtype keyword with
data and it'll still compile, indeed there's even a good chance your
program will still work. The converse is not true, however - data can
only be replaced with newtype if the type has exactly one constructor
with exactly one field inside it.
Some Examples:
newtype Fd = Fd CInt
-- data Fd = Fd CInt would also be valid
-- newtypes can have deriving clauses just like normal types
newtype Identity a = Identity a
deriving (Eq, Ord, Read, Show)
-- record syntax is still allowed, but only for one field
newtype State s a = State { runState :: s -> (s, a) }
-- this is *not* allowed:
-- newtype Pair a b = Pair { pairFst :: a, pairSnd :: b }
-- but this is:
data Pair a b = Pair { pairFst :: a, pairSnd :: b }
-- and so is this:
newtype Pair' a b = Pair' (a, b)
Sounds pretty limited! So why does anyone use newtype?
The short version The restriction to one constructor with one field
means that the new type and the type of the field are in direct
correspondence:
State :: (s -> (a, s)) -> State s a
runState :: State s a -> (s -> (a, s))
or in mathematical terms they are isomorphic. This means that after
the type is checked at compile time, at run time the two types can be
treated essentially the same, without the overhead or indirection
normally associated with a data constructor. So if you want to declare
different type class instances for a particular type, or want to make
a type abstract, you can wrap it in a newtype and it'll be considered
distinct to the type-checker, but identical at runtime. You can then
use all sorts of deep trickery like phantom or recursive types without
worrying about GHC shuffling buckets of bytes for no reason.
See the article for the messy bits...
Simple version for folks obsessed with bullet lists (failed to find one, so have to write it by myself):
data - creates new algebraic type with value constructors
Can have several value constructors
Value constructors are lazy
Values can have several fields
Affects both compilation and runtime, have runtime overhead
Created type is a distinct new type
Can have its own type class instances
When pattern matching against value constructors, WILL be evaluated at least to weak head normal form (WHNF) *
Used to create new data type (example: Address { zip :: String, street :: String } )
newtype - creates new “decorating” type with value constructor
Can have only one value constructor
Value constructor is strict
Value can have only one field
Affects only compilation, no runtime overhead
Created type is a distinct new type
Can have its own type class instances
When pattern matching against value constructor, CAN be not evaluated at all *
Used to create higher level concept based on existing type with distinct set of supported operations or that is not interchangeable with original type (example: Meter, Cm, Feet is Double)
type - creates an alternative name (synonym) for a type (like typedef in C)
No value constructors
No fields
Affects only compilation, no runtime overhead
No new type is created (only a new name for existing type)
Can NOT have its own type class instances
When pattern matching against data constructor, behaves the same as original type
Used to create higher level concept based on existing type with the same set of supported operations (example: String is [Char])
[*] On pattern matching laziness:
data DataBox a = DataBox Int
newtype NewtypeBox a = NewtypeBox Int
dataMatcher :: DataBox -> String
dataMatcher (DataBox _) = "data"
newtypeMatcher :: NewtypeBox -> String
newtypeMatcher (NewtypeBox _) = "newtype"
ghci> dataMatcher undefined
"*** Exception: Prelude.undefined
ghci> newtypeMatcher undefined
“newtype"
Off the top of my head; data declarations use lazy evaluation in access and storage of their "members", whereas newtype does not. Newtype also strips away all previous type instances from its components, effectively hiding its implementation; whereas data leaves the implementation open.
I tend to use newtype's when avoiding boilerplate code in complex data types where I don't necessarily need access to the internals when using them. This speeds up both compilation and execution, and reduces code complexity where the new type is used.
When first reading about this I found this chapter of a Gentle Introduction to Haskell rather intuitive.