function :: Type1 Type2
Are Type1 and Type2 return values (tuples)?
data Loc = Loc String Int Int
data Parser b a = P (b -> [(a, b)])
parse :: Parser b a -> b -> [(a, b)]
parse (P p) inp = p inp
type Lexer a = Parser (Loc, String) a
item :: Lexer Char
item = ????
How should I return Lexer and Char from item function?
Could you please give me some simple example.
No, it is not a tuple, types can be parameterized as well. In imperative languages like Java, this concept is usually know as generic types (although there is no one-on-one mapping of the two concepts).
In Java for instance you have classes like:
class LinkedList<E> {
// ...
}
Now here we can see LinkedList as a function that takes as input a parameter E, and then returns a real type (for example LinkedList<String> is a linked list that stores Strings). So we can see such abstract type as a function.
This is a concept that is used in Haskell as well. We have for instance the Maybe type:
data Maybe a = Nothing | Just a
Notice the a here. This is a type parameter that we need to fill in. A function can not return a Maybe, it can only return a Maybe a where a is filled in. For example a Maybe Char: a Maybe type that is a Nothing, or a Just x with x a Char.
Related
In the following:
data DataType a = Data a | Datum
I understand that Data Constructor are value level function. What we do above is defining their type. They can be function of multiple arity or const. That's fine. I'm ok with saying Datum construct Datum. What is not that explicit and clear to me here is somehow the difference between the constructor function and what it produce. Please let me know if i am getting it well:
1 - a) Basically writing Data a, is defining both a Data Structure and its Constructor function (as in scala or java usually the class and the constructor have the same name) ?
2 - b) So if i unpack and make an analogy. With Data a We are both defining a Structure(don't want to use class cause class imply a type already i think, but maybe we could) of object (Data Structure), the constructor function (Data Constructor/Value constructor), and the later return an object of that object Structure. Finally The type of that Structure of object is given by the Type constructor. An Object Structure in a sense is just a Tag surrounding a bunch value of some type. Is my understanding correct ?
3 - c) Can I formally Say:
Data Constructor that are Nullary represent constant values -> Return the the constant value itself of which the type is given by the Type Constructor at the definition site.
Data Constructor that takes an argument represent class of values, where class is a Tag ? -> Return an infinite number of object of that class, of which the type is given by the Type constructor at the definition site.
Another way of writing this:
data DataType a = Data a | Datum
Is with generalised algebraic data type (GADT) syntax, using the GADTSyntax extension, which lets us specify the types of the constructors explicitly:
{-# LANGUAGE GADTSyntax #-}
data DataType a where
Data :: a -> DataType a
Datum :: DataType a
(The GADTs extension would work too; it would also allow us to specify constructors with different type arguments in the result, like DataType Int vs. DataType Bool, but that’s a more advanced topic, and we don’t need that functionality here.)
These are exactly the types you would see in GHCi if you asked for the types of the constructor functions with :type / :t:
> :{
| data DataType a where
| Data :: a -> DataType a
| Datum :: DataType a
| :}
> :type Data
Data :: a -> DataType a
> :t Datum
Datum :: DataType a
With ExplicitForAll we can also specify the scope of the type variables explicitly, and make it clearer that the a in the data definition is a separate variable from the a in the constructor definitions by also giving them different names:
data DataType a where
Data :: forall b. b -> DataType b
Datum :: forall c. DataType c
Some more examples of this notation with standard prelude types:
data Either a b where
Left :: forall a b. a -> Either a b
Right :: forall a b. b -> Either a b
data Maybe a where
Nothing :: Maybe a
Just :: a -> Maybe a
data Bool where
False :: Bool
True :: Bool
data Ordering where
LT, EQ, GT :: Ordering -- Shorthand for repeated ‘:: Ordering’
I understand that Data Constructor are value level function. What we do above is defining their type. They can be function of multiple arity or const. That's fine. I'm ok with saying Datum construct Datum. What is not that explicit and clear to me here is somehow the difference between the constructor function and what it produce.
Datum and Data are both “constructors” of DataType a values; neither Datum nor Data is a type! These are just “tags” that select between the possible varieties of a DataType a value.
What is produced is always a value of type DataType a for a given a; the constructor selects which “shape” it takes.
A rough analogue of this is a union in languages like C or C++, plus an enumeration for the “tag”. In pseudocode:
enum Tag {
DataTag,
DatumTag,
}
// A single anonymous field.
struct DataFields<A> {
A field1;
}
// No fields.
struct DatumFields<A> {};
// A union of the possible field types.
union Fields<A> {
DataFields<A> data;
DatumFields<A> datum;
}
// A pair of a tag with the fields for that tag.
struct DataType<A> {
Tag tag;
Fields<A> fields;
}
The constructors are then just functions returning a value with the appropriate tag and fields. Pseudocode:
<A> DataType<A> newData(A x) {
DataType<A> result;
result.tag = DataTag;
result.fields.data.field1 = x;
return result;
}
<A> DataType<A> newDatum() {
DataType<A> result;
result.tag = DatumTag;
// No fields.
return result;
}
Unions are unsafe, since the tag and fields can get out of sync, but sum types are safe because they couple these together.
A pattern-match like this in Haskell:
case someDT of
Datum -> f
Data x -> g x
Is a combination of testing the tag and extracting the fields. Again, in pseudocode:
if (someDT.tag == DatumTag) {
f();
} else if (someDT.tag == DataTag) {
var x = someDT.fields.data.field1;
g(x);
}
Again this is coupled in Haskell to ensure that you can only ever access the fields if you have checked the tag by pattern-matching.
So, in answer to your questions:
1 - a) Basically writing Data a, is defining both a Data Structure and its Constructor function (as in scala or java usually the class and the constructor have the same name) ?
Data a in your original code is not defining a data structure, in that Data is not a separate type from DataType a, it’s just one of the possible tags that a DataType a value may have. Internally, a value of type DataType Int is one of the following:
The tag for Data (in GHC, a pointer to an “info table” for the constructor), and a reference to a value of type Int.
x = Data (1 :: Int) :: DataType Int
+----------+----------------+ +---------+----------------+
x ---->| Data tag | pointer to Int |---->| Int tag | unboxed Int# 1 |
+----------+----------------+ +---------+----------------+
The tag for Datum, and no other fields.
y = Datum :: DataType Int
+-----------+
y ----> | Datum tag |
+-----------+
In a language with unions, the size of a union is the maximum of all its alternatives, since the type must support representing any of the alternatives with mutation. In Haskell, since values are immutable, they don’t require any extra “padding” since they can’t be changed.
It’s a similar situation for standard data types, e.g., a product or sum type:
(x :: X, y :: Y) :: (X, Y)
+---------+--------------+--------------+
| (,) tag | pointer to X | pointer to Y |
+---------+--------------+--------------+
Left (m :: M) :: Either M N
+-----------+--------------+
| Left tag | pointer to M |
+-----------+--------------+
Right (n :: N) :: Either M N
+-----------+--------------+
| Right tag | pointer to N |
+-----------+--------------+
2 - b) So if i unpack and make an analogy. With Data a We are both defining a Structure(don't want to use class cause class imply a type already i think, but maybe we could) of object (Data Structure), the constructor function (Data Constructor/Value constructor), and the later return an object of that object Structure. Finally The type of that Structure of object is given by the Type constructor. An Object Structure in a sense is just a Tag surrounding a bunch value of some type. Is my understanding correct ?
This is sort of correct, but again, the constructors Data and Datum aren’t “data structures” by themselves. They’re just the names used to introduce (construct) and eliminate (match) values of type DataType a, for some type a that is chosen by the caller of the constructors to fill in the forall
data DataType a = Data a | Datum says:
If some term e has type T, then the term Data e has type DataType T
Inversely, if some value of type DataType T matches the pattern Data x, then x has type T in the scope of the match (case branch or function equation)
The term Datum has type DataType T for any type T
3 - c) Can I formally Say:
Data Constructor that are Nullary represent constant values -> Return the the constant value itself of which the type is given by the Type Constructor at the definition site.
Data Constructor that takes an argument represent class of values, where class is a Tag ? -> Return an infinite number of object of that class, of which the type is given by the Type constructor at the definition site.
Not exactly. A type constructor like DataType :: Type -> Type, Maybe :: Type -> Type, or Either :: Type -> Type -> Type, or [] :: Type -> Type (list), or a polymorphic data type, represents an “infinite” family of concrete types (Maybe Int, Maybe Char, Maybe (String -> String), …) but only in the same way that id :: forall a. a -> a represents an “infinite” family of functions (id :: Int -> Int, id :: Char -> Char, id :: String -> String, …).
That is, the type a here is a parameter filled in with an argument value given by the caller. Usually this is implicit, through type inference, but you can specify it explicitly with the TypeApplications extension:
-- Akin to: \ (a :: Type) -> \ (x :: a) -> x
id :: forall a. a -> a
id x = x
id #Int :: Int -> Int
id #Int 1 :: Int
Data :: forall a. a -> DataType a
Data #Char :: Char -> DataType Char
Data #Char 'x' :: DataType Char
The data constructors of each instantiation don’t really have anything to do with each other. There’s nothing in common between the instantiations Data :: Int -> DataType Int and Data :: Char -> DataType Char, apart from the fact that they share the same tag name.
Another way of thinking about this in Java terms is with the visitor pattern. DataType would be represented as a function that accepts a “DataType visitor”, and then the constructors don’t correspond to separate data types, they’re just the methods of the visitor which accept the fields and return some result. Writing the equivalent code in Java is a worthwhile exercise, but here it is in Haskell:
{-# LANGUAGE RankNTypes #-}
-- (Allows passing polymorphic functions as arguments.)
type DataType a
= forall r. -- A visitor with a generic result type
r -- With one “method” for the ‘Datum’ case (no fields)
-> (a -> r) -- And one for the ‘Data’ case (one field)
-> r -- Returning the result
newData :: a -> DataType a
newData field = \ _visitDatum visitData -> visitData field
newDatum :: DataType a
newDatum = \ visitDatum _visitData -> visitDatum
Pattern-matching is simply running the visitor:
matchDT :: DataType a -> b -> (a -> b) -> b
matchDT dt visitDatum visitData = dt visitDatum visitData
-- Or: matchDT dt = dt
-- Or: matchDT = id
-- case someDT of { Datum -> f; Data x -> g x }
-- f :: r
-- g :: a -> r
-- someDT :: DataType a
-- :: forall r. r -> (a -> r) -> r
someDT f (\ x -> g x)
Similarly, in Haskell, data constructors are just the ways of introducing and eliminating values of a user-defined type.
What is not that explicit and clear to me here is somehow the difference between the constructor function and what it produce
I'm having trouble following your question, but I think you are complicating things. I would suggest not thinking too deeply about the "constructor" terminology.
But hopefully the following helps:
Starting simple:
data DataType = Data Int | Datum
The above reads "Declare a new type named DataType, which has the possible values Datum or Data <some_number> (e.g. Data 42)"
So e.g. Datum is a value of type DataType.
Going back to your example with a type parameter, I want to point out what the syntax is doing:
data DataType a = Data a | Datum
^ ^ ^ These things appear in type signatures (type level)
^ ^ These things appear in code (value level stuff)
There's a bit of punning happening here. so in the data declaration you might see "Data Int" and this is mixing type-level and value-level stuff in a way that you wouldn't see in code. In code you'd see e.g. Data 42 or Data someVal.
I hope that helps a little...
The data and type keywords always confuse me.
I want to know what is the difference between data and type and how to use them.
type declares a type synonym. A type synonym is a new name for an existing type. For example, this is how String is defined in the standard library:
type String = [Char]
String is another name for a list of Chars. GHC will replace all usages of String in your program with [Char] at compile-time.
To be clear, a String literally is a list of Chars. It's just an alias. You can use all the standard list functions on String values:
-- length :: [a] -> Int
ghci> length "haskell"
7
-- reverse :: [a] -> [a]
ghci> reverse "functional"
"lanoitcnuf"
data declares a new data type, which, unlike a type synonym, is different from any other type. Data types have a number of constructors defining the possible cases of your type. For example, this is how Bool is defined in the standard library:
data Bool = False | True
A Bool value can be either True or False. Data types support pattern matching, allowing you to perform a runtime case-analysis on a value of a data type.
yesno :: Bool -> String
yesno True = "yes"
yesno False = "no"
data types can have multiple constructors (as with Bool), can be parameterised by other types, can contain other types inside them, and can recursively refer to themselves. Here's a model of exceptions which demonstrates this; an Error a contains an error message of type a, and possibly the error which caused it.
data Error a = Error { value :: a, cause :: Maybe (Error a) }
type ErrorWithMessage = Error String
myError1, myError2 :: ErrorWithMessage
myError1 = Error "woops" Nothing
myError2 = Error "myError1 was thrown" (Just myError1)
It's important to realise that data declares a new type which is apart from any other type in the system. If String had been declared as a data type containing a list of Chars (rather than a type synonym), you wouldn't be able to use any list functions on it.
data String = MkString [Char]
myString = MkString ['h', 'e', 'l', 'l', 'o']
myReversedString = reverse myString -- type error
There's one more variety of type declaration: newtype. This works rather like a data declaration - it introduces a new data type separate from any other type, and can be pattern matched - except you are restricted to a single constructor with a single field. In other words, a newtype is a data type which wraps up an existing type.
The important difference is the cost of a newtype: the compiler promises that a newtype is represented in the same way as the type it wraps. There's no runtime cost to packing or unpacking a newtype. This makes newtypes useful for making administrative (rather than structural) distinctions between values.
newtypes interact well with type classes. For example, consider Monoid, the class of types with a way to combine elements (mappend) and a special 'empty' element (mempty). Int can be made into a Monoid in many ways, including addition with 0 and multiplication with 1. How can we choose which one to use for a possible Monoid instance of Int? It's better not to express a preference, and use newtypes to enable either usage with no runtime cost. Paraphrasing the standard library:
-- introduce a type Sum with a constructor Sum which wraps an Int, and an extractor getSum which gives you back the Int
newtype Sum = Sum { getSum :: Int }
instance Monoid Sum where
(Sum x) `mappend` (Sum y) = Sum (x + y)
mempty = Sum 0
newtype Product = Product { getProduct :: Int }
instance Monoid Product where
(Product x) `mappend` (Product y) = Product (x * y)
mempty = Product 1
With data you create new datatype and declare a constructor for it:
data NewData = NewDataConstructor
With type you define just an alias:
type MyChar = Char
In the type case you can pass value of MyChar type to function expecting a Char and vice versa, but you can't do this for data MyChar = MyChar Char.
type works just like let: it allows you to give a re-usable name to something, but that something will always work just as if you had inlined the definition. So
type ℝ = Double
f :: ℝ -> ℝ -> ℝ
f x y = let x2 = x^2
in x2 + y
behaves exactly the same way as
f' :: Double -> Double -> Double
f' x y = x^2 + y
as in: you can anywhere in your code replace f with f' and vice versa; nothing would change.
OTOH, both data and newtype create an opaque abstraction. They are more like a class constructor in OO: even though some value is implemented simply in terms of a single number, it doesn't necessarily behave like such a number. For instance,
newtype Logscaledℝ = LogScaledℝ { getLogscaled :: Double }
instance Num LogScaledℝ where
LogScaledℝ a + LogScaledℝ b = LogScaledℝ $ a*b
LogScaledℝ a - LogScaledℝ b = LogScaledℝ $ a/b
LogScaledℝ a * LogScaledℝ b = LogScaledℝ $ a**b
Here, although Logscaledℝ is data-wise still just a Double number, it clearly behaves different from Double.
How can you create a type that is a subset of an other type? I want a string type that only contains alphanumeric characters.
So I want something like this
type AlphNumString = [AlphaNumChar]
data AlphaNumChar = ???? filter (isAlphaNum) Char ????
The standard way to do this is with so-called "smart constructors".
First, you define a new type that's identical to the old one:
newtype AlphNumString = X String
Next, you write the smart constructor itself:
toAlphNumString :: String -> AlphNumString
toAlphNumString txt = X (filter isAlphNum txt)
Finally, you make it so toAlphNumString is the only way to create an AlphNumString.
module Foo (AlphNumString (), toAlphNumString, ...) where ...
Note that this does not allow you to use an AlphNumString like a normal String; you can't create "subtypes" like that in Haskell. So you'll also need another function
fromAlphNumString :: AlphNumString -> String
fromAlphNumString (X txt) = txt
This concept of types that are "subsets" of other types based on some predicate is called refinement types. For Haskell, this is implemented as LiquidHaskell.
However, I would consider this an ongoing research. In practice, I would go with a newtype and dynamic checks, as MathematicalOrchid describes in their answer.
In a dependently typed language (which Haskell is not, yet) you could use dependent pairs for this.
data Sigma : (t : Type) -> (t -> Type) -> Type where
MkSigma : (x : t) -> p x -> Sigma t p
data IsAlphaNum : (c : Char) -> Type where
MkIsAlphaNum : isAlphaNum c = True -> IsAlphaNum c
Then Sigma Char IsAlphNum would be a type representing alphanumeric characters. Each element of that type would consist of a character and a proof that the character is alphanumeric.
I want to write a Haskell function that takes a custom type with eleven fields and returns either a list of all the fields' values, or a map associating the fields' names with their values. I don't want to have to explicitly get every field because that would be verbose and less versatile. Is there any way to do this?
What you write would be possible to some degree, but it wouldn't be very useful.
Let's imagine we insist on writing this function for a moment. Given that the fields' values may have different types, you probably rather want to yield a tuple. I.e.
data MyType = MyType Int String Bool
getFields :: MyType -> (Int, String, Bool)
getFields (MyType a b c) = (a,b,c)
So you could now call it like
let v = MyType 1 "Hello" True
let (x, y, z) = getFields v
Now, this isn't actually very useful, because you could use pattern matching in all of these cases, e.g.
let v = MyType 1 "Hello" True
let (MyType x y z) = v
Alright, but what if you wanted to address individual fields? Like
let x = fst (getFields v)
...how to do that without a 'getFields' function? Well, you can simply assign field names (as you probably already did):
data MyType = MyType
{ i :: Int
, s :: String
, b :: Bool
}
Now you could functions for accessing indivial fields for free:
let x = i v
...since assigning names ot fields actually generates functions like i :: MyType -> Int or s :: MyType -> String.
I'm following this introduction to Haskell, and this particular place (user defined types 2.2) I'm finding particularly obscure. To the point, I don't even understand what part of it is code, and what part is the thoughts of the author. (What is Pt - it is never defined anywhere?). Needless to say, I can't execute / compile it.
As an example that would make it easier for me to understand, I wanted to define a type, which is a pair of an Integer and a String, or a String and an Integer, but nothing else.
The theoretical function that would use it would look like so:
combine :: StringIntPair -> String
combine a b = (show a) ++ b
combine a b = a ++ (show b)
If you need a working code, that does the same, here's CL code for doing it:
(defgeneric combine (a b)
(:documentation "Combines strings and integers"))
(defmethod combine ((a string) (b integer))
(concatenate 'string a (write-to-string b)))
(defmethod combine ((a integer) (b string))
(concatenate 'string (write-to-string a) b))
(combine 100 "500")
Here's one way to define the datatype:
data StringIntPair = StringInt String Int |
IntString Int String
deriving (Show, Eq, Ord)
Note that I've defined two constructors for type StringIntPair, and they are StringInt and IntString.
Now in the definition of combine:
combine :: StringIntPair -> String
combine (StringInt s i) = s ++ (show i)
combine (IntString i s) = (show i) ++ s
I'm using pattern matching to match the constructors and select the correct behavior.
Here are some examples of usage:
*Main> let y = StringInt "abc" 123
*Main> let z = IntString 789 "a string"
*Main> combine y
"abc123"
*Main> combine z
"789a string"
*Main> :t y
y :: StringIntPair
*Main> :t z
z :: StringIntPair
A few things to note about the examples:
StringIntPair is a type; doing :t <expression> in the interpreter shows the type of an expression
StringInt and IntString are constructors of the same type
the vertical bar (|) separates constructors
a well-written function should match each constructor of its argument's types; that's why I've written combine with two patterns, one for each constructor
data StringIntPair = StringInt String Int
| IntString Int String
combine :: StringIntPair -> String
combine (StringInt s i) = s ++ (show i)
combine (IntString i s) = (show i) ++ s
So it can be used like that:
> combine $ StringInt "asdf" 3
"asdf3"
> combine $ IntString 4 "fasdf"
"4fasdf"
Since Haskell is strongly typed, you always know what type a variable has. Additionally, you will never know more. For instance, consider the function length that calculates the length of a list. It has the type:
length :: [a] -> Int
That is, it takes a list of arbitrary a (although all elements have the same type) and returns and Int. The function may never look inside one of the lists node and inspect what is stored in there, since it hasn't and can't get any informations about what type that stuff stored has. This makes Haskell pretty efficient, since, as opposed to typical OOP languages such as Java, no type information has to be stored at runtime.
To make it possible to have different types of variables in one parameter, one can use an Algebraic Data Type (ADT). One, that stores either a String and an Int or an Int and a String can be defined as:
data StringIntPair = StringInt String Int
| IntString Int String
You can find out about which of the two is taken by pattern matching on the parameter. (Notice that you have only one, since both the string and the in are encapsulated in an ADT):
combine :: StringIntPair -> String
combine (StringInt str int) = str ++ show int
combine (IntString int str) = show int ++ str