Haskell Guide: Types, Lambda Functions and Type Classes
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Function types
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- Ordinary data types are for primitive data (like (Int) and (Char)) and basic data structures (like ([Int]) and ([Char])).
- Algebraic data types are types that combine other types either as records (‘products’), e.g.
data Pair = Pair Int Double
or as variants (‘sums’), e.g.
data Bool = False | True
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- Functions have types containing an arrow, e.g. (Int rightarrow String).
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- We now look at function types in more detail.
Lambda expressions
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- Lambda expressions (named after the greek letter (lambda)) play a very important role in functional programming in general and Haskell in particular.
Named and anonymous expressions
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- You can give a name (sum) to an expression (2+2):
sum = 2+2
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- But you can also write anonymous expressions — expressions that just
appear, but are not given names.
- But you can also write anonymous expressions — expressions that just
(-b) + sqrt (b^2 - 4*a*c)
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- Without anonymous expressions, writing this would almost be like
assembly language:
- Without anonymous expressions, writing this would almost be like
e1 = (-b)
e2 = b^2
e3 = 4*a
e4 = e3*c
e5 = e2-e4
e6 = sqrt e5
e7 = e1+e6
Some background
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Sometimes in a mathematics or physics book, there are statements
like “the function (x^2) is continuous(ldots)”
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Sometimes in a mathematics or physics book, there are statements
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This is ok when the context makes it clear what (x) is.
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This is ok when the context makes it clear what (x) is.
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But it can lead to problems. What does (x*y) mean?
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Is it a constant, because both (x) and (y) have fixed values?
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Is it a constant, because both (x) and (y) have fixed values?
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Is it a function of (x), with a fixed value of (y)?
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Is it a function of (x), with a fixed value of (y)?
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Is it a function of (y), with a fixed value of (x)?
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Is it a function of (y), with a fixed value of (x)?
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Is it a function of both (x) and (y)?
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Is it a function of both (x) and (y)?
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But it can lead to problems. What does (x*y) mean?
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In mathematical logic (and computer programming) we need to be
precise about this!
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In mathematical logic (and computer programming) we need to be
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A lambda expression (backslash x rightarrow e) contains
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An explicit statement that the formal parameter is (x), and
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An explicit statement that the formal parameter is (x), and
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the expression (e) that defines the value of the function.
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the expression (e) that defines the value of the function.
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A lambda expression (backslash x rightarrow e) contains
Anonymous functions
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- A function can be defined and given a name using an equation:
f :: Int -> Int
f x = x+1
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Since functions are “first class”, they are ubiquitous, and it’s
often useful to denote a function anonymously.
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Since functions are “first class”, they are ubiquitous, and it’s
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This is done using lambda expressions.
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This is done using lambda expressions.
x -> x+1
Pronounced “lambda x arrow x+1”.
There may be any number of arguments:
x y z -> 2*x + y*z
Using a lambda expression
Functions are first class: you can use a lambda expression wherever a
function is needed. Thus
f = x -> x+1
is equivalent to
f x = x+1
But lambda expressions are most useful when they appear inside larger
expressions.
map (x -> 2*x + 1) xs
Monomorphic and polymorphic functions
Monomorphic functions
Monomorphic means “having one form”.
f :: Int -> Char
f i = "abcdefghijklmnopqrstuvwxyz" !! i
x :: Int
x = 3
f :: Char->String
f x = x:" There is a kind of character in thy life"
Polymorphic functions
Polymorphic means “having many forms”.
fst :: (a,b) -> a
fst (x,y) = x
snd :: (a,b) -> b
snd (x,y) = y
fst :: (a,b) -> a
fst (a,b) = a
snd :: (a,b) -> b
snd (a,b) = b
Currying
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Most programming languages allow functions to have any number of
arguments.
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Most programming languages allow functions to have any number of
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But this turns out to be unnecessary: we can restrict all functions
to have just one argument, without losing any expressiveness.
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But this turns out to be unnecessary: we can restrict all functions
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This process is called Currying, in honor of Haskell Curry.
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The technique makes essential use of higher order functions.
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The technique makes essential use of higher order functions.
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It has many advantages, both practical and theoretical.
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It has many advantages, both practical and theoretical.
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This process is called Currying, in honor of Haskell Curry.
A function with two arguments
You can write a definition like this, which appears to have two
arguments:
f :: Int -> Int -> Int
f x y = 2*x + y
But it actually means the following:
f :: Int -> (Int -> Int)
f 5 :: Int -> Int
The function takes its arguments one at a time:
f 3 4 = (f 3) 4
g :: Int -> Int
g = f 3
g 10 -- > (f 3) 10 -- > 2*3 + 10
Grouping: arrow to the right, application left
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The arrow operator takes two types (a rightarrow b), and gives the
type of a function with argument type (a) and result type (b)
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The arrow operator takes two types (a rightarrow b), and gives the
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An application (e_1; e_2) applies a function (e_1) to an argument
(e_2)
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An application (e_1; e_2) applies a function (e_1) to an argument
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Note that for both types and applications, a function has only one
argument
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Note that for both types and applications, a function has only one
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To make the notation work smoothly, arrows group to the right, and
application groups to the left.
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To make the notation work smoothly, arrows group to the right, and
f :: a -> b -> c -> d
f :: a -> (b -> (c -> d))
f x y z = ((f x) y) z
Type classes and ad-hoc polymorphism
The type of ((+))
Note that (fst) has the following type, and there is no restriction on
what types (a) and (b) could be.
fst :: (a,b) -> a
What is the type of ((+))? Could it be(ldots)
(+) :: Int -> Int -> Int
(+) :: Integer -> Integer -> Integer
(+) :: Ratio Integer -> Ratio Integer -> Ratio Integer
(+) :: Double -> Double -> Double
(+) :: a -> a -> a -- Wrong! has to be a number
Type classes
Answer: ((+)) has type (a rightarrow a rightarrow a) for any type (a)
that is a member of the type class (Num).
(+) :: Num a => a -> a -> a
- The class (Num) is a set of types for which ((+)) is defined
- It includes (Int), (Integer), (Double), and many more.
- But (Num) does not contain types like (Bool), ([Char]), (Intrightarrow Double), and many more.
Two kinds of polymorphism
- Parametric polymorphism.
- A polymorphic type that can be instantiated to any type.
- Represented by a type variable. It is conventional to use (a), (b), (c), (ldots)
- Example: (length :: [a] rightarrow Int) can take the length of a list whose elements could have any type.
- Ad hoc polymorphism.
- A polymorphic type that can be instantiated to any type chosen from a set, called a “type class”
- Represented by a type variable that is constrained using the (Rightarrow) notation.
- Example: ((+) :: Num, a Rightarrow a rightarrow a rightarrow a) says that ((+)) can add values of any type (a), provided that (a) is an element of the type class (Num).
Type inference
- Type checking takes a type declaration and some code, and determines whether the code actually has the type declared.
- Type inference is the analysis of code in order to infer its type.
- Type inference works by
- Using a set of type inference rules that generate typings based on the program text
- Combining all the information obtained from the rules to produce the types.
Type inference rules
The type system contains a number of type inference rules, with the form
[frac {hbox{assumption — what you’re given}} {hbox{consequence — what you can infer}}]
Context
- Statements about types are written in the form similar to (Gamma vdash e :: alpha)
- This means “if you are given a set (Gamma) of types, then it is proven that (e) has type (alpha).
Type of constant
[frac {hbox{$c$ is a constant with a fixed type $T$}} {Gamma vdash c :: T}]
If we know the type (T) of a constant (c) (for example, we know that (‘a’ :: Char)), then this is expressed by saying that there is a given theorem that (c :: T). Furthermore, this holds given any context (Gamma).
Type of application
[frac {Gamma vdash e_1 :: (alpha rightarrow beta) qquad Gamma vdash e_2 :: alpha } {Gamma vdash (e_1 e_2) :: beta}]
If (e_1) is a function with type (alpha rightarrow beta), then the application of (e_1) to an argument of type (alpha) gives a result of type (beta).
Type of lambda expression
[frac {Gamma, x :: alpha quad vdash quad e :: beta} {Gamma vdash (lambda x rightarrow e) :: (alpha rightarrow beta)}]
We have a context (Gamma). Suppose that if we’re also given that (x :: alpha), then it can be proven that an expression (e :: beta). Then we can infer that the function (lambda x rightarrow e) has type (alpha rightarrow beta).
Functional Programming in Haskell: Supercharge Your Coding
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Functional Programming in Haskell: Supercharge Your Coding
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