CS321 Functional Programming 2 © JAS 2005 1-1 CS321 Functional Programming 2 Dr John A Sharp 10...

© JAS 2005 1-1

CS321 Functional Programming 2


Dr John A Sharp

10 credits

Tuesday 10am Robert Recorde Room

Thursday 11am Robert Recorde Room

© JAS 2005 1-2


Assessment

80% written examination in May/June

20% coursework

probably two assignments – roughly weeks 4 and 7

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• Type Classes• Programming with Streams• Lazy Data Structures• Memoization • Lambda Calculus • Type Checking and Inference• Implementation Approaches

Syllabus(Provisional)

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Recommended Reading

Course will not follow any specific textS Thompson, Haskell: The Craft of Functional Programming, Second Edition, Addison-Wesley, 1999

P Hudak, The Haskell School of Expression – Learning Functional Programming through Multimedia, Cambridge University press, 2000

R Bird, Introduction to Functional Programming using Haskell, Second Edition, Prentice-Hall, 1998

A J T Davie, An Introduction to Functional Programming using Haskell, Cambridge University Press, 1992

www.haskell.org

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Notes will be handed out in sections

mainly copies of PowerPoint slides

some Haskell programs

They will also be available on a course web page

www.cs.swan.ac.uk/~csjohn/cs321/main.html

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Assumptions from CS221 Functional Programming 1

• Familiar with principles of Functional Programming

• Able to write and run simple Haskell programs/scripts

• Familiar with concepts of types and higher-order functions

• Able to define own structured types

• Familiar with basic λ calculus

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Type Classes

Monomorphic functions

add1 :: Int -> Int

add1 x = x + 1

Parametric polymorphic functions

map :: (a -> b) -> [a] -> [b]

map f [] = []

map f (x:xs) = f x : (map f xs)

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Ad-hoc polymorphic functions

add :: Int -> Int -> Int

add x y = x + y

add :: Float -> Float -> Float

add :: Double -> Double -> Double

add :: Num a => a -> a -> a

add x y = x + y

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This sort of type expression is sometimes referred to as a “Qualified Type”.

Num a is termed a predicate which limits the types for which a type expression is valid. It is also termed the context for the type.

Multiple predicates can be defined

contrived :: (Num a, Eq b)=>a->b->b->a

contrived x y z = if y == z

then x + x

else x + x + x

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For completeness we could specify an empty context for any polymorphic function

map :: () => (a -> b) -> [a] -> [b]

map f [] = []

map f (x:xs) = f x : (map f xs)

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The predicate restricts the set of types for which the expression is valid.

Num a should be read as “for all types a that are members of the class Num.

Members of a class (which are types) are also called instances of a class.

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Type classes allow us to group together types which have common properties (or more accurately common operations that can be applied to them).

An obvious example is the types for which equality can be defined. An appropriate Class is defined in the standard prelude:-

class Eq a where(==), (/=) :: a -> a -> Boolx /= y = not ( x == y )

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class Eq a whereheader introducing name of class and parameters (a)

(==), (/=) :: a -> a -> Boolsignature listing functions applicable to instances of

class and their type(==) and (/=) are termed member functions

x /= y = not ( x == y )default definitions of member functions that can be

overridden

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Having defined the concept of an equality class the next step is to define various instances of the class.

The following examples are again taken from the standard prelude.

instance Eq Int where (==) = primEqInt-- primEqint is a primitive Haskell function

instance Eq Bool whereTrue == True = TrueFalse == False = True_ == _ = False-- defined by pattern matching

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instance Eq Char wherec == d = ord c == ord d-- note the second == is defined on integers

instance (Eq a,Eq b) => Eq(a,b) where(x,y) == (u,v) = x==u && y==v-- pairwise equality

instance Eq a => Eq [a] where[] == [] = True[] == (y:ys) = False(x:xs)== [] = False(x:xs)== (y:ys) = x==y && xs==ys-- extend to equality on lists

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These declarations allow us to obtain definitions for equality on an infinite family of types involving pairs, and lists of (pairs and lists of) Int, Bool, Char.

The general format of an instance declaration is

instance context => predicate where

definitions of member functions

We can define instances of Eq for user defined types.

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Consider a definition of Sets

data Set a = Set [a]

To make the type Set a a member of the class Eq we define an instance of Eq.

Note that we do not use the standard (==) on the type a as we do not require the elements of the Sets to be in the same order.

instance Eq a => Eq (Set a) whereSet xs == Set ys =

xs 'subset' ys && ys 'subset xswhere xs 'subset' ys =

all ('elem' ys) xs

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Some types (such as functions) can not be defined to be members of the class Eq (for obvious reasons, I hope). The error message that results from a mistaken attempt to test for equality can be obscure.

It is possible to make a test for equality on functions type check ok and produce a run-time error message using the standard error function defined in the standard prelude.

instance Eq (a->b) where(==) = error "== not defined on fns"

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An aIf the functions are defined to operate on a finite set of elements then a form of equality can be defined.

instance Eq a => Eq (Bool->a) wheref == g = f False == g False &&

f True == g True

It is possible to have instances for both Eq (a->b) and Eq (Bool->a) as long as both are not required at the same time in type checking some expression.

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Inheritance, Derived Classes, Superclasses, and Subclasses

In general a class declaration has the formclass context => Class a1 .. an wheretype declarations for member functionsdefault declarations of member functions

where Class is the name of a new type class which takes n arguments (a1 .. an ).

As with instances the context must be satisfied in order to construct any instance of the Class.

The predicates in the context part of the declaration are called the superclasses of Class. Class is a subclass of the classes in the context.

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Consider a definition of a Class Ord whose instances have both strict (<), (>) and non-strict (<=), (>=) versions of an ordering defined on their elements.

class Eq a => Ord a where(<),(<=),(>),(>=) :: a->a->Boolmax,min :: a->a->ax < y = x <= y && x /= yx >= y = y <= xx > y = y < xmax x y | x >= y = x

| y >= x = ymin x y | x <= y = x

| y <= x = y

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Why define Eq as a superclass of Ord?

• the default definition of (<) relies on the use of (/=) taken from the class Eq. Therefore every instance of Ord must be an instance of Eq.

• given the definition of non-strict ordering (<=) it is always possible to define (==) and hence (/=) using

x==y = x<=y && y<= x

so there will be no loss of generality in requiring Eq to be a superclass of Ord.

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It is possible for some types to require a type to be an instance of class Ord in order to define it to be an instance of the class Eq.

For example consider an alternative way of defining equality on Sets.

instance Ord (Set a) => Eq (Set a) where

x == y = x <= y && y <= x

instance Eq a => Ord (Set a) where

Set xs <= Set ys = all ('elem' ys) xs

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Numeric Class Hierarchy

Ord Enum Integral

Num Real RealFrac

Fractional Floating

EqAll but IO, (->) Int, IntegerAll but IO, (->),

IOError

Int, Integer,Float, Double

(), Bool, Char, Int, Integer, Float,Double, Ordering

Int, Integer,Float, Double Float, Double

RealFloatFloat, DoubleFloat, Double Float, Double

ShowAll Prelude Types

ReadAll but IO, (->)

BoundedInt, Char, Bool, (),Ordering, tuples

MonadIO, [], Maybe

MonadPlusIO, [], Maybe

FunctorIO, [], Maybe

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Operations defined in these classes (red means must be provided)

Eq == /=Ord < <= > >= max min compareNum + - * negate abs signum fromInteger

Real toRationalEnum succ pred toEnum fromEnum enumFrom

enumFromThen enumFromToenumFromThenTo

Integral quot rem div mod quotRem divModtoInteger

Fractional / recip fromRationalRealFrac properFraction truncate round

ceiling floor

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Floating pi exp log sqrt ** logBase sin costan asin acos atan sinh cosh tanh

asinh acosh atanhRealFloat floatRadix floatDigits floatRange

decodeFloat encodeFloat exponentsignificand scaleFloat isNaN

isInfinite isDenormalized isIEEEisNegativeZero atan2

Show showsPrec showList showRead readsPrec readListBounded minBound maxBoundMonad >>= >> return failFunctor fmap

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By introducing type classes Haskell has enabled programmers to use numerical types in a manner similar to the way they are used to in traditional imperative languages.

But what else are type classes useful for?

They can provide a mechanism for adding your own numerical types eg complex numbers

They can provide a mechanism for a sort of object-oriented programming

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Imperative O-O Classes = Data + Methods

Objects contain values which can be changed by methods

Classes inherit data fields and methods

FP Classes = Methods

Methods/Functions can be applied to variables whose type is an Instance of a Class

Classes inherit methods/functions

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Implementing Type Classes – An Approach using Dictionaries

A function with a qualified type context=>type is implemented by a function which takes an extra argument for every predicate in the context.

When the function is used each of these parameters is filled by a 'dictionary' which gives the values of each of the member functions in the appropriate class.

For example, for the class Eq each dictionary has at least two elements containing the definitions of the functions for (==) and (/=).

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We will write(#n d) to select nth element of dictionary{dict} to denote a specific dictionary

( contents not displayed)dnnn for a dictionary variable representing an unknown

dictionaryd:: Class Inst

to denote that d is dictionary for the instanceInst of a class Class

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The member functions of the class Eq thus behave as if defined(==) d1 = (#1 d1)(/=) d1 = (#2 d1)

The dictionary for Eq Int contains two entries:

d1 :: Eq Int

primEqInt defNeq d1

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Evaluating 2 == 3=> (==) d1 2 3 => (#1 d1) 2 3

=> primEqInt 2 3=> False

Evaluating 2 /= 3=> (/=) d1 2 3 => (#2 d1) 2 3

=> defNeq d1 23=> not ((==) d1 2 3)=> not ((#1 d1) 2 3)=> not (primEqInt 2

3)=> not False=> True

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Now consider a more complex example

We have seen definitions of (==) in the instances of the class Eq defined for pairs and lists.

eqPair d (x,y) (u,v) =

(==) (#3 d) x u && (==) (#4 d) y v

eqList d [] [] = True

eqList d [] (y:ys) = False

eqList d (x:xs) [] = False

eqList d (x:xs) (y:ys) =

(==) (#3 d) x y && (==) d xs ys

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The dictionary structure for Eq (Int, [Int]) is

d3 :: Eq (Int, [Int])

eqPair d3 defNeq d3

d2 :: Eq [Int]

d1 :: Eq InteqList d3 defNeq d3

defNeq d3primEqInt

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(2,[1]) == (2,[1.3])

(#1 d3)(==) d3 (2,[1]) (2,[1,3]) => eqPair d3

=> (==) (#3 d3) 2 2 && (#4 d3) [1] [1,3]d1(#1 d1)primEqIntTrue (==) d2(#1 d2)eqList d2

=> (==) (#3 d2) 1 1 && (==) [] [3]d2d1(#1 d1)primEqIntTrue (#1 d2)eqList d2False

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Dictionaries for superclasses can be defined in a similar way as instance dictionaries.

For example for the Ord class which has Eq as a context

class Eq => Ord a where

(<),(<=),(>),(>=) :: a->a->Bool

max,min :: a->a->Bool

the dictionary would contain the following

(<) (<=) (>) (>=) max min Eq a

defLessThan d x y = (<=) d x y && (/=) (#7 d) x y

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Combining ClassesA dictionary consists of three (possibly empty) components:

• Superclass dictionaries• Instance specific dictionaries• Implementation of class members

instances of Eq have no superclass dictionaries

Eq Int has no instance specific dictionary

Classes with no member functions can be used as abbreviations for lists of predicates

class C a where cee :: a -> aclass D a where dee :: a -> a class (C a, D a) => CandD a

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Contexts and Derived Types

eg1 x = [x] == [x] || x == x

Since the (==) operation is applied to both lists and an argument x if x::a then we would seem to require instances Eq a and Eq [a] giving a type signature

(Eq [a], Eq a) => a -> Bool

with translation

eg1 d1 d2 x = (==) d1 [x] [x] || (==) d2 x x

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However, given d1::Eq[a] we can always find Eq a by taking the third element of d1 ((#3 d1)::Eq a).

Since it is more efficient to select an element from a dictionary than to complicate both type and translation with extra parameters the type of eg1 is (by default) derived as

Eq [a] => a -> Bool

with translation

eg1 d1 x = (==) d1 [x] [x] ||(==) (#3 d1) x x

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If you definitely required the tuple context then this can be produced by explicitly defining the type and context of eg1

eg2 = (\x y-> x ==x || y==y)

The type derived for this is

(Eq b, Eq a) => a -> b -> Bool

with translation

\d1 d2 x y -> (==) d2 x x || (==) d1 y y

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If you wished to ensure that only one dictionary parameter is used you could explicitly type is using

Eq (a,b) => a -> b -> Bool

with translation

\d1 x y -> (==) (#3 d1) x x || (==) (#4 d1) y y

CS321 Functional Programming 2 © JAS 2005 1-1 CS321 Functional Programming 2 Dr John A Sharp 10...

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