CSE-321 Programming Languages

Introduction to Functional Programming(Part II)

POSTECH

March 13, 2006

박성우

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Outline• Expressions and values V• Variables V• Functions V• Types

– Polymorphism• Recursion• Datatypes• Pattern matching• Higher-order functions• Exceptions• Modules

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What is the Type of ?

true,

false)

(

fn f => (f true, f false)

f

f

f

All we know about f is that it takes booleans as arguments.

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f : bool -> ?

fn f => (f true, f false) : (bool -> ?) -> ? * ?

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f : bool -> 'a

fn f => (f true, f false) : (bool -> 'a) -> 'a * 'a

• 'a– type variable– usually read as alpha– means 'for any type alpha'

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Polymorphic Types• Types involving type variables 'a, 'b, 'c, ...• E.g.

– fn x => x : 'a -> 'a

– fn x => fn y => (x, y) : 'a -> 'b -> ('a * 'b)

– fn (x : 'a) => fn (y : 'a) => x = y :'a -> 'a -> bool(* actually does not typecheck! *)

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Equality Types• Motivation

– Equality (=) is not defined on every type.– E.g.

• comparing two functions for equality?

• Type variables with equality– ''a, ''b, ''c, ...– ''a means 'for any type alpha

for which equality is defined'

– fn (x : ''a) => fn (y : ''a) => x = y :''a -> ''a -> bool

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Outline• Expressions and values V• Variables V• Functions V• Types V• Recursion• Datatypes• Pattern matching• Higher-order functions• Exceptions• Modules

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Recursion vs. Iteration• Recursion in SML

fun sum n =if n = 0 then 0else sum (n - 1) + n

• Iteration in C

int i, sum;for (i = 0, sum = 0;

i <= n; i++)

sum += n;

• Recursion is not an awkward toolif you are used to functional programming.

• Recursion seems elegant but inefficient!

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Recursion in Actionfun sum n =

if n = 0 then 0else sum (n - 1) + n

call stack

evaluation

f 10

f 8

f 9

...

f 1

f 0

55

36 + 9

45 + 10

...0 + 1

1 + 2

0

further computation

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Funny Recursionfun zero n =

if n = 0 then 0else zero (n - 1)

call stack

evaluation

f 10

f 9

f 8

...

f 1

f 0

0

0

0

0

0

...0

no further computation

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Funny Recursion Optimizedfun zero n =

if n = 0 then 0else zero (n - 1)

call stack

evaluation

f 10

f 9

f 8

...

f 1

f 0

0

0

0

0

0...

0

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Funny Recursion Further Optimized

fun zero n =if n = 0 then 0else zero (n - 1)

call stack

evaluation

f 10 f 9 f 8 ...

f 1 f 0 0

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Tail Recursive Function• A tail recursive function f:

– A recursive call to f is the last step in evaluating the function body.

– That is, no more computation remains after calling f itself.

• A tail recursive call needs no stack!• A tail recursive call is as efficient as iteration!

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Example• Tail recursive sum

fun sum' accum k =if k = 0 then accumelse sum' (accum + k) (k - 1)

fun sum n = sum' 0 n

• Non-tail recursive sum

fun sum n =if n = 0 then 0else sum (n - 1) + n

• Think about the invariant of sum:– given:

sum' accum k– invariant:

accum = (k + 1) + ... + n

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Mathematics vs. Computer Science

• CS ½ Math? • CS Math?

Math

CS

Math

CS

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Factorial Function fac 1 = 1 fac n = n * fac (n - 1)

• Makes sense in– Math– CS

fac 1 = 1 fac n = fac (n + 1) / n

• Makes sense in math.• But not in CS:

– it always diverges.

• "Then what kind of recursive functions are computationally meaningful?"

) domain theory in computer science• Other examples

– real numbers vs. computable real numbers– logic in math vs. logic in computer science

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Outline• Expressions and values V• Variables V• Functions V• Types V• Recursion V• Datatypes• Pattern matching• Higher-order functions• Exceptions• Modules

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Enumeration Types in Cenum shape { Circle, Rectangle, Triangle};

• Great flexibility– e.g.

Circle + 1 == RectangleTriangle - 1 == Rectangle(Circle + Triangle) / 2 == Rectangle

• But is this good or bad?

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Datatypes in SMLdatatype shape = Circle | Rectangle | Triangle

• No flexibility– e.g.

Circle + 1 (x)Triangle - 1 (x)Circle + Triangle (x)

• But high safety.

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Primitive Setdatatype set =

Empty | Singleton | Pair |Many

- Empty;val it = Empty : set- Many;val it = Many : set

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Primitive Set with Argumentsdatatype set =

Empty | Singleton | Pair |Many of int

- Many 5;val it = Many 5 : set

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Primitive Set with Type Parameters

datatype 'a set = Empty | Singleton of 'a | Pair of 'a * 'a |Many of int

- Singleton 0;val it = Singleton 0 : int set- Pair (0, 1);val it = Pair (0, 1) : int set

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Primitive Set with Type Parameters

datatype 'a set = Empty | Singleton of 'a | Pair of 'a * 'a |Many of int

- Pair (Singleton 0, Pair (0, 1))val it = Pair (Singleton 0, Pair (0, 1)) :

int set set

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Primitive Set with Type Parameters

datatype 'a set = Empty | Singleton of 'a | Pair of 'a * 'a |Many of int

- Empty;val it = Empty : 'a set- Many 5;val it = Many 5 : 'a set

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Primitive Set with Type Parametersdatatype 'a set =

Empty | Singleton of 'a | Pair of 'a * 'a |Many of int

- Pair (0, true);stdIn:27.1-27.15 Error: operator and operand

don't agree [literal] operator domain: int * int operand: int * bool in expression: Pair (0,true)

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Recursive Set with Type Parameters

datatype 'a set = Empty | NonEmpty of 'a * 'a set

• This is essentially the definition of datatype list.

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Datatype listdatatype 'a list =

nil | :: of 'a * 'a list

- nil;val it = [] : 'a list- 2 :: nil;val it = [2] : int list- 1 :: (2 :: nil);val it = [1,2] : int list- 1 :: 2 :: nil;val it = [1,2] : int list

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Datatype listdatatype 'a list =

nil | :: of 'a * 'a list

- 1 :: [2] :: nil;- [1] :: 2 :: nil;- [1] :: [2] :: nil;val it = [[1],[2]] : int list list- [1] :: [[2]];val it = [[1],[2]] : int list list

(X)(X)

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Using Datatypes• We know how to create values of various datatypes:

– shape– set– int set– int list– ...

• But how do we use them in programming?• What is the point of creating datatype values that

are never used?

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Outline• Expressions and values V• Variables V• Functions V• Types V• Recursion V• Datatypes V• Pattern matching• Higher-order functions• Exceptions• Modules

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Simple Patterndatatype shape = Circle | Rectangle |

Triangle

(* convertToEnum : shape -> int *)fun convertToEnum (x : shape) : int =

case x ofCircle => 0

| Rectangle => 1| Triangle => 2

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Pattern with Argumentsdatatype 'a set =

Empty | Singleton of 'a | Pair of 'a * 'a |Many of int

fun size (x : 'a set) : int =case x of

Empty => 0| Singleton e => 1| Pair (e1, e2) => 2| Many n => n

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Wildcard Pattern _ : "don't care"datatype 'a set =

Empty | Singleton of 'a | Pair of 'a * 'a |Many of int

fun isEmpty (x : 'a set) : bool =case x ofEmpty => true

| _ => false

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Pattern with Type Annotationdatatype 'a list =

nil | :: of 'a * 'a list

fun length (x : 'a list) : int =case x of

(nil : 'a list) => 0| (_ : 'a) :: (tail : 'a list) =>

1 + length tail

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Outline• Expressions and values V• Variables V• Functions V• Types V• Recursion V• Datatypes V• Pattern matching V• Higher-order functions• Exceptions• Modules

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Higher-order Functions• Take functions as arguments.• Return functions as the result.

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Why "Higher-order"?• T0 ::= int | bool | real | unit | ...

• T1 ::= T0 -> T0 | T0 (* 1st order *)

• T2 ::= T1 -> T1 | T1 (* 2nd order *)

• T3 ::= T2 -> T2 | T 2 (* higher order *)

• ...

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Higher-order Functions in List• val exists : ('a -> bool) -> 'a list -> bool• val all : ('a -> bool) -> 'a list -> bool• val map : ('a -> 'b) -> 'a list -> 'b list• val filter : ('a -> bool) -> 'a list -> 'a list• val app : ('a -> unit) -> 'a list -> unit

(* printInt : int -> unit *)fun printInt i = TextIO.print ((Int.toString i) ^ "\n" );List.app printInt [1, 2, 3];

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List Fold Function• foldl : ('a * 'b -> 'b) -> 'b -> 'a list -> 'b

foldr : ('a * 'b -> 'b) -> 'b -> 'a list -> 'b

• foldl f b0 [a0, a1, a2, ..., an-1]

b0

a0

f b1

a1

f b2

an-1

f bn

a2

f b3 bn-2

an-2

f bn-1...

...

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ExamplesList.exists f l =

List.foldl (fn (a, b) => b orelse f a) false lList.all f l =

List.foldl (fn (a, b) => b andalso f a) true lList.app f l =

List.foldl (fn (a, _) => f a) () l List.map f l =

List.foldr (fn (a, b) => f a :: b) nil lList.filter f l =

List.foldr (fn (a, b) => if f a then a :: b else b) nil l

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More Examplesfun sum (l : int list) =

List.foldl (fn (a, accum) => a + accum) 0 l

fun reverse (l : 'a list) =List.foldl (fn (a, rev) => a :: rev) nil l

• Whenever you need iterations over lists,first consider foldl and

foldr.

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Outline• Expressions and values V• Variables V• Functions V• Types V• Recursion V• Datatypes V• Pattern matching V• Higher-order functions V• Exceptions

– exception: please see the course notes.• Modules

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Outline• Expressions and values V• Variables V• Functions V• Types V• Recursion V• Datatypes V• Pattern matching V• Higher-order functions V• Exceptions V• Modules

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Structures and Signatures• Structure

– collection of type declarations, exceptions, values, and so on.

structure Set =struct

type 'a set = 'a listval emptySet = nilfun singleton x = [x]fun union s1 s2 = s1 @ s2

end

• Signature– conceptually type of

structures.

signature SET =sig

type 'a setval emptySet : 'a setval singleton : 'a -> 'a setval union : 'a set -> 'a set -> 'a set

end

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Structures + Signatures• Transparent constraint

– Type definition is exported to the outside.

structure Set : SET =struct

type 'a set = 'a listval emptySet = nilfun singleton x = [x]fun union s1 s2 = s1 @ s2

end

- Set.singleton 1 = [1];val it = true : bool

• Opaque constraint– No type definition is

exported.

structure Set :> SET =struct

type 'a set = 'a listval emptySet = nilfun singleton x = [x]fun union s1 s2 = s1 @ s2

end

- Set.singleton 1 = [1];(* Error! *)

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I need to test your 57 structures!structure BetOne :> HW_ONE = ...structure Bigh2000One :> HW_ONE = ...structure BoongOne :> HW_ONE = structure BrandonOne :> HW_ONE = ......structure ZiedrichOne :> HW_ONE = ...

• How can I test 57 structures?– They all conform to the same signature.

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Simple (but Ugly) Solutionfun sumTest f = f nil = 0 andalso f [1, 2, 3] = 6fun facTest f = f 1 = 1 andalso f 3 = 6 andalso f 7 = 5040...

fun allTest (sum, fac, ...) = let

val sumScore = if sumTest sum then 10 else 0val facScore = if facTest fac then 4 else 0...

insumScore + facScore + ...

end

allTest (BetOne.sum, BetOne.fac, ...);allTest (Bigh2000One.sum, Bigh2000One.fac, ...);allTest (BoongOne.sum, BoongOne.fac, ...);allTest (BrandonOne.sum, BrandonOne.fac, ...);...allTest (ZiedrichOne.sum, ZiedrichOne.fac, ...);

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Functors• Functions on structures

– takes a structure as an argument– returns a structure as the result

structure

structure

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HW1 Test Functorsignature HW_TEST =sig

val score : intend

functor Hw1TestFn (P : HW_ONE) : HW_TEST =struct

structure S = Hw1Solution val sumScore =

if P.sum nil = S.sum nil andalso P.sum [1, 2, 3] = S.sum [1, 2, 3] then 10 else 0

val facScore = if P.fac 1 = S.fac 1 andalso P.fac 3 = S.fac 3 andalso

P.fac 7 = S.fac 7 then 4 else 0

...val score = sumScore + facScore + ...

end

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HW1 Test Functor in Actionstructure BetOneTest = Hw1TestFn (BetOne)structure Bigh2000OneTest = Hw1TestFn

(Bigh2000One)structure BoongOneTest = Hw1TestFn (BoongOne)structure BrandonOneTest = Hw1TestFn

(BrandonOne)...structure ZiedrichOneTest = Hw1TestFn (ZiedrichOne)

BetOneTest.score;Bigh2000OneTest.score;BoongOneTest.score;BrandonOneTest.score;...ZiedrichOneTest.score;

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So it is YOU who will grade your homework!