CSE-321 Programming Languages

Introduction to Functional Programming

POSTECH

March 8, 2006

박성우

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Programming Paradigms• Structural programming

– C, Pascal, …• Object-oriented programming

– C++, Java, …• Logic programming

– Prolog, …• Functional programming

– SML, Haskell, Objective Caml, Lisp, Scheme, …

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

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C• A program consists of commands.

– command = “do something”

• Nothing wrong:

if (x == 1) thenx = x + 1;

elsex = x - 1;

• Nothing wrong either:

if (x == 1) thenx = x + 1;

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SML• A program consists of expressions.

– expression = “obtain a value”

• Nothing wrong:if (x = 1) then

x + 1else

x - 1

• But this does not make sense:if (x = 1) then

x + 1– what is the value if x <> 1?

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Evaluation

Expression Value

• An expression “evaluates” to a value.• We “evaluate” an expression to obtain a value.

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Integer Evaluation

1 + 1 2

1 - 1 0

…1 * 1 1

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Boolean Evaluation

1 = 1 true

1 <> 1 false

…1 <> 0 true

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An Integer Expression

if 1 = ~1 then 10 else ~10

~10

if false then 10 else ~10

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Values as Expressions

1 ???

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Everything is an Expression!

• 1• ~1• 1 = ~1• 10• ~10• if 1 = ~1 then 10 else ~10

if 1 = ~1 then 10 else ~10

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Actually Not Everything

• Ill-formed expressions

– if 1 = ~1 then 10 (x)

– if 1 = ~1 then 10 else ~10(x)

if 1 = ~1 then 10 else ~10

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

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Variable Declaration- val x = 1 + 1;val x = 2 : int

• A variable x is “bound” to value 2.• From now on, any occurrence of x is replaced by 2.

- val y = x + x;val y = 4 : int

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Local Declaration

letval x = 1val y = x + xval z = y + y

inz + z

end

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Nested Local Declaration

letval x = 1

inx + x

end

letval y = <expression>

iny + y

end

letval y = let val x = 1 in x + x end

iny + y

end

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Why “Local”?

letval y = let

val x = 1 in x + x end

inx + y

end

okay???

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Variables are NOT variable.• The contents of a variable never change.• Surprise?

– That’s because you are thinking about variables in imperative programming.

variables in SML <> variables in C

• Once you get used to functional programming,immutability of variables in functional programming will be taken for granted!

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Then Why Variables?• Any advantage in using variables at all?

letval x = 1val y = x + xval z = y + y

inz + z

end

((1 + 1) + (1 + 1)) + ((1 + 1) + (1 + 1))VS.

What if it takes 10 hours to evaluate 1?

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

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When is the first time you learned the concept of

function?

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즐거운 낱말 공부 시간• 함수 ( 函數 ): 두 변수 x, y 간에 어떤 관계가 있어 x

의 변화에 따라 y 가 일정한 법칙으로 변화할 때 y를 x 의 함수라 함 . (function)

( 동아 마스타 국어사전 )

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즐거운 한자 공부 시간

• 函1. 함 ( 함 ).

2. 편지 ( 함 )

3. 갑옷 ( 함 )

4. 넣을 , 들일 ( 함 )

예 :

( 書函 ) 서함 : 책을 넣는 상자

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Function = 函數 = Box Number!

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Using a Box Number

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Using a Box Number - Generalized

…

…

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Function in SML = Box Number

(fn x => x + 1) n

=

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Function Application

• We “apply” (fn x => x + 1) to n.• x is called a formal argument/parameter.• n is called an actual argument/parameter.

…

(fn x => x + 1) n

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Evaluating a Function Application…

…

(fn x => x + 1) n

n + 1

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Functions in SML• Nameless function

– fn x => x + 1;

• Storing a nameless function to a variable

– val incr = fn x => x + 1;

• Function declaration

– fun incr x = x + 1;

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Function Applications

(fn x => x + 1) 1

2

1 + 1

incr 1

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So far,

So good,

So easy.

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First-class Objects• First-class objects = primitive objects

– can be stored in a variable.– can be passed as an argument to a function.– can be returned as a return value of a function.

• Examples:– integers– booleans– characters– floating-point numbers– …

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First-class Objects in C• First-class objects

– integers– characters– floating-point numbers– pointers– structures– …

• Functions?– Function pointers are first-class objects.– But functions are not.

• Why? You cannot create new functions on the fly!

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Functions = First-class Objects in SML

• Functions:– can be passed as an argument to a function.– can be returned as a return value of a function.

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Box Number as Output

…

…

… …

such that

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Box Number as Output

…

…

x

+x

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x

y+x

y

Box Number as Output

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x

y

fn y => y+x

y+x

Box Number as Output

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x

y

fn y => y+x

fn y => y+x

y+x

Box Number as Output

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x

y

fn y => y+x

fn y => y+x

fn x => (fn y => y+x)

y+x

Box Number as Output

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In SML• Recall the following declarations are equivalent:

– val incr = fn x => x + 1;– fun incr x = x + 1;

• Then:

– val add = fn x => (fn y => y + x);– fun add x = fn y => y + x;– fun add x y = y + x;

• add takes only one argument, not two!• In fact, every function in SML takes only a single

argument.

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Adding Two Integers

(fn x => (fn y => y + x)) 1 2

3

add 1 2

(fn y => y + 1) 2

2 + 1

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Box Number as Input

true,

false)

(

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Box Number as Input

true,

false)

(

fn f => (f true, f false)

f

f

f

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

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Types• A type specifies what kind of value a given

expression evaluates to.– 1 + 1 : int– true andalso false : bool– #”A” : char– ”hello” : string– (1, true) : int * bool– (1, ~1, true) : int * int * bool– 1.0 : real– () : unit

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Type Preservation

Expression : T Value : T

• An evaluation preserves the type of a given expression.

• Type preservation will be discussed in detail later in this course. – It’s a lot more important than you might think!

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Example

letval x = 1val y = x + xval z = y + y

inz + z

end

8 : int: int

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Function Types• T -> T’

– type of functions:• taking arguments of type T• returning values of type T’

• Example:– val incr = fn x => x + 1;

val incr = fn : int -> int

– fun incr x = x + 1;val incr = fn : int -> int

• Explicit type annotation– val incr = fn (x:int) => x + 1;

val incr = fn (x:int) : int => x + 1;

– fun incr (x:int) = x + 1;fun incr (x:int) : int = x + 1;

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x

fn y => y+x

fn x => (fn y => y+x)

Type of add

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fn y => y+x

fn x => (fn y => y+x)

Type of add

int

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fn x => (fn y => y+x)

Type of add

int

int -> int

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int -> (int -> int)

Type of add

int

int -> int

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What is the Type?

true,

false)

(

fn f => (f true, f false)

f

f

f

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f : bool -> int ?

true,

false)

(

fn f => (f true, f false)

f

f

f

: (bool -> int) -> int * int

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But why is it

f : bool -> int ?

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Why not f : bool -> char ?

true,

false)

(

fn f => (f true, f false)

f

f

f

: (bool -> char) -> char * char

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Then why not

f : bool -> string ?f : bool -> int * string ?f : bool -> unit ?f : bool -> (int -> int) ?f : bool -> <crazy type> ?…

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So we need

Polymorphism.

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Polymorph in Warcraft 2

• Cost: 2000 GoldMana 200Range: 10

Time to Upgrade: 200 Time Units.

• Perhaps the most fearsome of the Mage spells, Polymorph alters the physical form of its target. This metamorphosis changes man to beast, forever warping both mind and body. This sorcery wholly transforms the victim into a creature of the wolds, thus losing their reasoning - making commands for direction or battle impossible.

• This turns any living creature into the critter for that map type such as Pig for Wasteland, Seal for Snow, Sheep for Forest, Boar for Swamp. Although the unit hasn't been killed it has been in effect killed as it cannot be turned back into the previous unit or commanded in anyway.

• Polymorph is good for instantly taking out units, especially costly and dangerous units. The primary use for Polymorph is for hit and run attacks against enemy Mages, Death Knights, and Gryphons.

• Polymorph is really helpful on defense, especially on water maps. When the enemy lands you can quickly Polymorph the most dangerous enemy units right after they unload from the Transport.

• Make a Mage with at least 200 Mana Invisible, sneak up to an enemy Mage or Dragon and Polymorph it. If it is sea, do the pop out of the Transport Polymorph then run.

• Often you can cast Polymorph before the enemy even notices it, there is no warning to enemy players that their units have been Polymorphs. Against inattentive players you might be able to Polymorph quite a few units using hit and run attacks before they catch on.

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Polymorphism in Warcraft 3 will be discussed in the next lecture.