Download x + 1 - Postech

CSE-321 Programming Languages
Introduction to Functional Programming
박성우
POSTECH
March 8, 2006
Programming Paradigms
• Structural programming
– C, Pascal, …
• Object-oriented programming
– C++, Java, …
• Logic programming
– Prolog, …
• Functional programming
– SML, Haskell, Objective Caml, Lisp, Scheme,
…
2
Outline
•
•
•
•
•
•
•
•
•
•
Expressions and values
Variables
Functions
Types
Recursion
Datatypes
Pattern matching
Higher-order functions
Exceptions
Modules
3
C
• A program consists of commands.
– command = “do something”
• Nothing wrong:
if (x == 1) then
x = x + 1;
else
x = x - 1;
• Nothing wrong either:
if (x == 1) then
x = x + 1;
4
SML
• A program consists of expressions.
– expression = “obtain a value”
• Nothing wrong:
if (x = 1) then
x + 1
else
x - 1
• But this does not make sense:
if (x = 1) then
x + 1
– what is the value if x <> 1?
5
Evaluation
Expression
Value
• An expression “evaluates” to a value.
• We “evaluate” an expression to obtain a value.
6
Integer Evaluation
1 + 1
2
1 - 1
0
1 * 1
1
…
7
Boolean Evaluation
1 = 1
true
1 <> 1
false
1 <> 0
true
…
8
An Integer Expression
if 1 = ~1 then 10
else ~10
if false then 10
else ~10
~10
9
Values as Expressions
1
???
10
Everything is an Expression!
if 1 = ~1 then 10
else ~10
•
•
•
•
•
1
~1
1 = ~1
10
~10
• if 1 = ~1 then 10 else ~10
11
Actually Not Everything
if 1 = ~1 then 10
else ~10
• Ill-formed expressions
– if 1 = ~1 then 10
(x)
– if 1 = ~1 then 10 else ~10 (x)
12
Outline
•
•
•
•
•
•
•
•
•
•
Expressions and values V
Variables
Functions
Types
Recursion
Datatypes
Pattern matching
Higher-order functions
Exceptions
Modules
13
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
14
Local Declaration
let
val x = 1
val y = x + x
val z = y + y
in
z + z
end
8
15
Nested Local Declaration
let
let
val x = 1
in
val y = <expression>
in
x + x
y + y
end
end
let
val y = let val x = 1 in x + x end
in
y + y
end
16
Why “Local”?
let
val y = let
val x = 1
in
x + x
end
in
x + y
end
okay???
17
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!
18
Then Why Variables?
• Any advantage in using variables at all?
let
val x = 1
val y = x + x
val z = y + y
in
z + z
end
VS.
((1 + 1) + (1 + 1)) +
((1 + 1) + (1 + 1))
What if it takes 10 hours to evaluate 1?
19
Outline
•
•
•
•
•
•
•
•
•
•
Expressions and values V
Variables V
Functions
Types
Recursion
Datatypes
Pattern matching
Higher-order functions
Exceptions
Modules
20
When is the first time you learned the concept of
function?
21
즐거운 낱말 공부 시간
• 함수(函數): 두 변수 x, y간에 어떤 관계가 있어 x의
변화에 따라 y가 일정한 법칙으로 변화할 때 y를 x의
함수라 함. (function)
(동아 마스타 국어사전)
22
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
23
Function = 函數 = Box Number!
24
Using a Box Number
25
Using a Box Number - Generalized
…
…
26
Function in SML = Box Number
(fn x => x + 1) n
=
27
Function Application
…
(fn x => x + 1) n
• We “apply” (fn x => x + 1) to n.
• x is called a formal argument/parameter.
• n is called an actual argument/parameter.
28
Evaluating a Function Application
…
(fn x => x + 1) n
n+ 1
…
29
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;
30
Function Applications
incr 1
(fn x => x + 1) 1
1 + 1
2
31
So far,
So good,
So easy.
32
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
–…
33
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!
34
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.
35
Box Number as Output
…
…
such that
…
…
36
Box Number as Output
x
…
…
+x
37
Box Number as Output
x
y
y+x
38
Box Number as Output
x
y
fn y => y+x
y+x
39
Box Number as Output
x
y
fn y => y+x
y+x
fn y => y+x
40
Box Number as Output
x
y
fn x => (fn y => y+x)
fn y => y+x
y+x
fn y => y+x
41
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.
42
Adding Two Integers
add 1 2
(fn x => (fn y => y + x)) 1 2
(fn y => y + 1) 2
2 + 1
3
43
Box Number as Input
(
true,
false)
44
Box Number as Input
f
fn f => (f true, f false)
( f
true,
f
false)
45
Outline
•
•
•
•
•
•
•
•
•
•
Expressions and values V
Variables V
Functions V
Types
Recursion
Datatypes
Pattern matching
Higher-order functions
Exceptions
Modules
46
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
47
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!
48
Example
let
val x = 1
val y = x + x
val z = y + y
in
z + z
end
: int
8 : int
49
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;
50
Type of add
x
fn x => (fn y => y+x)
fn y => y+x
51
Type of add
int
fn x => (fn y => y+x)
fn y => y+x
52
Type of add
int
fn x => (fn y => y+x)
int -> int
53
Type of add
int
int -> (int -> int)
int -> int
54
What is the Type?
f
fn f => (f true, f false)
( f
true,
f
false)
55
f : bool -> int ?
f
: (bool -> int) ->
fn f => (f true, f false)
int * int
( f
true,
f
false)
56
But why is it
f : bool ->
int ?
57
Why not f : bool -> char ?
f
: (bool -> char) ->
fn f => (f true, f false)
char * char
( f
true,
f
false)
58
Then why not
f :
f :
f :
f :
f :
…
bool
bool
bool
bool
bool
->
->
->
->
->
string ?
int * string ?
unit ?
(int -> int) ?
<crazy type> ?
59
So we need
Polymorphism.
60
Polymorph in Warcraft 2
•
Cost: 2000 Gold
Mana 200
Range: 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.
61
Polymorphism in Warcraft 3 will be
discussed in the next lecture.
62