Introduction to Functional Programming

1
Introduction to Functional Programming
Programming Languages
???

• Kangwon National University

These slides were originally created by Prof.
Sungwoo Park at POSTECH.
2
Programming Paradigms

• Paradigm In science, a paradigm describes
  distinct concepts or thought patterns in some
  scientific discipline.
• Main programming paradigms
• Imperative programming C, Pascal,
• Logic programming Prolog,
• Functional programming SML, Haskell, OCaml,
  Lisp, Scheme, F,
• Orthogonal concept
• Object-oriented programming C, Java, OCaml,

3
Outline

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

4
Imperative Language 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

5
Functional Language OCaml

• 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 ltgt 1?

6
Evaluation
Expression
Value

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

7
Integer Evaluation
1 1
2
1 - 1
0
1 1
1

8
Boolean Evaluation
1 1
true
1 ltgt 1
false
1 ltgt 0
true

9
An Integer Expression
if 1 -1 then 10 else -10
if false then 10 else -10
-10
10
Values as Expressions
1
???
11
Everything is an Expression!
if 1 -1 then 10 else -10

• 1
• -1
• 1 -1
• 10
• -10
• if 1 -1 then 10 else -10

12
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)

13
Outline

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

14
Variable Declaration

• - let x 1 1
• val x int 2
• A variable x is bound to value 2.
• From now on, any occurrence of x is replaced by
  2.
• - let y x x
• val y int 4

15
Local Declaration
let x 1 in let y x x in let z y y in z
z
8
16
Nested Local Declaration
let x 1 in x x
let y ltexpressiongt in y y
let y let x 1 in x x in y y
17
Why Local?
let y let x 1 in x x
in x y
okay???
18
Variables are NOT variable.

• The contents of a variable never change.
• immutability of variables
• Surprise?
• Thats because you are thinking about variables
  in imperative programming.
• variables in OCaml ltgt variables in C

19
Then Why Variables?

• Any advantage in using variables at all?

let x 1 in let y x x in let z y y in z
z
((1 1) (1 1)) ((1 1) (1 1))
VS.
What if it takes 10 hours to evaluate 1?
20
Outline

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

21

• When is the first time you learned the concept of
  function?

22
??? ?? ??

• ?? function, ?? ?? x? y ??? x? ?? ???? ??? y??
  ????? ??? ?? ?, y? x? ???? ??.
• (????)

23
?? ?

• ?
• 1. ?(?).
• 2. ??(?)
• 3. ??(?)
• 4. ??, ??(?)
• ?
• (??) ?? ?? ?? ??

24
Function ?? Box Number!
25
Using a Box Number
26
Using a Box Number - Generalized


27
Function in OCaml Box Number
fun x -gt x 1

28
Function Application

(fun x -gt x 1)n

• We apply (fun x -gt x 1) to n.
• x is called a formal argument/parameter.
• n is called an actual argument/parameter.

29
Evaluating a Function Application
(fun x -gt x 1)n
n 1
30
Functions in OCaml

• Nameless function
• fun x -gt x 1
• Storing a nameless function to a variable
• let incr fun x -gt x 1
• Function declaration
• let incr x x 1

31
Function Applications
incr 1
(fun x -gt x 1) 1
1 1
2
32
First-Class Stamps
33
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

34
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!

35
Functions First-Class Objects in OCaml

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

36
Box Number as Output
such that
37
Box Number as Output
x
x
38
Box Number as Output
x
y
yx
39
Box Number as Output
x
y
fun y -gt yx
yx
40
Box Number as Output
x
y
fun y -gt yx
yx
fun y -gt yx
41
Box Number as Output
x
y
fun x -gt (fun y -gt yx)
fun y -gt yx
yx
fun y -gt yx
42
In OCaml

• Recall the following declarations are equivalent
• let incr fun x -gt x 1
• let incr x x 1
• Then
• let add fun x -gt (fun y -gt y x)
• let add x fun y -gt y x
• let add x y y x
• add can take a single argument to return a
  function.

43
Adding Two Integers
add 1 2
(fun x -gt (fun y -gt y x)) 1 2
(fun y -gt y 1) 2
2 1
3
44
Box Number as Input
true,
false)
(
45
Box Number as Input
f
fun f -gt (f true, f false)
true,
false)
(
f
f
46
Outline

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

47
Types

• A type specifies what kind of value a given
  expression evaluates to.
• 1 1 int
• true false bool
• A char
• hello string
• (1, true) int bool
• (1, -1, true) int int bool
• 1.0 float
• () unit

48
Type Preservation
Expression T
Value T

• An evaluation preserves the type of a given
  expression.

49
Example
let x 1 in let y x x in let z y y in z
z
int
8 int
50
Function Types

• T -gt T
• type of functions
• taking arguments of type T
• returning values of type T
• Example
• let incr fun x -gt x 1
• val incr int -gt int ltfungt
• let incr x x 1
• val incr int -gt int ltfungt
• Explicit type annotation
• let incr fun (xint) -gt x 1
• val incr int -gt int ltfungt
• let incr (xint) x 1
• val incr int -gt int ltfungt

51
Type of add
52
Type of add
int
fun x -gt (fun y -gt yx)
fun y -gt yx
53
Type of add
int
fun x -gt (fun y -gt yx)
int -gt int
54
Type of add
int
int -gt (int -gt int)
int -gt int
55
What is the Type?
f
fun f -gt(f true,f false)
true,
false)
(
f
f
56
f bool -gt int ?
f
(bool -gt int) -gt int int
fun f -gt(f true,f false)
true,
false)
(
f
f
57
But why is it f bool -gt int ?
58
Why not f bool -gt char ?
f
(bool -gt char) -gt char char
fun f -gt(f true,f false)
true,
false)
(
f
f
59
Then why not f bool -gt string ? f bool -gt
int string ? f bool -gt unit ? f bool -gt
(int -gt int) ? f bool -gt ltcrazy typegt ?
60
So We Need Polymorphism

• Poly many
• Morphism forms/shapes
• Polymorphic function a function that can be
  applied to values of different types
• E.g., C templates, Java generics

61
What is the Type of ?
All we know about f is that it takes booleans
as arguments.
62
All we know about f is that it takes booleans
as arguments.
bool
?
63
f bool -gt ?
fun f -gt (f true, f false) (bool -gt ?)
-gt ? ?
64
f bool -gt 'a
fun f -gt (f true, f false) (bool -gt 'a) -gt
'a 'a

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

65
Polymorphic Types

• Types involving type variables 'a, 'b, ...
• E.g.
• fun x -gt x
• 'a -gt 'a
• fun x -gt fun y -gt (x, y)
• 'a -gt 'b -gt ('a 'b)
• fun (x 'a) -gt fun (y 'a) -gt x y
• 'a -gt 'a -gt bool

66
Outline

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

67
Recursion vs. Iteration

• Recursion in OCaml
• let rec sum n if n 0 then 0else sum (n - 1)
  n

• Iteration in C
• int i 0, sum 0for ( i lt n i) sum
  i

• Recursion is not an awkward tool if you are used
  to functional programming.
• Recursion seems elegant but inefficient!

68
Recursion in Action

• let rec sum n if n 0 then 0else sum (n - 1)
  n

call stack
0
further computation
...
...
f 10
evaluation
69
Funny Recursion

• let rec zero n if n 0 then 0else zero (n - 1)

call stack
no further computation
evaluation
70
Funny Recursion Optimized
let rec zero n if n 0 then 0else zero (n - 1)
call stack
0
0
...
...
f 10
evaluation
71
Funny Recursion Further Optimized
let rec zero n if n 0 then 0else zero (n - 1)
call stack
f 10
...
0
evaluation
72
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!

73
Example

• Tail recursive sum
• let rec sum' accum k if k 0 then accumelse
• sum' (accum k) (k - 1)
• let sum n sum' 0 n

• Non-tail recursive sum
• let rec sum n if n 0 then 0else sum (n - 1)
  n

• Think about the invariant of sum'
• given
• sum' accum k
• invariant
• accum n (n - 1) ... (k 1)

74
Outline

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

75
Enumeration Types in C

• enum shape Circle, Rectangle, Triangle
• Great flexibility
• e.g.
• Circle 1 Rectangle
• Triangle - 1 Rectangle
• (Circle Triangle) / 2 Rectangle
• But is this good or bad?

76
Datatypes in OCaml

• type shape Circle Rectangle Triangle
• No flexibility
• e.g.
• Circle 1 (x)
• Triangle - 1 (x)
• Circle Triangle (x)
• But high safety.

77
Set

• type set
• Empty
• Many
• Empty
• - set Empty
• Many
• - set Many

78
Set with Arguments

• type set
• Empty
• Many of int
• Many 5
• - set Many 5

79
Set with Type Parameters

• type 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Singleton 0
• - int set Singleton 0
• Pair (0, 1)
• - int set Pair (0, 1)

80
Set with Type Parameters

• type 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Pair (Singleton 0, Pair (0, 1))
• - int set set Pair (Singleton 0, Pair (0,
  1))

81
Set with Type Parameters

• type 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Pair (0, true)

82
Set with Type Parameters

• type 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Pair (0, true)
• Error This expression has type bool but an
  expression was expected of type int

83
Recursive Set with Type Parameters

• type 'a set
• Empty
• NonEmpty of 'a 'a set
• This is essentially the definition of datatype
  list.

84
Datatype list

• type 'a list
• ( defined internally )
• of 'a 'a list ( infix )
• - 'a list
• 2 ( infix )
• - int list 2
• 2
• - int list 2
• 1 2
• - int list 1 2
• 1 2 ( right associative )
• int list 1 2
• 1 (2 )
• - int list 1 2

85
Datatype list

• type 'a list
• ( defined internally )
• of 'a 'a list ( infix )
• 1 2
• 1 2
• 1 2
• - int list list 1 2
• 1 2
• - int list list 1 2

(X)
(X)
86
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?

87
Outline

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

88
Simple Pattern

• type shape Circle Rectangle Triangle
• ( convertToEnum shape -gt int )
• let convertToEnum (x shape) int
• match x with
• Circle -gt 0
• Rectangle -gt 1
• Triangle -gt 2

89
Pattern with Arguments

• type set
• Empty
• Many of int
• let size (x set) int
• match x with
• Empty -gt 0
• Many n -gt n

90
Wildcard Pattern _ "don't care"

• type 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• let isEmpty (x 'a set) bool
• match x with
• Empty -gt true
• _ -gt false

91
Pattern with Type Annotation

• type 'a list
• ( defined internally )
• of 'a 'a list ( infix )
• let rec length (x 'a list) int
• match x with
• ( 'a list) -gt 0
• (_ 'a) (tail 'a list) -gt
• 1 length tail

92
Outline

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

93
Higher-order Functions

• Take functions as arguments.
• Return functions as the result.

94
Why "Higher-order"?

• T0 int bool float unit ...
• T1 T0 -gt T0 T0 ( 1st order )
• T2 T1 -gt T1 T1 ( 2nd order )
• T3 T2 -gt T2 T2 ( higher order )
• ...

95
Higher-Order Functions in List

• val exists ('a -gt bool) -gt 'a list -gt bool
• val for_all ('a -gt bool) -gt 'a list -gt bool
• val map ('a -gt 'b) -gt 'a list -gt 'b list
• val filter ('a -gt bool) -gt 'a list -gt 'a list
• val iter ('a -gt unit) -gt 'a list -gt unit
• ( print_int int -gt unit )
• let print_int i
• print_string ((string_of_int i) "\n" )
• List.iter print_int 1 2 3

96
List Fold Function - Left

• fold_left ('a -gt 'b -gt 'a) -gt 'a -gt 'b list -gt
  'a
• fold_left f a0 b0 b1 b2 ... bn-1

current result
current element
new result
initial result
elements
final result
...
a0
an-2
...
97
List Fold Function - Right

• fold_right (b -gt a -gt a) -gt b list -gt a -gt
  a
• fold_right f b0 b1 b2 ... bn-1 an

current result
current element
new result
initial result
elements
final result
b2
b0
b1
bn-1
bn-2
...
an
98
Summation

• let sum (l int list)
• List.fold_left (fun accum a -gt a accum) 0 l
• fold_left () 0 a0 a1 a2 ... an-1

()
a0
a1
an-1
a2
an-2
...
0
...

99
List Reversal

• let reverse (l 'a list)
• List.fold_left (fun rev a -gt a rev) l
• Whenever you need iterations over lists, first
  consider fold_left and fold_right.

100
More Examples

• let exists f l
• List.fold_left (fun a b -gt a f b) false l
• let for_all f l
• List.fold_left (fun a b -gt a f b) true l
• let map f l
• List.fold_right (fun a b -gt f a b) l
• let filter f l
• List.fold_right (fun a b -gt if f a then a b
  else b) l

101
Outline

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

102
Exception

• To signal and handle exceptional conditions
• Non-local control structures (e.g., goto in C)
• Exception declaration
• exception Empty_list
• exception Empty_list
• Exceptions are used throughout the standard
  library to signal cases where the library
  functions cannot complete normally.

103
Signaling the exception

• With the raise operator
• e.g., a function for taking the head of a list
• let head l
• match l with
• -gt raise Empty_list
• hd tl -gt hd
• val head 'a list -gt 'a ltfungt
• head 12
• - int 1
• head
• Exception Empty_list

104
Catching the exception

• Exceptions can be trapped with the try with
  construct.
• The with part is a regular pattern-matching on
  the exception value.
• let print_head_list l
• try print_string (head l \n)
• with Empty_list -gt print_string Empty_list\n
• val print_head_list string list -gt unit ltfungt
• print_head_list abc, def
• abc
• - unit ()
• print_head_list
• Empty_list
• - unit ()

105
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

106
Structures and Signatures

• Structure
• collection of type declarations, exceptions,
  values, and so on.
• Structures a program and provides a new namespace
• module Set
• struct
• type 'a set 'a list
• let empty_set
• let singleton x x
• let union s1 s2 s1 _at_ s2
• end

• Signature
• conceptually type of structures.
• module type SET
• sig
• type 'a set
• val empty_set 'a set
• val singleton 'a -gt 'a set
• val union
• 'a set -gt 'a set -gt 'a set
• end

107
Transparent Constraints

• Type definition is exported to the outside.
• module Set
• struct
• type 'a set 'a list
• let empty_set
• let singleton x x
• let union s1 s2 s1 _at_ s2
• end
• Set.singleton 1 1
• - bool true

108
If youre a library developer

• What if you later improve your implementation
  using a binary search tree instead of a list?
• module Set ( your library code )
• struct
• type 'a set 'a bst
• let empty_set ...
• let singleton x ...
• ...
• End
• ( library users code )
• Set.singleton 1 1
• Error This expression has type 'a list but an
  expression was expected of type int bst

109
Opaque Constraints

• module Set SET
• struct
• type 'a set 'a list
• let empty_set
• let singleton x x
• let union s1 s2 s1 _at_ s2
• end
• Set.singleton 1 1
• Error This expression has type int list but an
  expression was expected of type int Set.set

• module type SET
• sig
• type 'a set
• val empty_set 'a set
• val singleton 'a -gt 'a set
• val union
• 'a set -gt 'a set -gt 'a set
• end

• No type definition is exported (Information
  hiding)

110
Overriding Opaque Constraints

• Use the with keyword.
• Furthermore, only definitions specified in the
  signature are exported to outside.

• module Set
• SET with
• type 'a set 'a list
• struct
• type 'a set 'a list
• let empty_set
• let temp x x
• let singleton x temp x
• let union s1 s2 s1 _at_ s2
• end
• Set.singleton 1 1
• - bool true

• module type SET
• sig
• type 'a set
• val empty_set 'a set
• val singleton 'a -gt 'a set
• val union
• 'a set -gt 'a set -gt 'a set
• end

111
Functors

• Functions on structures
• takes a structure as an argument
• returns a structure as the result
• The most powerful tool for structuring programs

structure
structure
112
Code Reuse by Functors
Variation points (Set elements)
Int
String
Float
User defined type
Reusable code (Set operations)
Functors
Set for user defined type
Int Set
String Set
Float Set
Functor applications (Set implementations)
113
Generic Set Functor

• Code reuse using a functor
• Ex. Different set implementations w.r.t. the type
  of set elements
• type comparison LESS Equal Greater
• module type ORDERED sig
• type t
• val compare t -gt t -gt comparison
• end
• module Set
• functor (Elt ORDERED) -gt (
  functor declaration )
• struct
• type elem Elt.t
• type set elem list
• let empty
• ...
• end
• module OrderedString struct type t string
  ... end
• module StringSet Set(OrderedString) (
  functor application )
• module OrderedADT struct type t
  user_defined_t ... end

114
For more information on OCaml, visithttp//caml
.inria.fr/pub/docs/manual-ocaml/index.htmlhttps
//realworldocaml.org/http//ocaml.org
115
Reading Assignment

• Notes on Programming Languages 1?, 2? ??
• If youre comfortable with English, Read
  POSTECH PL Course Note Chapters 1-2