CS 2104 : Prog. Lang. Concepts.
Functional Programming I
Lecturer : Dr. Abhik Roychoudhury
School of Computing
From Dr. Khoo Siau Cheng’s lecture notes
Outline
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What is Functional Programming?
Basic Expressions
Using Higher-order Functions
List Processing
Value and Type Constructions
Features of Functional Programming
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Expressions: The main construct in the language
Functions : First-class citizens of the language
Types : for specification and verification
List Processing
Computation
• Achieved via function application
• Functions are mathematical functions without
side-effects.
– Output is solely dependent of input.
States
Pure function
Impure function
with assignment
Outline
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•
•
•
•
What is Functional Programming?
Basic Expressions
Using Higher-order Functions
List Processing
Value and Type Constructions
Basic SML Expressions
<Exp> ::= <Constants> | <Identifiers>
| <Exp1> <op> <Exp2> |
| if <Exp0> then <Exp1> else <Exp2>
| let {<Decl>} in <Exp> end
| fn <Pattern> {<Pattern>} => <Exp>
| <Exp1> <Exp2>
| (<Exp1>, <Exp2> {, <Exp3>})
| <List-Exp>
<Constants>
::= all constants
<Identifiers> ::= all identifiers
<op>
::= all binary operators
If-expression, Not If-statement
if <e0> then <e1> else <e2>
if (3 > 0)
then 4+x
else x-4 ;
4+x
If-expression evaluates to a value.
Constructing a Linear List
<List-Exp> ::=[[<Exp> {, <Exp>}] ]
| nil | <Exp> :: <Exp>
nil ;
Note : A list must end with nil.
1::nil ;
1::(2::nil);
(x+2)::(y-3)::(x*3)::nil ;
Constructing a Linear List
<List-Exp> ::=[[<Exp> {, <Exp>}] ]
| nil | <Exp> :: <Exp>
nil ;
Operator :: is right associative.
1::nil ;
1::(2::nil);
(x+2)::(y-3)::(x*3)::nil ;
head
tail
[] ; [1] ; [1,2]; [x+2,y-3,x*3] ;
Constructing a Tuple
(<Exp> , <Exp> {, <Exp>})
Tuple is like a record without field names.
(1,2) ;
(1,2,True,4,False) ;
(x+2,y=3,x*3) ;
Giving An Expression A Name
Declaring an expression with a name enables easy reference.
<decl> ::= val <pattern> = <exp>
Identifiers are legally declared after proper pattern matching.
val
val
val
val
x = 3
(y,z)
[a,b]
[a,b]
;
= (x+3,x-3) ;
= [x+2,y-3] ;
= [x+2] ;
val (x::xs) = [1,2,3,4,5] ;
Pattern-Match a List
val (x::xs) = [1,2,3,4,5] ;
x  1
xs  [2,3,4,5] ;
val (x::_::y::xs) = [1,2,3,4,5] ;
x  1
y  3
xs  [4,5] ;
Local Declaration/Definition
let {<decl>} in <exp> end
let
val x = (3,4)
val (a,b) = x
in
a + a * b
end ;
15
Identifiers are statically scoped
Nested bindings of same var.
• Let val x =2 in let val x = x +1 in x*x end end
• Let val x = 2 in let val y = x +1 in y * y end end
• Let val x = 2 in (x+1)*(x+1) end
• 9
Function Declaration and
Function Application
<Decl> ::= fun <var> <para> = <exp>
<para> ::= <pattern> {<pattern>}
fun fac x = if (x > 0)
then x * (fac (x-1)) else 1
fac 2  if (2 > 0) then 2 * (fac (2-1)) else 1
 2 * (fac 1)
 2 * (if (1 > 0) then 1 * (fac (1-1)) else 1)
 2 * (1 * (fac 0))
 2 * (1 * 1)
 2
Function as Equations
A function can be defined by a set of equations.
<Decl> ::= fun <var> <para> = <exp>
{| <var> <para> = <exp>}
fun fac 0
|
fac n
= 1
= n * (fac (n-1))
Function application is accomplished via pattern matching.
fac 2  2 * (fac (2-1))
 2 * (1 * (fac (1-1)))
 2 * (1 * 1)
 2 * (fac 1)
 2 * (1 * (fac 0))
 2
Outline
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What is Functional Programming?
Basic Expressions
Using Higher-order Functions
List Processing
Value and Type Constructions
Functions as First-Class Citizens
Lambda abstraction: A nameless function
fn <para> => <Exp>
(fn (x,y) => x + y) (2,3)  2+3  5
Definition
Application
val newfn = fn (x,y) => x + y ;
newfn(2,4) + newfn(4,3)
Functions as First-Class Citizens
Passing function as argument
fun apply (f, x) = f x
apply (fac, 3)


fac 3
6
apply (fn x => x*2, 3)
 (fn x => x*2) 3
 3*2
 6
Functions as First-Class Citizens
Returning function as result
fun choose (x,f,g) = if x then f else g ;
choose(a = b,fn x => x*2, fac)(4)
returns either 8 or 24
fun compose (f,g) = fn x => f (g x)
(compose (add3,fac)) 4
 (fn x => add3 (fac x))(4)
 (add3 (fac 4))
compose (f,g) is denoted as f o g
 27
Functions as First-Class Citizens
Storing function in data structure
val
flist = [ fac, fib, add3] ;
let val [f1,f2,f3] = flist
in (f1 2) + (f2 3) + (f3 4)
end ;
Functions as Algorithms
• A function describes a way for computation.
• Self-Recursive function
fun
factorial (x) = if (x <= 1) then 1
else x * factorial (x-1) ;
• Mutual-Recursive functions
fun
|
|
and
even(0)
even(1)
even(x)
odd(x)
= true
= false
= odd(x-1)
= even(x-1) ;
Passing Arguments to a Function
How many argument does this function really take?
fun f (x,y) = x + 2 * y ;
a pair
Calling f with argument (1,2) yields 5.
fun g x y = x + 2 * y ;
pattern 1
pattern 2
Calling g with arguments 1 and 2 yields 5.
Outline
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•
•
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What is Functional Programming?
Basic Expressions
Using Higher-order Functions
List Processing
Value and Type Constructions
Exploring a List
fun length []
= 0
|
length (x::xs) = 1 + length(xs) ;
length([10,10,10]) ==>
==>
==>
==>
==>
1
1
1
1
3
+
+
+
+
length([10,10])
1 + length([10])
1 + 1 + length([])
1 + 1 + 0
Note how a list is being consumed from head to tail.
fun append([],ys)
= ys
|
append(x::xs,ys) = x :: append(xs,ys);
append([1,2],[3]) ==> 1::append([2],[3])
==> 1::2::append([],[3])
==> 1::2::[3]
==> [1,2,3]
Note how a new list is being produced from head to tail.
append(x,y) is denoted by x @ y.
fun rev(xs) =
let fun r([],ys) = ys
|
r(x::xs,ys)= r(xs,x::ys)
in r(xs,[])end
rev([1,2,3]) ==> r([1,2,3],[])
==> r([2,3],1::[])
==> r([3],2::1::[])
==> r([],3::2::1::[])
==> [3,2,1]
Note how lists are being produced and consumed
concurrently.
Map operation
map square [1,2,3,4]
==> [square 1, square 2,.., square 4]
==> [1,4,9,16]
fun map f [] = []
|
map f (x::xs) =
(f x) :: (map f xs)
map square [1,2]
==> (square 1) :: (map square [2])
==> 1 :: (square 2) :: (map square [])
==> 1 :: 4 :: []
== [1,4]
Map operation
fun map f [] = []
|
map f (x::xs) =
(f x) :: (map f xs)
Law for Map :
map f (map g list) = map (f o g) list
map f (map g [a1,..,an])
= map f [g a1,..,g an]
= [f (g a1),.., f (g an)]
= [(f o g) a1,.., (f o g) an]
= map (f o g) [a1,..an]
Reduce Operation
reduce  [a1,..,an] a0 =
a1  (a2  (..  (an  a0)))
reduce f [a1,..,an] a0 =
f(a1,f(a2,f(..,f(an,a0))))

:
a1
a2

a1
:
a2
:


:
an
[]
an
a0
reduce (op *) [2,4,6] 1
==> 2 * (4 * (6 * 1))
==> 48
reduce (fn (x,y)=>1+y) [2,4,6] 0
==> 1 + (1 + (1 + 0))
==> 3
+
:
2
1
:
4
1
:
6
+
[]
+
1
0
Outline
•
•
•
•
•
What is Functional Programming?
Basic Expressions
Using Higher-order Functions
List Processing
Value and Type Constructions
Types:
Classification of Values and Their
Operators
Basic Types
Type
bool
int
real
string
Values
true,false
…,~1,0,1,2,…
..,0.0,.,3.14,..
“foo”,”\”q\””,…
Operations
=, <>, …
=,<>,<,+,div,…
=,<>,<,+,/,…
=,<>,…
Boolean Operations:
e1 andalso e2
e1 orelse e2
Types in ML
• Every expression used in a program must be welltyped.
– It is typable by the ML Type system.
• Declaring a type :
3 : int
[1,2] : int list
• Usually, there is no need to declare the type in
your program – ML infers it for you.
Structured Types
Structured Types consist of structured values.
• Structured values are built up through
expressions. Eg : (2+3, square 3)
• Structured types are denoted by type
expressions.
<type-expr>
::= <type-name> | <type-constant>
| <type-expr> * <type-expr>
| <type-expr>  <type-expr>
| <type-expr> list
|…
Type of a Tuple
(1,2) : int * int
(3.14159, x+3,true) : real * int * bool
A * B = set of ordered pairs (a,b)
Data Constructor :
Type Constructor : *
In general,
(a1,a2,…,an)
(,)
as in
belongs to
as in (a,b)
A* B
A1*A2*…*An.
Type of A List
Type Constructor :
list
[1,2,3] : int list
[3.14, 2.414] : real list
Not well-typed!!
[1, true, 3.14] : ??
A list = set of all lists of A -typed values.
A in A-list refers to any types:
(int*int) list : [ ], [(1,3)], [(3,3),(2,1)], …
int list list
: [ ], [[1,2]], [[1],[0,1,2],[2,3],…
Function Types
Declaring domain & co-domain
fac : int -> int
A -> B = set of all functions from A to B.
Type Constructor :
->
Data Construction via :
1. Function declaration :
2. Lambda abstraction :
fun f x = x + 1 ;
fn x => x + 1;
Value Selection via function application:
f3

4
(fn x => x + 1) 3

4
Sum of Types
Enumerated Types
datatype Days
New Type
= Mo | Tu | We | Th | Fr | Sa | Su ;
data / data constructors
Selecting a summand via pattern matching:
case d
of
Sa => “Go to cinema”
|
Su => “Extra Curriculum”
|
_ => “Life goes on”
Combining Sum and Product of Types:
Algebraic Data Types
Defining an integer binary tree:
datatype IntTree = Leaf int |
Node of (IntTree, int, IntTree) ;
fun height (Leaf x) = 0
|
height (Node(t1,n,t2))=
1 + max(height(t1),height(t2)) ;
Conclusion
• A functional program consists of an expression,
not a sequence of statements.
• Higher-order functions are first-class citizen in the
language.
– It can be nameless
• List processing is convenient and expressive
• In ML, every expression must be well-typed.
• Algebraic data types empowers the language.