CS 2104 : Prog. Lang. Concepts. Functional Programming I

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CS 2104 Prog. Lang. Concepts.Functional
Programming I

• Lecturer Dr. Abhik Roychoudhury
• School of Computing
• From Dr. Khoo Siau Chengs lecture notes

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Outline

• What is Functional Programming?
• Basic Expressions
• Using Higher-order Functions
• List Processing
• Value and Type Constructions

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Features of Functional Programming

• Expressions The main construct in the language
• Functions First-class citizens of the language
• Types for specification and verification
• List Processing

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Computation

• Achieved via function application
• Functions are mathematical functions without
  side-effects.
• Output is solely dependent of input.

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Outline

• What is Functional Programming?
• Basic Expressions
• Using Higher-order Functions
• List Processing
• Value and Type Constructions

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Basic SML Expressions
ltExpgt ltConstantsgt ltIdentifiersgt ltExp1gt
ltopgt ltExp2gt if ltExp0gt then ltExp1gt else
ltExp2gt let ltDeclgt in ltExpgt end fn
ltPatterngt ltPatterngt gt ltExpgt ltExp1gt
ltExp2gt (ltExp1gt, ltExp2gt , ltExp3gt)
ltList-Expgt ltConstantsgt all
constants ltIdentifiersgt all identifiers ltopgt
all binary operators
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If-expression, Not If-statement
if lte0gt then lte1gt else lte2gt
if (3 gt 0) then 4x else x-4
4x
If-expression evaluates to a value.
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Constructing a Linear List
ltList-Expgt ltExpgt , ltExpgt nil
ltExpgt ltExpgt
nil 1nil 1(2nil) (x2)(y-3)(x3)
nil
Note A list must end with nil.
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Constructing a Linear List
ltList-Expgt ltExpgt , ltExpgt nil
ltExpgt ltExpgt
nil 1nil 1(2nil) (x2)(y-3)(x3)
nil
Operator is right associative.
head
1 1,2 x2,y-3,x3
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Constructing a Tuple
(ltExpgt , ltExpgt , ltExpgt)
Tuple is like a record without field names.
(1,2) (1,2,True,4,False) (x2,y3,x3)
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Giving An Expression A Name
Declaring an expression with a name enables easy
reference.
ltdeclgt val ltpatterngt ltexpgt
Identifiers are legally declared after proper
pattern matching.
val x 3 val (y,z) (x3,x-3) val a,b
x2,y-3
val a,b x2
val (xxs) 1,2,3,4,5
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Pattern-Match a List
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Local Declaration/Definition
let ltdeclgt in ltexpgt end
let val x (3,4) val (a,b) x in a a b
end
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Identifiers are statically scoped
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Nested bindings of same var.

• Let val x 2 in let val x x 1 in xx end end
• Let val x 2 in let val y x 1 in y y end
  end
• Let val x 2 in (x1)(x1) end
• 9

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Function Declaration and Function Application
ltDeclgt fun ltvargt ltparagt ltexpgt ltparagt
ltpatterngt ltpatterngt
fun fac x if (x gt 0) then x (fac
(x-1)) else 1
fac 2 ? if (2 gt 0) then 2 (fac (2-1)) else 1 ?
2 (fac 1) ? 2 (if (1 gt 0) then 1 (fac
(1-1)) else 1) ? 2 (1 (fac 0)) ? 2 (1
1) ? 2
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Function as Equations
A function can be defined by a set of equations.
ltDeclgt fun ltvargt ltparagt ltexpgt
ltvargt ltparagt ltexpgt
fun fac 0 1 fac n n (fac (n-1))
Function application is accomplished via pattern
matching.
fac 2 ? 2 (fac (2-1)) ? 2 (fac 1) ? 2
(1 (fac (1-1))) ? 2 (1 (fac 0)) ? 2 (1
1) ? 2
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Outline

• What is Functional Programming?
• Basic Expressions
• Using Higher-order Functions
• List Processing
• Value and Type Constructions

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Functions as First-Class Citizens

• Lambda abstraction A nameless function
• fn ltparagt gt ltExpgt

(fn (x,y) gt x y) (2,3) ? 23 ? 5
Definition Application val newfn
fn (x,y) gt x y newfn(2,4) newfn(4,3)
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Functions as First-Class Citizens

• Passing function as argument
• fun apply (f, x) f x

apply (fac, 3) ? fac 3 ? 6
apply (fn x gt x2, 3) ? (fn x gt x2) 3 ?
32 ? 6
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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 gt x2, fac)(4) returns
either 8 or 24
fun compose (f,g) fn x gt f (g x)
(compose (add3,fac)) 4 ? (fn x gt add3 (fac
x))(4) ? (add3 (fac 4)) ? 27
compose (f,g) is denoted as f o g
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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
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Functions as Algorithms

• A function describes a way for computation.
• Self-Recursive function
• fun factorial (x) if (x lt 1) then 1
• else x factorial (x-1)

• Mutual-Recursive functions
• fun even(0) true
• even(1) false
• even(x) odd(x-1)
• and odd(x) even(x-1)

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Passing Arguments to a Function
How many argument does this function really take?
fun f (x,y) x 2 y
Calling f with argument (1,2) yields 5.
fun g x y x 2 y
Calling g with arguments 1 and 2 yields 5.
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Outline

• What is Functional Programming?
• Basic Expressions
• Using Higher-order Functions
• List Processing
• Value and Type Constructions

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Exploring a List
fun length 0 length (xxs) 1
length(xs)
length(10,10,10) gt 1 length(10,10)
gt 1 1 length(10) gt 1 1 1
length() gt 1 1 1 0 gt 3
Note how a list is being consumed from head to
tail.
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fun append(,ys) ys append(xxs,ys)
x append(xs,ys)
append(1,2,3) gt 1append(2,3) gt 1
2append(,3) gt 123 gt 1,2,3
Note how a new list is being produced from head
to tail.
append(x,y) is denoted by x _at_ y.
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fun rev(xs) let fun r(,ys) ys
r(xxs,ys) r(xs,xys) in r(xs,)end
rev(1,2,3) gt r(1,2,3,) gt
r(2,3,1) gt r(3,21) gt
r(,321) gt 3,2,1
Note how lists are being produced and consumed
concurrently.
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Map operation
map square 1,2,3,4 gt square 1, square
2,.., square 4 gt 1,4,9,16
fun map f map f (xxs) (f x)
(map f xs)
map square 1,2 gt (square 1) (map square
2) gt 1 (square 2) (map square ) gt
1 4 1,4
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Map operation
fun map f map f (xxs) (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
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Reduce Operation
reduce ? a1,..,an a0 a1 ? (a2 ? (.. ? (an
? a0))) reduce f a1,..,an a0 f(a1,f(a2,f(..,
f(an,a0))))
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reduce (op ) 2,4,6 1 gt 2 (4 (6
1)) gt 48
reduce (fn (x,y)gt1y) 2,4,6 0 gt 1 (1
(1 0)) gt 3
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Outline

• What is Functional Programming?
• Basic Expressions
• Using Higher-order Functions
• List Processing
• Value and Type Constructions

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Types Classification of Values and Their
Operators
Basic Types
Type Values Operations bool true,false ,
ltgt, int ,1,0,1,2, ,ltgt,lt,,div, real ..
,0.0,.,3.14,.. ,ltgt,lt,,/, string foo,\q\
, ,ltgt,
Boolean Operations e1 andalso e2 e1
orelse e2
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Types in ML

• Every expression used in a program must be
  well-typed.
• 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.

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Structured Types
Structured Types consist of structured values.

• Structured values are built up through
• expressions. Eg (23, square 3)

• Structured types are denoted by type
• expressions.

lttype-exprgt lttype-namegt lttype-constantgt
lttype-exprgt lttype-exprgt
lttype-exprgt ? lttype-exprgt lttype-exprgt list

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Type of a Tuple
(1,2) int int (3.14159, x3,true) real
int bool
A B set of ordered pairs (a,b)
Data Constructor ( , ) as in (a,b) Type
Constructor as in A B
In general, (a1,a2,,an) belongs to A1A2An.
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Type of A List
Type Constructor list
1,2,3 int list 3.14, 2.414 real
list 1, true, 3.14 ??
Not well-typed!!
A list set of all lists of A -typed values.
A in A-list refers to any types (intint)
list , (1,3), (3,3),(2,1), int list
list , 1,2, 1,0,1,2,2,3,
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Function TypesDeclaring domain co-domain
fac int -gt int
A -gt B set of all functions from A to B.
Type Constructor -gt Data Construction via
1. Function declaration fun f x x 1 2.
Lambda abstraction fn x gt x 1
Value Selection via function application f 3
? 4 (fn x gt x 1) 3 ? 4
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Sum of Types Enumerated Types
datatype Days Mo Tu We Th Fr Sa
Su
New Type
data / data constructors
Selecting a summand via pattern matching case d
of Sa gt Go to cinema Su gt Extra
Curriculum _ gt Life goes on
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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))

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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.