CS%202104%20

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CS 2104 Prog. Lang. ConceptsFunctional
Programming II

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

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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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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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Some remarks

• 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.

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Outline

• More about Higher-order Function
• Type inference and Polymorphism
• Evaluation Strategies
• Exception Handling

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Function with Multiple Arguments

• Curried functions accept multiple arguments
• fun twice f x f (f x)

Take 2 arguments
Curried function enables partial application.
let val inc2 twice (fn x gt x x) in (inc2 1)
(inc2 2) end val it 12
Apply 1st argument
Apply 2nd argument
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Curried vs. Uncurried

• Curried functions
• fun twice f x f (f x)
• twice (fn x gt xx) 3 ? 12

Uncurried functions fun twice (f, x) f (f
x) twice (fn x gt xx, 3) ? 12
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Curried Functions

• Curried functions provide extra flexibility
• to the language.

compose f g fn x gt f (g x) ? compose f g
x f (g x) ? compose f fn g gt fn x gt f
(g x) ? compose fn f gt fn g gt fn x gt f (g
x)
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Types of Multi-Argument Funs
fun f(x,y) x y f intint -gt int
fun g x y x y g int -gt int -gt int
(g 3) int -gt int ((g 3) 4) int
Function application is left associative -gt is
right associative
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Outline

• More about Higher-order Function
• Type inference and Polymorphism
• Evaluation Strategies
• Exception Handling

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Type Inference

• ML expressions seldom need type declaration.
• ML cleverly infers types without much help from
  
• the user.

2 2 val it 4 int fun succ n n 1
val succ fn int -gt int
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Helping the Type Inference

• Explicit types are needed when type coercion is
  needed.

fun add(x,y real) x y
fun add(x,y) (xreal) y
val add fn realreal -gt real
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Every Expression has only One Type
fun f x if x gt 0 then x else 1,2,3 val f
fn Int -gt ???
This is not type-able in ML.

• Conditional expression has the same type at
• both branch.

fun abs(x) if xgt0 then x else 0-x val abs
fn int -gt int
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Example of Type Inference
fun f g g (g 1)
type(g) t int?t2 t2?t3 int t2, t2
t3 type(g) t int ? int
type(f) t ? t1 t ? t3 (int ? int) ? int
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Three Type Inference Rules
(Application rule) If f x t, then x t
and f t -gt t for some new type t.
(Equality rule) If both the types x t and
x t can be deduced for a variable
x, then t t.
(Function rule) If t ? u t ? u, then t
t and u u.
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Example of Type Inference
fun f g g (g 1)
Let g tg (g (g 1)) trhs So, by function
declaration, we have f tg -gt trhs By
application rule, let (g 1) t(g 1) g (g 1)
trhs ? g t(g 1) -gt trhs. By application rule,
(g 1) t(g 1) ? g int -gt t(g 1). By equality
rule t(g 1) int trhs. By equality rule tg
int -gt int Hence, f (int -gt int) -gt int
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Parametric Polymorphism
Type parameter
fun I x x val I fn a -gt a

• A Polymorphic function is one whose type
  contains
• type parameters.

• A poymorphic function can be applied to
  arguments
• of more than one type.

(I 3) (I 1,2) (I square)

• Interpretation of val I fn a -gt a
• for all type a, function I takes an input of
  type a
• and returns a result of the same type a.

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Polymorphic Functions

• A polymorphic function is one whose type contains
  type parameter.
• fun map f
• map f (xxs) (f x) (map f xs)
• Type of map (a-gtb) -gt a -gt b
• map (fn x gt x1) 1,2,3 gt 2,3,4
• map (fn x gt x) 1,2,3 gt 1,2,3
• map (fn x gt x) y,n gt y, n

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Examples of Polymorphic Functions
fun compose f g (fn x gt f (g x))
type(f) t1 t5?t6 type(g) t2
t4?t5 range(compose) t t4?t6 type(compose)
t1?t2?t (t5?t6) ? (t4?t5) ? (t4?t6)
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Examples of Polymorphic Functions
fun compose f g (fn x gt f (g x))
Let xtx ftf gtg so compose tf -gt tg -gttrhs
(fn xgtf (g x))trhs gt trhs tx-gtt(f(gx)) (g
x)t(gx) gt g tx-gtt(gx) and tg tx-gtt(g x)
(f (g x))t(f(gx)) gt ft(gx)-gtt(f(gx))
and tf t(gx)-gtt(f(gx)) compose
(t(gx)-gtt(f(gx)))-gt(tx-gtt(gx))-gt(tx-gtt(f(gx))) Ren
ame the variables compose (a-gtb)-gt(c-gta)-gt(
c-gtb)
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Outline

• More about Higher-order Function
• Type inference and Polymorphism
• Evaluation Strategies
• Exception Handling

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Approaches to Expression Evaluation

• Different approaches to evaluating an expression
  may change the expressiveness of a programming
  language.
• Two basic approaches
• Innermost (Strict) Evaluation Strategy
• SML, Scheme
• Outermost (Lazy) Evaluation Strategy
• Haskell, Miranda, Lazy ML

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Innermost Evaluation Strategy

• To Evaluate the call ltnamegtltactualsgt
• (1) Evaluate ltactualsgt
• (2) Substitute the result of (1) for the formals
  in the body
• (3) Evaluate the body of ltnamegt
• (4) Return the result of (3) as the answer.

let fun f x x 1 x in f (2 3) end
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fun f x x 2 x
f (23) gt f (5) gt 5 2 5 gt 12
fun g x y if (x lt 3) then y else x
g 3 (4/0) gt g 3 ? gt ?

• Also referred to as call-by-value evaluation.
• Occasionally, arguments are evaluated
  unnecessarily.

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Outermost Evaluation Strategy

• To Evaluate ltnamegtltactualsgt
• (1) Substitute actuals for the formals in the
  body
• (2) Evaluate the body
• (3) Return the result of (2) as the answer.

fun f x x 2 x f (23) gt (23) 2
(23) gt 12 fun g x y if x lt 3 then y
else x g 3 (4/0) gt if 3 lt 3 then (4/0) else
3 gt 3
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It is possible to eliminate redundant computation
in outermost evaluation strategy.
fun f x x 2 x f (23) gt x 2 x
gt 5 2 x gt 7 x gt
7 5 gt 12
x(23)
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Note Arguments are evaluated only when they are
needed.
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Why Use Outermost Strategy?

• Closer to the meaning of mathematical functions

fun k x y x val const1 k 1 val const2 k
2

• Better modeling of real mathematical objects

val naturalNos let fun inf n n inf
(n1) in inf 1 end
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Hamming Number

• List, in ascending order with no repetition, all
  positive integers with no prime factors other
  than 2, 3, or 5.
• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
  15,...

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n as a prime factor
fun scale n scale n (xxs)
(nx) (scale n xs)
scale 2 1,2,3,4,5 2,4,6,8,10 scale 3
1,2,3,4,5 3,6,9,12,15
scale 3 (scale 2 1,2,3,4,5) 6,12,18,24,30
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Merging two Streams
fun merge merge (xxs)
(yys) if x lt y then x merge xs
(yys) else if x gt y then y merge (xxs)
ys else x merge xs ys
merge 2,4,6 3,6,9 2,3,4,6,9
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Hamming numbers
val hamming 1 merge (scale 2 hamming)
(merge (scale 3 hamming) (scale 5
hamming))
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Outline

• More about Higher-order Function
• Type inference and Polymorphism
• Evaluation Strategies
• Exception Handling

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Exception Handling

• Handle special cases or failure (the
  exceptions)
• occurred during program execution.
• hd
• uncaught exception hd

• Exception can be raised and handled in the
  program.
• exception Nomatch
• exception Nomatch exn

fun member(a,x) if null(x) then raise
Nomatch else if a hd(x) then x else
member(a,tl(x))
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fun member(a,x) if null(x) then raise
Nomatch else if a hd(x) then x else
member(a,tl(x))
member(3,1,2,3,1,2,3) val it 3,1,2,3
int list member(4,) uncaught exception
Nomatch member(5,1,2,3) handle Nomatchgt
val it int list
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Conclusion

• More about Higher-order Function
• Curried vs Uncurried functions
• Full vs Partial Application
• Type inference and Polymorphism
• Basic Type inference rules
• Polymorphic functions
• Evaluation Strategies
• Innermost
• Outermost
• Exception Handling is available in ML