CSC301CPE425SC433: Programming Languages

1
CSC301/CPE425/SC433 Programming Languages

• Topic 3 Programming Paradigms
• Functional Programming (FP)
  Paradigm
• SubTopic 3.2 Example FP Language SCHEME
• Reference Louden, Ch. 11

2
Topic 3.2 SCHEME

• Scheme
• Basic components
• Objects
• Binding forms lambda, let, define, set
• Data structures lists, vectors
• Functions primitive, user-defined
• b) Comparison of Functional and Imperative
  Languages

3
Functions

• Any function application is written in prefix
  form
• e.g.
• ( 10 20) 1020
• (/ ( ( 10 20) 5) 2) (1020) 5 / 2

(function_name arg1 arg2 arg3 )
f(10, 20)
g( )
h( )
4
Objects

• Scheme (like LISP) is a language for symbolic
  computation (AI applications)
• Values are represented by symbolic expressions
  (or S-expressions)
• An expression is either
• atoms e.g. a, 1, hello world
• lists e.g. (a b c d), (hello world)
• Vector e.g. (a b c d)

5
Binding Forms

• Syntactic forms used to bind or assign
  identifiers
• Lambda
• Let
• Definitions
• Assignments

6
? - Calculus

• Developed by Alonzo Church
• Mathematical theory for anonymous functions
• A fundamental goal
• To describe what can be computed
• LISP is based on ?-calculus

7
? - Function

• A mathematical function may be expressed in
  ?-form
• e.g.
• The lambda form clearly shows which variables are
  bound or free
• e.g.

f(x) x 3 (?x . x3)
f(x) x 3a (?x . x3a)
8
Lambda Expression in Scheme
(lambda (idspec) expr1 expr2 )

• Aka lambda expression
• Analogous to ?-calculus expressions
• idspec (list of the) formal parameters of
  procedure
• The expressions expr1, expr2, are evaluated in
  sequence
• Creates (returns) an anonymous procedure

9
Lambda Expression in Scheme (contd)

• Examples

f(x) x 3
f(x) x 3 x7
f(x) x x
x11
((lambda (f x) (f x x )) 11)
((lambda ( x) ( x x )) 11)
f( ) 3 4
nothing
((lambda ( ) ( 3 4)) )
10
Lambda Expression in Scheme (contd)

• Examples

((lambda (x) ((lambda (y) (- x y)) 15)) 20)
The variable x is free in the body of the inner
lambda expr, but its binding is found in the
local environment for the outer lambda expr
5
Global environment
?x . (?y . x-y 15) 20
X -15
Local environment 1 x
20 -15
int f1 (int x) int f2( int y)
return (x-y) f2(15)
public static void main ( ) Console.Write(out
put is 1, f1(20)
Local environment 2 y
11
Local Binding using Let

• Temporary (local) binding of identifiers to
  values in the body of let
• Order of evaluation of val_, expr_ is at
  discretion of Scheme implementation
• Any free (i.e. unbound) variable appearing in
  val1, val2, is looked up in a non-local
  environment

(let ( (id1 val1) (id2 val2) ) expr1 expr2 )
12
Local Binding using Let (contd)

• Examples
• (let ((a 2) (b 3))
• ( a b))
• (let ((sum ( 2 4)))
• ( sum sum))
• (let ((b 3))
• (let ((b 10) (c b))
• c))

Global environment
Local environment 1-a a, b
Local environment 1-b sum
Local environment 1-c b
Local environment 2 b, c
13
Let and Lambda

• The let construct does not add any new semantic
  facility to Scheme
• Every let expression can be rewritten as a lambda
  expression applied to arguments
• (let ((x 3) (y ( 2 5))) ( x y))

((lambda (x y) ( x y)) 3 ( 2 5))
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Using Let
(let ( (id1 val1) (id2 val2) ) expr1 expr2 )

• Syntax is similar to let
• Behaves as though the successive bindings are in
  nested let expressions
• i.e. equivalent to

(let ((id1 val1)) (let ((id2 val2))
expr1 expr2 ) )
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(No Transcript)
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(let ((a 0) (lower (- a 3)) (upper (
a 4))) )
lower ?? upper ??
(let ((a 0) (let ((lower (- a 3))
(upper ( a 4))) )) )
a 0 lower -3 upper 4
(let ((a 0) (lower (- a 3)) (upper
( a 4))) )
a 0 lower -3 upper 4
17
Definitions

• Binding of an identifier id to a value of expr
• Binding of an identifier id to a procedure

(define id expr)
(define (id idspec) expr1 expr2 )
equivalent to
(define id (lambda (idspec) expr1 expr2 ))
In short
Give a name to a lambda expression
(define id lambda_expression)
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Example Bindings

• e.g.
• (define a 2) top-level definition
• a 2
• (let ((b 3) (c 4)) ( b c)) 7
• b reference to undefined identifier b
• ( a a) 4
• (define (double x) ( 2 x))
• (double 3) 6

19
Example Bindings (contd)

• e.g.
• (define add_b
• (let ((b 100))
• (lambda (x) ( x b))
• )
• )
• (let ((b 10)) (add_b 25)) 125

(define id (lambda (idspec) expr1 expr2 ))
20
Example Bindings (contd)

• e.g.
• (define (func1 x)
• (define (func2 y) ( x y))
• (define (func3 x) (func2 ( 2 x)))
• (func3 x)
• )
• (func1 3) 9

(define id (lambda (idspec) expr1 expr2 ))
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Assignment
(set! id expr)

• Assign a new value to an existing identifier
• Imperative feature
• Before the assignment, the identifier must be
  bound
• example

(let ((x 3) (y 4)) (set! x 5) ( x y))
9
22
Lists and Pairs

• The pair is the most fundamental object type in
  Scheme
• A pair is a record structure with two fields
  called the car and cdr fields
• Lists are built up from pairs
• Lists are ordered sequences of pairs linked to
  the next by the cdr field
• Linked list
• The elements of the list occupy the car field of
  each pair

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Proper and Improper Lists

• Proper list
• The cdr of the last pair is the empty list ( )
• e.g. (a b c)
• Improper list
• The cdr of the last pair can be anything other
  than the empty list ( )
• e.g. (a . b)

( )
Z
first pair
24
Vectors

• Elements in a vector are accessed directly by
  their positions in the vector
• May have heterogeneous data
• Elements have indices from 0 to (length of
  vector) 1
• A prefix n specifies the length of the vector
• Examples
• (a b c)
• (a a a) and 3(a a a) and 3(a) are equivalent
• (define v '(1 (a b) hello))
• (vector-ref v 1) (a b)
• (vector-length v) 3

25
Data and Functions

• How do we represent data?
• e.g.
• ( 2 3) 5
• S-expression represents function application
• To prevent function evaluation, use quote
  function
• e.g.
• (quote ( 2 3)) or '( 2 3) ( 2 3)

26
Data Type
A strongly-typed language is one in which each
variable has a single type associated with it,
and that type is known at compile time

• Scheme is weakly typed
• Aka dynamic typing
• Uses run-time type checking, i.e. a variable
  assumes the type of the value bound to it during
  run-time
• (define (square x) ( x x))
• (square 3) 9
• (square 3.3) 10.8899999999999
• (square hello)
• expects type ltnumbergt as 1st argument, given
  hello other arguments were hello

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Functions

• Function application
• S-expression is used to express function
  applications
• e.g.
• ( 6 9) 15
• (Anonymous) function definition is based on
  lambda expression

(function_name param )
(lambda (idspec) expr1 expr2 )
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Functions (contd)

• Building a name to a function
• Use defined
• or
• example
• (define (add x y) ( x y))
• or
• (define add (lambda (x y) ( x y)))

(define (id idspec) ltfunction_bodygt)
(define id (lambda_expression))
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Built-in Functions List Operations

• Use car to return the first element of the list
• Use cdr to return the remaining of the list with
  the first element removed
• examples
• (car '(a b c)) a
• (cdr '(a b c)) (b c)
• (car '((a b) c d)) (a b)
• (cdr '(a)) nil
• (car (cdr '(a b c))) b
• (car '(cdr (a b c))) cdr

30
Built-in Functions List Operations (contd)

• Abbreviate composition of car and cdr
  applications
• Example
• (cdr (cdr (car '((a b c) (d e)))))
• can be abbreviated as
• (cddar '((a b c) (d e))) (c)

31
Built-in Functions List Operations (contd)

• cons creates a new pair
• Construct a list using cons
• Example

Z
(define Z (cons 'a (cons 'b (cons 'c '() )))) (a
b c)
32
Built-in Functions List Operations (contd)
(cons 'a '(b c)) (a b c)
(cons '(a) '(b c)) ((a) b c)
33
Cell Model Representation
(a (b (c) d))
34
Built-in Functions List Operations (contd)

• Creating lists using list
• Return a list of objects

(list 'a 'b 'c) (cons 'a (cons 'b (cons 'c '()
)))
Examples (list) ( ) (list 1 2 3) (1 2
3) (list 1 '(2 3)) (1 (2 3)) (cons 1 (cons
'(2 3) '() )) (list 1 '(2 3) '() ) (1 (2 3)
()) (cons 1 (cons '(2 3) (cons '() '() )))
35
Built-in Functions Vector Operations
Examples (vector) 0() (vector 1 2
3) 3(1 2 3) (vector 1 'a 3) 3(1 a
3) (make-vector 5) 5() (make-vector 5
\space) 5(\space) (vector? (1 2
3)) t (vector-length (1 2 3)) 3 (vector-ref
(10 20 30) 2) 30 (vector-fill! (10 20 30)
222) 3(222)
36
Built-in Functions Predicates

• A predicate is a function whose result is always
  t or f

Examples (even? 4) t (integer?
4) t (number? 14/3) t (null? '(1 2
3)) f (real? 4.3) t (string?
hello) t (eq? a 3) f use for
atoms (equal? '(1 3 4) '(1 3 4)) t use for
lists ( 3 4) f use for numbers
37
Built-in Functions Numbers

• A number in Scheme can be exact or inexact
• Depends on the operations used to derive the
  number and the inputs to these operations
• An exact integer is normally written as a
  sequence of numerals preceded by an optional sign
• An exact rational number is normally written as
  two sequences of numerals separated by a slash
  and preceded by an optional sign
• Inexact integers and rational numbers are
  normally written in either floating-point or
  scientific notation

38
Built-in Functions Numbers (contd)
Examples (quotient 30 4) 7 (remainder 30
4) 2 (truncate 30/4) 7 (round
30/4) 8 (floor 30/4) 7 (ceiling
30/4) 8 (abs -30/4) 30/4 (numerator
30/4) 15 (denominator 30/4) 2 (max 4 -2 1 30
21) 30
39
Built-in Functions Sequencing

• Begin
• Each expr is evaluated from left to right, in
  sequence
• Used to package a sequence of expressions into a
  single expression

(begin ltexprgt )
(define (begin_ex) (define number 0)
(newline) (display Enter number) (set!
number (read)) (if (zero? number) (display
You enter 0.) (begin (display Number
is) (display number) ) ) )
(if ltpredicategt ltexpr_if_truegt
ltexpr_if_falsegt )
40
Built-in Functions Conditionals

• Test function using if
• Test function using cond

(if ltpredicategt ltexpr_if_truegt
ltexpr_if_falsegt )
(let ((x 0)) (cond ((lt x 0) (list
'minus (abs x))) ((gt x 0) (list 'plus
x)) (else (list 'zero x)) ) )
(cond (lttest-1gt ltexpr-1gt) (lttest-2gt
ltexpr-2gt) (lttest-ngt ltexpr-ngt) (else
ltexpr-pgt) )
41
Built-in Functions Conditionals (contd)

• Test function using case

(case val (ltkey-1gt ltexpr-1gt) (ltkey-2gt
ltexpr-2gt) (else ltexpr-pgt) )
(let ((x 4) (y 5)) (case ( x y)
((1 3 5 7 9) 'odd) ((0 2 4 6 8) 'even)
(else 'out-of-range) ) )
42
Built-in Functions Conditionals (contd)
(and expr )

• Evaluates each arg in sequence from left to right
  and stops if any expr evaluates to false
• Example
• (let ((x 3))
• (and (gt x 2) (lt x 4) ( x 5))) f

(or expr )

• Evaluates each arg in sequence from left to right
  and stops if any expr evaluates to true
• Example
• (let ((x 3))
• (or (gt x 2) (lt x 4) ( x 5))) t

43
Built-in Functions Controls

• Iteration using do

(do ((id val update) ) (test res ) expr )
(define factorial (lambda (x) (do
( ( i x (- i 1)) initial value of i is
x, decrement i (a 1 ( a i)) initial
value of a is 1, a a i ) ((zero? i)
a) termination test if i 0, return a
) ) ) (factorial 3)
44
Built-in Functions Controls (contd)
(map proc list1 list2)

• Applies proc to corresponding elements of the
  lists list1, list2,
• Each list1, list2, must have the same length
• Returns a list

Examples (map abs '(1 -2 3 -4)) (1 2 3
4) (map (lambda (x y) ( x y)) '(1 2) '(4 5)) (4
10)
45
Built-in Functions Controls (contd)
(for-each proc list1 list2)

• Applies proc to corresponding elements of the
  lists list1, list2,
• Elements are processed in left-to-right sequence
• Does not return a list
• Mainly used for side-effects (e.g. display)

(let ((same-count 0)) (for-each (lambda (x
y) (cond (( x y) (set! same-count (
same-count 1))) ) ) '(1 2 3 4 5) '(2 4 5 7
0) ) same-count or (display same-count) )
46
Recursive Functions

• Example to count number of elements in formal
  parameter

(define (mylength list) (if (null? list)
0 ( 1 (mylength (cdr list)))
) )
mylength '(a (b c) d)
Function call (mylength '(a (b c) d)
47
Recursive Functions (contd)

• Example factorial

(define (factorial x) (cond ((lt x 0) ())
(( x 0) 1) (else ( x
(factorial (- x 1)))) ) )
factorial 3
Function call (factorial 3)
48
Tail Recursion

• A call is tail-recursive
• If nothing has to be done after the call returns
• When the call returns, the returned value is
  immediately returned from the calling function
• A call is not tail-recursive
• If some operation is performed on the result of
  the recursive call
• Scheme guarantees to optimize tail recursion into
  loops
• So tail recursion does not build up any sort of
  control stack that might be associated with
  normal (not tail) recursion (which gives poor
  performance)

49
Tail Recursion (contd)

• Example factorial (tail-recursive version)

(define (factorial n) (define (fact result
n) (if (zero? n) result (fact ( result
n) (- n 1)) ) ) (fact 1 n) )
factorial 3
helper function
Function call (factorial 3)
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Tail Recursion (contd)
factorial 3
factorial 3
Non tail-recursive
6
3 factorial 2
2
2 factorial 1
1
1 factorial 0
Tail-recursive
51
Topic 3.2 SCHEME

• Scheme
• Basic components
• Objects
• Binding forms lambda, let, define, set!
• Data structures lists, vectors
• Functions primitive, user-defined
• b) Comparison of Functional and Imperative
  Languages

52
Comparison of Functional and Imperative Languages

• Imperative languages
• Based on conventional computer
• More efficient in terms of execution time because
  they reflect structure and operations of machines
• Programmer must pay attention to machine level
  details
• Presence of side-effects

53
Comparison of Functional and Imperative
Languages (contd)

• Functional languages
• Based on mathematical functions
• Allows (problem-oriented) design of data
  structures without concern for memory cells
• No mutation (or side effects)
• E.g. an object passed as an arg to a proc, a new
  object is created but the original object remains
  unchanged
• Appears to be at a higher level than imperative
  languages
• Inefficient to execute due to numerous function
  calls and dynamic object creation and
  deallocation
• Natural way of exploiting parallel machine
  architecture

54
Summary

• Scheme departs from functional model established
  by FP in two ways
• Use of variables concept with let, set!
• Ability to specify sequential execution of
  functions
• S-expression used for expressing function
  applications in Scheme
• Cell model used by Scheme to represent
  S-expressions

55
Summary (contd)

• Scheme constructs
• cons
• predicates
• car
• cdr
• let
• define
• lambda

56
Lambda Expression in Scheme (contd)

• Examples

(let ((a 2) (b 3)) ( a b))
int f1 (int a, int b) return (ab)
((lambda(a b) ( a b)) 2 3)
5
Global environment
public static void main ( ) Console.Write(out
put is 1, f1(2, 3)
Local environment 1 a 2 b 3
57
Lambda Expression in Scheme (contd)

• Examples

(let ((sum ( 2 4))) ( sum sum))
int f1 (int sum) return (sumsum)
36
((lambda(sum) ( 2 4)) ( sum sum) )
Global environment
public static void main ( ) Console.Write(out
put is 1, f1(24)
Local environment 1 sum 6
58
Lambda Expression in Scheme (contd)

• Examples

(let ((b 3)) (let ((b 10) (c b)) c))
public static void main ( ) Console.Write(out
put is 1, f1(3)
(let ((b 3)) (let ((z 10) (c b)) c))
Global environment
3
Local environment 1 b 3
int f1 (int b) int f2(int b, int c)
return (c) return f2(10, b)
z,
Local environment 2 z (renamed b), c note that
here z (renamed b) 10 c3
(let ((b 3)) (let ( (c b) (b 10) ) c))
59
Lambda Expression in Scheme (contd)

• Examples

(let ((a 0) (lower (- a 3)) (upper (
a 4))) )
int f1 (int a, int lower, int upper) return
()
Undefined lower , upper ..
((lambda (a lower upper)) 0 (- a 3) ( a 4))
Global environment a .. should have somevalue
here ..
Local environment 1 a0 lower upper
public static void main ( ) Console.Write(out
put is 1, f1(0, a-3, a4)
60
Lambda Expression in Scheme (contd)

• Examples

public static void main ( ) Console.Write(out
put is 1, f1(0)
(let ((a 0) (let ((lower (- a 3))
(upper ( a 4))) .. .. )) )
ok..
Global environment
Local environment 1 a 0
int f1 (int a) int f2(int lower, int
upper) .. .. return () return(
f2(a-3, a4) )
Local environment 2 lower -3 upper 4
61
Lambda Expression in Scheme (contd)

• Examples

public static void main ( ) Console.Write(out
put is 1, f1(0)
(let ((a 0)) (let ((lower (- a 3))) (let
(upper ( a 4))) ) )
(let ((a 0) (lower (- a 3)) (upper
( a 4))) )
Global environment
ok..
int f1 (int a) int f2(int lower)
int f3(int upper) return
() f3(a4)
f2(a-3)
Local environment 1 a 0
Local environment 2 lower -3
Local environment 3 upper 4
62
Lambda Expression in Scheme (contd)

• Examples

(let ((a 0) (lower (- a 3)) (upper
( a 4))) )
int f1 (int a, int lower, int upper) return
()
??? Undefined a ..
Global environment a .. should have somevalue
here ..
Local environment 1 a lower upper
public static void main ( ) Console.Write(out
put is 1, f1(0, a-3, a4)
((lambda (a lower upper)) 0 (- a 3) ( a 4))
63
Local Binding using Let (contd)

• Examples
• (let ((a 2) (b 3))
• ( a b))
• (let ((sum ( 2 4)))
• ( sum sum))
• (let ((b 3))
• (let ((b 10) (c b))
• c))

Global environment
Local environment 1-a a, b
Local environment 1-b sum
Local environment 1-c b
Local environment 2 b, c
64
Lambda Expression in Scheme (contd)

• Examples

public static void main ( ) Console.Write(out
put is 1, f1(0)
(let ((a 0) (let ((lower (- a 3))
(upper ( a 4))) .. .. )) )
ok..
Global environment
((lambda (a (lambda (lower upper) ) (- a 3) ( a
4) ) 0)
Local environment 1 a 0
int f1 (int a) int f2(int lower, int
upper) .. .. return () return(
f2(a-3, a4) )
Local environment 2 lower -3 upper 4