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Programming Languages

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Title: Programming Languages

1
Programming Languages

COMP 321
Spring 2012
Practicum 4

2
Practicum 4

Functional Forms
Functional abstraction
Function Composition
Construction
Mapping
Dynamic code

3
reference

reference
Teach Yourself Scheme in Fixnum Days, section 6.4.

4
Fundamentals of Functional Programming
Languages

FPL objective to mimic mathematical functions
to the greatest extent possible
basic process of computation in a FPL vs an
imperative language
In an imperative language, operations are done
and the results are stored in variables for later
use
Management of variables is a source of complexity
for imperative programming
In an FPL variables are used as unknowns (as is
the case in mathematics)

5
Fundamentals of Functional Programming Languages

In an FPL, the evaluation of a function always
produces the same result given the same
parameters
This is called referential transparency
Is this true in an imperative language?

6
Abstraction

Programming is all about abstraction
Variables, Data structures abstract on memory
control structures abstract on the processor
How?

7
Abstraction

Programming is all about abstraction
OOP abstracts on data (ADTs)
Encapsulation
Polymorphism
OOP provides higher level abstractions
Inheritance
How is this abstraction?

8
OOP Abstraction
Class A Public f() Private x, y

inheritance

Class B Public g() Private a,b
Class C Public h() Private i,j
These classes build on class A dont know how it
is implemented.
9
Abstraction

Programming is all about abstraction
Functional programming languages (FPL) abstract
on functions

(define A (lambda (b) )
Receive new function B that performs some task.
Call function A and pass function f (A f)
10
Mathematical Functions

Functional Forms
Def A higher-order function, or functional form,
is one that either takes functions as parameters
or yields a function as its result, or both

11
Mathematical Functions

Functional Forms
Functional abstraction
Function Composition
Construction
Mapping
Dynamic code

12
Functional Forms

0. Functional abstraction
Passing functions as arguments
Returning functions as results
Powerful technique that can be used in
conjunction with other techniques

13
Functional Forms

0. Functional abstraction
In an OOP language (like Java) we abstract over
classes
Example Javas abstract classes
An abstract class cannot be implemented
It represents the structure of the class

14
Functional Forms

0. Functional abstraction
(define doubler
(lambda (f)
(lambda (x) (f x x))))
(define double (doubler ))
(double 13/2) 13
(double 5) 10

15
Functional Forms

0. Functional abstraction
(define doubler
(lambda (f)
(lambda (x) (f x x))))
(define double-cons (doubler cons))
(double-cons a) (a . a)

16
Functional Forms

0. How does doubler work?
((doubler ) 2)
1. (define doubler
(lambda (f)
(lambda (x) (f x x))))

f
1. create closure
(lambda (x) (f x x))))
2. doubler returns the closure (which contains
the lambda)
3. returned function is executed in the context
of the closure
2. ((doubler ) 2)
x 2
(lambda (x) (f x x))))
17
Functional Forms

0. Functional abstraction
(define doubler
(lambda (f)
(lambda (x) (f x x))))
(define double-any
(lambda (f x)
((doubler f) x)))

18
Functional Forms

0. How does double-any work?
(define doubler
(lambda (f)
(lambda (x) (f x x))))
(define double-any
(lambda (f x)
((doubler f) x)))

Doubler returns a function that takes one
parameter. The function is defined within the
scope of doubler where f is bound.
19
Functional Forms

0. How does double-any work?
(double-any 2)
1. (define double-any
(lambda (f x)
((doubler f) x)))

f x 2
((doubler f) x)))
4. Call the returned function
3
2
g
2. (define doubler (lambda (g) (lambda (x)
(g x x))))
(lambda (x) (g x x))))
20
Functional Forms

0. Functional abstraction
Exercise create a function that takes a
one-argument function f and a list, applies f to
the first item in the list, and returns the
result.

21
Functional Forms

Solution
(define change-first
(lambda (f)
(lambda (x)
(f (car x)))))
gt (add1_first '(2 3 4))
3
gt

(define incr (lambda (x) ( 1
x))) (define add1_first (lambda (x)
((change-first incr) x)))
22
Functional Forms

1. Function Composition
A functional form that takes two functions as
parameters and yields a function whose value is
the first actual parameter function applied to
the application of the second
Form h ? f g
which means h (x) ? f ( g ( x))
For f (x) ? x x x and g (x) ? x
3,
h ? f g yields (x 3) (x 3) (x
3)

23
Functional Forms

1. Function Composition
Most scheme functions use composition.
Example the list-copy function applies cons to
list-copy and car
(define list-copy
(lambda (ls)
(if (null? ls)
( )
(cons (car ls)
(list-copy (cdr ls)))
)
)
)

24
Functional Forms

1. Function Composition
Can abstract composition also
(define compose
(lambda (f g)
(lambda (x)
(f (g x)))
))
gt (define safeSqr (compose sqrt abs))
gt (safeSqr -100)
10
gt

25
Functional Forms

2. Construction
A functional form that takes a list of functions
as parameters and yields a list of the results of
applying each of its parameter functions to a
given parameter
Form f, g
For f (x) ? x x x and g (x) ? x 3,
f, g (4) yields (64, 7)

26
construction

construct applies two functions to an argument
and puts the result in a list.
(define construct
(lambda (x)
(cons (sqrt x) (cons (abs x) ()))
)
)

27
construction

Can abstract construction
(define construct2
(lambda (f g)
(lambda (x)
(cons (f x) (cons (g x) ()))
)
))
gt (define test (construct2 sqrt abs))
gt (test 100)
(10 100)
gt

28
Functional Forms

3. Apply-to-all or mapping
A functional form that takes a single function as
a parameter and yields a list of values obtained
by applying the given function to each element of
a list of parameters
Form ?
For h (x) ? x x x
?( h, (3, 2, 4)) yields (27, 8, 64)

29
mapping

mapping applies a function over all elements of a
list
abs-all maps abs over the input list to produce
the output list
(define abs-all
(lambda (ls)
(if (null? ls)
( )
(cons (abs (car ls))
(abs-all (cdr ls)))
)
)
)

30
mapping

Scheme supplies a built-in map function
(define abs-all2
(lambda (ls)
(map abs ls)
)
)

(map abs (1 -2 3 -4 5 -6)) (map (lambda (x) ( x
x)) (1 -3 -5 7)) (map cons (a b c) (1 2 3))
31
mapping

Can abstract mapping, but need a letrec
(define mapper
(lambda (f)
(lambda (ls)
(if (null? Ls)
()
(cons (f (car ls))
(?? (cdr ls)))
)
)
)

How do we call ourselves?
32
mapping

Can abstract mapping, but need a new construct (a
letrec)
(define mapper
(lambda (f)
(lambda (ls)
(letrec ( (mapRec
(lambda (lst)
(if (null? lst)
()
(cons (f (car ls))
(mapRec (cdr lst)))
) end if
) end
lambda
) end mapRec
) end definitions
(mapRec ls)
) end letrec
)))

33
Functional Forms

4. Dynamic Code
It is possible in Scheme to define a function
that builds Scheme code and requests its
interpretation
This is possible because the interpreter is a
user-available function, EVAL

34
Functional Forms