BAB XII
Functional Programming Languages
The design of the imperative languages is based directly on
the von Neumann architecture
The design of the functional languages is based on
mathematical functions
Mathematical functions
· A
mathematical function is a mapping of members of one set, called the domain
set, to another set, called the range set
· A lambda
expression specifies the parameter(s) and the mapping of a function in the
following form
l(x) x * x * x
for the
function cube(x) = x * x * x
· Lambda
expressions
o Lambda expressions
describe nameless functions
o Lambda expressions
are applied to parameter(s) by placing the parameter(s) after the expression
e.g., (l(x) x * x * x)(2)
which
evaluates to 8
· Functional
forms
o A higher-order
function, or functional form, is one that either takes functions as parameters
or yields a function as its result, or both
o 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 º f ° g
which means h (x) º f ( g ( x))
For f (x) º x + 2 and g (x) º 3 * x,
h º f ° g yields (3 * x)+ 2
o Apply-to-all
§ 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: a
For h(x) º x * x
a(h, (2, 3, 4)) yields (4, 9, 16)
LISP Functional programming
· LISP Data
Types and Structures
o Data object types:
originally only atoms and lists
o List form: parenthesized
collections of sublists and/or atoms
e.g.,
(A B (C D) E)
o Originally, LISP
was a typeless language
o LISP lists are
stored internally as single-linked lists
· LISP
interpretation
o Lambda notation is
used to specify functions and function definitions. Function applications and
data have the same form.
e.g.,
If the list (A B C) is interpreted as data it is
a
simple list of three atoms, A, B, and C
If it is interpreted as a
function application,
it
means that the function named A is
applied to the two parameters, B and C
o The first LISP
interpreter appeared only as a
demonstration of the universality of the computational capabilities of
the notation
· Primitive
Function & LAMBDA Expressions
o Primitive
Arithmetic Functions: +, -, *, /, ABS, SQRT, REMAINDER, MIN, MAX
e.g.,
(+ 5 2) yields 7
o Lambda Expressions
§ Form is based on l
notation
e.g., (LAMBDA (x)
(* x x)
x is called a
bound variable
o Lambda expressions
can be applied to parameters
e.g.,
((LAMBDA (x) (* x x)) 7)
o LAMBDA expressions
can have any number of parameters
(LAMBDA (a b x) (+
(* a x x) (* b x)))
· Special form
function: DEFINE
o DEFINE - Two
forms:
1. To bind a symbol to an expression
e.g.,
(DEFINE pi 3.141593)
Example use: (DEFINE two_pi (* 2 pi))
These symbols are not variables – they are like the names
bound by Java’s final declarations
2. To bind names to lambda expressions (LAMBDA is implicit)
e.g.,
(DEFINE (square x) (* x x))
Example use: (square 5)
- The evaluation process for DEFINE is different! The first
parameter is never evaluated. The second parameter is evaluated and bound to
the first parameter.
· Output
functions
o Usually not
needed, because the interpreter always displays the result of a function
evaluated at the top level (not nested)
o Scheme has PRINTF,
which is similar to the printf function of C
o Note: explicit
input and output are not part of the pure functional programming model, because
input operations change the state of the program and output operations are side
effects
· Numeric
predicate functions
o #T (or #t) is true
and #F (or #f) is false (sometimes () is used for false)
o =, <>, >,
<, >=, <=
o EVEN?, ODD?,
ZERO?, NEGATIVE?
o The NOT function
inverts the logic of a Boolean expression
· Control flow
o Selection- the
special form, IF
(IF
predicate then_exp else_exp)
(IF (<>
count 0)
(/ sum
count)
)
o Recall from
Chapter 8 the COND function:
(DEFINE (leap? year)
(COND
((ZERO? (MODULO
year 400)) #T)
((ZERO? (MODULO
year 100)) #F)
(ELSE (ZERO?
(MODULO year 4)))
))
· List
functions
o QUOTE - takes one
parameter; returns the parameter without evaluation
§ QUOTE is required
because the Scheme interpreter, named EVAL, always evaluates parameters to
function applications before applying the function. QUOTE is used to avoid parameter evaluation
when it is not appropriate
§ QUOTE can be
abbreviated with the apostrophe prefix operator
o '(A B) is
equivalent to (QUOTE (A B))
o Examples:
(CAR ′((A B) C D))
returns (A B)
(CAR ′A) is an error
(CDR ′((A B) C D))
returns (C D)
(CDR ′A) is an error
(CDR ′(A)) returns
()
(CONS ′() ′(A B))
returns (() A B)
(CONS ′(A B) ′(C D))
returns ((A B) C D)
(CONS ′A ′B) returns
(A . B) (a dotted pair)
o LIST is a function
for building a list from any number of parameters
(LIST ′apple
′orange ′grape) returns
(apple orange
grape)
· Predicate
function: EQ?
o EQ? takes two
expressions as parameters (usually two atoms); it returns #T if both parameters
have the same pointer value; otherwise #F
(EQ?
'A 'A) yields #T
(EQ? 'A 'B) yields
#F
(EQ?
'A '(A B)) yields #F
(EQ? '(A B) '(A B))
yields #T or #F
(EQ? 3.4 (+ 3 0.4)))
yields #T or #F
· Predicate
function: EQV?
o EQV? is like EQ?,
except that it works for both symbolic and numeric atoms; it is a value
comparison, not a pointer comparison
(EQV? 3 3) yields
#T
(EQV? 'A 3) yields
#F
(EQV 3.4 (+ 3 0.4))
yields #T
(EQV? 3.0 3) yields
#F (floats and integers are different)
· Predicate
functions: LIST? and NULL?
o LIST? takes one
parameter; it returns #T if the parameter is a list; otherwise #F
(LIST? '()) yields
#T
o NULL? takes one
parameter; it returns #T if the parameter is the empty list; otherwise #F
(NULL? '(())) yields
#F
Scheme functional programming
· Example
Scheme function: member
o member takes an
atom and a simple list; returns #T if the atom is in the list; #F otherwise
DEFINE
(member atm a_list)
(COND
((NULL? a_list)
#F)
((EQ? atm (CAR
lis)) #T)
((ELSE (member
atm (CDR a_list)))
))
· Example
Scheme function: equalsimp
o equalsimp takes
two simple lists as parameters; returns #T if the two simple lists are equal;
#F otherwise
(DEFINE (equalsimp list1 list2)
(COND
((NULL? list1) (NULL? list2))
((NULL? list2)
#F)
((EQ? (CAR
list1) (CAR list2))
(equalsimp(CDR list1)(CDR list2)))
(ELSE #F)
))
· Example
Scheme function: equal
o equal takes two
general lists as parameters; returns #T
if the two lists are equal; #F otherwise
(DEFINE (equal list1 list2)
(COND
((NOT (LIST?
list1))(EQ? list1 list2))
((NOT (LIST?
lis2)) #F)
((NULL? list1)
(NULL? list2))
((NULL? list2)
#F)
((equal (CAR list1) (CAR
list2))
(equal (CDR list1) (CDR list2)))
(ELSE #F)
))
· Example
Scheme function: append
o append takes two
lists as parameters; returns the first parameter list with the elements of the
second parameter list appended at the end
(DEFINE (append list1 list2)
(COND
((NULL? list1)
list2)
(ELSE (CONS
(CAR list1)
(append (CDR list1) list2)))
))
· Example
Scheme function: LET
o LET is actually
shorthand for a LAMBDA expression applied to a parameter
(LET ((alpha 7))(*
5 alpha))
is the same as:
((LAMBDA (alpha) (*
5 alpha)) 7)
o Example
(DEFINE (quadratic_roots a b c)
(LET (
(root_part_over_2a
(/ (SQRT (- (*
b b) (* 4 a c)))(* 2 a)))
(minus_b_over_2a (/ (- 0 b) (* 2 a)))
(LIST
(+ minus_b_over_2a root_part_over_2a))
(-
minus_b_over_2a root_part_over_2a))
))
· Tail
recursion in Scheme
o Definition: A
function is tail recursive if its recursive call is the last operation in the
function
o A tail recursive
function can be automatically converted by a compiler to use iteration, making
it faster
o Scheme language
definition requires that Scheme language systems convert all tail recursive
functions to use iteration
o Example of
rewriting a function to make it tail recursive, using helper a function
Original: (DEFINE (factorial n)
(IF (<= n 0)
1
(* n (factorial (- n 1)))
))
Tail
recursive: (DEFINE (facthelper n
factpartial)
(IF (<= n 0)
factpartial
facthelper((- n 1) (* n factpartial)))
))
(DEFINE (factorial n)
(facthelper n 1))
· Functional
form – Composition
o Composition
§ If h is the
composition of f and g, h(x) = f(g(x))
(DEFINE (g x) (* 3
x))
(DEFINE (f x) (+ 2
x))
(DEFINE h x) (+ 2 (*
3 x))) (The composition)
§ In Scheme, the
functional composition function compose can be written:
(DEFINE (compose f
g) (LAMBDA (x) (f (g x))))
((compose CAR CDR)
'((a b) c d)) yields c
(DEFINE (third
a_list)
((compose CAR
(compose CDR CDR)) a_list))
is equivalent to
CADDR
o Apply-to-all
§ Apply to All - one
form in Scheme is map
· Applies the
given function to all elements of the given list;
(DEFINE
(map fun a_list)
(COND
((NULL? a_list) '())
(ELSE (CONS (fun (CAR a_list))
(map fun (CDR a_list))))
))
(map (LAMBDA (num) (* num num num)) '(3 4 2 6)) yields (27
64 8 216)
· Function
that build code
o It is possible in
Scheme to define a function that builds Scheme code and requests its
interpretation
o This is possible
because the interpreter is a user-available function, EVAL
· Adding a
list of numbers
((DEFINE (adder a_list)
(COND
((NULL? a_list) 0)
(ELSE (EVAL (CONS '+ a_list)))
))
o The parameter is
a list of numbers to be added; adder inserts a + operator and evaluates the
resulting list
§ Use CONS to insert
the atom + into the list of numbers.
§ Be sure that + is
quoted to prevent evaluation
§ Submit the new list
to EVAL for evaluation
Comparing functional and imperative languages
· Imperative
Languages:
o Efficient
execution
o Complex semantics
o Complex syntax
o Concurrency is
programmer designed
· Functional
Languages:
o Simple semantics
o Simple syntax
o Less efficient execution
o Programs can
automatically be made concurrent
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