Functional Programming

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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