15.1 Introduction

• The design of the imperative languages is based directly on the von Neumann architecture
• Efficiency is the primary concern, rather than the suitability of the language for software development

• The design of the functional languages is based on mathematical functions
• A solid theoretical basis that is also closer to  the user, but relatively unconcerned with the  architecture of the machines on which  programs will run

• Def: 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

       λ (x) x * x * x

   for the function  cube (x) = x * x * x

Lambda expressions describe nameless  functions

Lambda expressions are applied to parameter(s) by placing the parameter(s) after the expression

     e.g.   (λ(x) x * x * x)(3)     which evaluates to 27
 

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

1.  Function Composition

A functional form that takes two functions as  parameters and yields a function whose result  is a function whose value is the first actual parameter function applied to the result of the application of the second

       Form:  h =  f ° g
        which means h (x) =    f ( g ( x))

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

 3. 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: λ
      For h (x) =  x * x * x
      λ( h, (3, 2, 4))  yields  (27, 8, 64)

15.3 Fundamentals of Functional Programming Languages
 

• The objective of the design of a FPL is to mimic mathematical functions to the greatest extent possible
• The basic process of computation is   fundamentally different in a FPL than in 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 constant concern and source of complexity for   imperative programming
• In an FPL, variables are not necessary, as is the   case in mathematics
• In an FPL, the evaluation of a function always produces the same result given the same   parameters.
• This is called referential transparency

Data object types: originally only atoms and lists
List form: parenthesized collections of sublists and/or atoms
      e.g.,    (A B (C D) E)
Originally, LISP was a typeless language

LISP lists are stored internally as single-linked lists

• Lambda notation is used to specify functions   and function definitions, function applications, and data all 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 parmeters, B and C
•  The first LISP interpreter appeared only as a  demonstration of the universality of the  computational capabilities of the notation

• A mid-1970s dialect of LISP, designed to be a cleaner, more modern, and simpler version than the contemporary dialects of LISP
• Uses only static scoping
• Functions are first-class entities
• They can be the values of expressions and elements of lists
• They can be assigned to variables and passed as parameters

• Arithmetic: +, -, *, /, ABS, SQRT, REMAINDER,MIN, MAX     e.g., (+ 5 2) yields 7
• 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
•  e.g., '(A B) is equivalent to (QUOTE (A B))
• CAR  takes a list parameter; returns the first  element of that list

 e.g., (CAR '(A B C)) yields A   (CAR '((A B) C D)) yields (A B)
 
• CDR  takes a list parameter; returns the list  after removing its first element

 e.g., (CDR '(A B C)) yields (B C)           (CDR '((A B) C D)) yields (C D)
 
• CONS  takes two parameters, the first of which  can be either an atom or a list and the second of which is a list; returns a new list that   includes the first parameter as its first element and the second parameter as the remainder of its result
• e.g., (CONS 'A '(B C)) returns (A B C)

 
• LIST - takes any number of parameters; returns  a list with the parameters as elements

Form is based on  λ  notation

e.g.,   (LAMBDA (L) (CAR (CAR L)))

L is called a bound variable
Lambda expressions can be applied
   e.g.,  ((LAMBDA (L) (CAR (CAR L))) '((A B) C D))
 

e.g.,    (DEFINE pi 3.141593)
(DEFINE two_pi (* 2 pi))

e.g.,   (DEFINE (cube x) (* x x x))
Example use:  (cube 4)
Evaluation process (for normal functions):
1. Parameters are evaluated, in no particularorder
2. The values of the parameters are  substituted into the function body
3. The function body is evaluated
4. The value of the last expression in the body is the value of the function
        (Special forms use a different evaluation process)

Examples:

(DEFINE (square x) (* x x))
(DEFINE (hypotenuse side1 side1) (SQRT (+ (square side1) (square side2))))

1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same
  e.g., (EQ? 'A 'A) yields #T
           (EQ? 'A '(A B)) yields ()

 Note that if EQ? is called with list parameters, the result is not reliable
 Also, EQ? does not work for numeric atoms

2. LIST? takes one parameter; it returns #T if the  parameter is an list; otherwise ()
3. NULL? takes one parameter; it returns #T if the parameter is the empty list; otherwise ()

 Note that NULL? returns #T if the parameter is ()

4. Numeric Predicate Functions
      =, <>, >, <, >=, <=, EVEN?, ODD?, ZERO?,  NEGATIVE?

(DISPLAY expression)
(NEWLINE)

(IF predicate then_exp else_exp)
e.g.,  (IF (<> count 0) (/ sum count)  0)

General form:
    (COND
       (predicate_1  expr {expr})
       (predicate_1  expr {expr})

       (predicate_1  expr {expr})
(ELSE expr {expr}))

Returns the value of the last expr in the first  pair whose predicate evaluates to true

Example of COND:
  (DEFINE (compare x y)
   (COND
     ((> x y) (DISPLAY “x is greater than y”))
     ((< x y) (DISPLAY “y is greater than x”))
     (ELSE (DISPLAY “x and y are equal”))))
 

Example Scheme Functions:

 1. member - takes an atom and a list; returns #T if  the atom is in the list; () otherwise
       (DEFINE (member atm lis)
         (COND
           ((NULL? lis) '())
           ((EQ? atm (CAR lis)) #T)
           ((ELSE (member atm (CDR lis))) ))
 

2. equalsimp - takes two simple lists as parameters; returns #T if the two simple lists are equal; ()  otherwise
  (DEFINE (equalsimp lis1 lis2)
    (COND
      ((NULL? lis1) (NULL? lis2))
      ((NULL? lis2) '())
      ((EQ? (CAR lis1) (CAR lis2))
           (equalsimp (CDR lis1) (CDR lis2)))
      (ELSE '())))

3. equal - takes two lists as parameters; returns  #T if the two general lists are equal;  () otherwise
     (DEFINE (equal lis1 lis2)
       (COND
         ((NOT (LIST? lis1)) (EQ? lis1 lis2))
         ((NOT (LIST? lis2)) '())
         ((NULL? lis1) (NULL? lis2))
         ((NULL? lis2) '())
         ((equal (CAR lis1) (CAR lis2))
              (equal (CDR lis1) (CDR lis2)))
         (ELSE '()) ))

4. 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 lis1 lis2)
     (COND
       ((NULL? lis1) lis2)
       (ELSE (CONS (CAR lis1)
             (append (CDR lis1) lis2)))))

The LET function

General form:
(LET (
 (name_1 expression_1)
 (name_2 expression_2)
 ...
 (name_n expression_n))
 body
)

Semantics: Evaluate all expressions, then bind the values to the names; evaluate the body

(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)))

  (DISPLAY (+ minus_b_over_2a
              root_part_over_2a))
  (NEWLINE)
  (DISPLAY (- minus_b_over_2a
              root_part_over_2a))))

Functional Forms

1. Composition
     The previous examples have used it

 2. Apply to All - one form in Scheme is mapcar
      Applies the given function to all elements of  the given list; result is a list of the results
 
  (DEFINE (mapcar fun lis)
    (COND
      ((NULL? lis) '())
      (ELSE (CONS (fun (CAR lis))
                     (mapcar fun (CDR lis))))))

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

      e.g., suppose we have a list of numbers that must be added together

        ((DEFINE (adder lis)
         (COND
           ((NULL? lis) 0)
           (ELSE (EVAL (CONS '+ lis)))
       ))

The parameter is a list of numbers to be added; adder inserts a + operator and interprets the resulting list.

Scheme includes some imperative  features:
   1. SET! binds or rebinds a value to a name
   2. SET-CAR! replaces the car of a list
   3. SET-CDR! replaces the cdr part of a list

15.6 COMMON LISP

- A combination of many of the features of the  popular dialects of LISP around in the early 1980s

- A large and complex language--the opposite of  Scheme

- Includes:
   - records
   - arrays
   - complex numbers
   - character strings
   - powerful i/o capabilities
   - packages with access control
   - imperative features like those of Scheme
   - iterative control statements

- Example (iterative set membership, member)

   (DEFUN iterative_member (atm lst)
     (PROG ()
       loop_1
         (COND
           ((NULL lst) (RETURN NIL))
           ((EQUAL atm (CAR lst)) (RETURN T))
         )
       (SETQ lst (CDR lst))
       (GO loop_1)
   ))

A static-scoped functional language with syntax that is closer to Pascal than to LISP

Uses type declarations, but also does type   inferencing to determine the types of undeclared variables (See Chapter 4)

It is strongly typed (whereas Scheme is  essentially typeless) and has no type coercions

Includes exception handling and a module facility for implementing abstract data types

Includes lists and list operations

The val statement binds a name to a value
    (similar to DEFINE in Scheme)

Function declaration form:
   fun function_name (formal_parameters) =
            function_body_expression;

   e.g.,  fun cube (x : int) = x * x * x;

Functions that use arithmetic or relational  operators cannot be polymorphic--those with only list operations can be polymorphic
 

Similar to ML (syntax, static scoped, strongly

   typed, type inferencing)

Different from ML (and most other functional languages) in that it is PURELY functional (e.g., no variables, no assignment statements,  and no side effects of any kind)

Most Important Features

• Uses lazy evaluation (evaluate no  subexpression until the value is needed)
• Has “list comprehensions,” which allow it to  deal with infinite lists


Examples

  1. Fibonacci numbers (illustrates function definitions with different parameter forms)

    fib 0 = 1
    fib 1 = 1
    fib (n + 2) = fib (n + 1) + fib n

2. Factorial (illustrates guards)

       fact n
      | n == 0 = 1
      | n > 0 = n * fact (n - 1)

   The special word otherwise can appear as a guard

3. List operations

   List notation: Put elements in brackets  e.g., directions = [north, south, east, west]

     - Length: #         e.g.,  #directions   is 4

     - Arithmetic series with the .. operator
         e.g., [2, 4..10] is [2, 4, 6, 8, 10]

     - Catenation is with ++
         e.g., [1, 3] ++ [5, 7] results in
                                                               [1, 3, 5, 7]

     - CAR and CDR via the colon operator (as in   Prolog)
         e.g., 1:[3, 5, 7] results in [1, 3, 5, 7]

- Examples:
 
       product [] = 1
       product (a:x) = a * product x

       fact n = product [1..n]

4. List comprehensions: set notation
    e.g.,
      [n * n | n ? [1..20]]
 
      defines a list of the squares of the first 20
      positive integers

      factors n = [i | i [1..n div 2],
                          n mod i == 0]

   This function computes all of the factors of its  given parameter

     Quicksort:

       sort [] = []
       sort (a:x) = sort [b | b ? x; b <= a]
       ++ [a] ++
       sort [b | b ? x; b > a]

5. Lazy evaluation
    Infinite lists
       e.g.,

        positives = [0..]
        squares = [n * n | n ? [0..]]
        (only compute those that are necessary)
        e.g.,
 
          member squares 16       would return True

       The member function could be written as:

     member [] b = False
     member (a:x) b = (a == b) || member x b

      However, this would only work if the parameter to squares was a perfect square; if not, it will keep generating them forever. The following version will always work:
         member2 (m:x) n
        | m < n = member2 x n
        | m == n = True
        | otherwise = False
 
 

Applications of Functional Languages:

 - APL is used for throw-away programs

 - LISP is used for artificial intelligence
    - Knowledge representation
    - Machine learning
    - Natural language processing
    - Modeling of speech and vision

 - Scheme is used to teach introductory  programming at a significant number of   universities
 

Comparing Functional and Imperative Languages

Imperative Languages:
 - Efficient execution
 - Complex semantics
 - Complex syntax
 - Concurrency is programmer designed
 

- Functional Languages:
   - Simple semantics
   - Simple syntax
   - Inefficient execution
   - Programs can automatically be made concurrent