CS 363 Comparative Programming Languages Functional Languages: Scheme.

CS 363 Comparative Programming Languages

Functional Languages: Scheme

CS 363 Spring 2005 GMU 2

Fundamentals of Functional Programming Languages

• The objective of the design of a functional programming language (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


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


Functions

• A function is a rule that associates to each x from some set X of values a unique y from another set Y of values:– y = f(x) or f: X Y

• Functional Forms– Def: A higher-order function, or functional form, is one

that either takes functions as parameters, yields a function as its result, or both

domainrange


Lisp

• Lisp – based on lambda calculus (Church)– Uniform representation of programs and data

using single general data structure (list)– Interpreter based (written in Lisp)– Automatic memory management– Evolved over the years– Dialects: COMMON LISP, Scheme


Scheme

(define (gcd u v) (if ( = v 0) u (gcd v (remainder u v)) ))

(define (reverse l) (if (null? l) l (append (reverse (cdr l))(list (car l))) ))


Introduction to Scheme

• 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


Scheme


• Once defined in the interpreter: (gcd 25 10) 5


Scheme atoms

• Constants:– numbers, strings, #T = True, …

• Identifier names:

• Function/operator names– pre-defined & user defined


Scheme Expression vs. C

In Scheme:(+ 3 (* 4 5 ))

(and (= a b)(not (= a 0)))

(gcd 10 35)

In C:3 + 4 * 5

(a = = b) && (a != 0)

gcd(10,35)


Evaluation Rules for Scheme Expressions

1. Constant atoms (numbers, strings) evaluate to themselves

2. Identifiers are looked up in the current environment and replaced by the value found there (using dynamically maintained symbol table)

3. A list is evaluated by recursively evaluating each element in the list as an expression; the first expression must evaluate to a function. The function is applied to the evaluated values of the rest of the list.


Scheme Evaluation

To evaluate (* (+ 2 3)(+ 4 5))1. * is the function – must evaluate the two

expressions (+ 2 3) and (+ 4 5)2. To evaluate (+ 2 3)

1. + is the function – must evaluation the two expressions 2 and 3

2. 2 evaluates to the integer 23. 3 evaluates to the integer 34. + 2 3 = 5

3. To evaluate (+ 4 5) follow similar steps4. * 5 9 = 45

*

+ +

2 3 4 5


Scheme Conditionals

If statement:

(if ( = v 0)

u

(gcd v (remainder u v))

)

(if (= a 0) 0 (/ 1 a))

Cond statement:(cond (( = a 0) 0)

((= a 1) 1)

(else (/ 1 a))

)

Both if and cond functions usedelayed evaluation for the expressions in their bodies (i.e. (/ 1 a) is not evaluated unlessthe appropriate branch is chosen).


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”)) ) )


Predicate Functions

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– EQ? does not work for numeric atoms (use = )


Predicate Functions2. LIST? takes one parameter; it returns #T if the

parameter is a 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?


let function

• Allows values to be given temporary names within an expression– (let ((a 2 ) (b 3)) ( + a b))– 5

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


Quote (‘) function• A list that is preceeded by QUOTE or a quote mark (‘) is

NOT evaluated.• 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

• Can be used to provide function arguments– (myfunct ‘(a b) ‘(c d))


Output functions

• Output Utility Functions:

(DISPLAY expression)

(NEWLINE)


define function

Form 1: Bind a symbol to a expression:

(define a 2)

(define emptylist ‘( ))

(define pi 3.141593)


define function

Form 2: To bind names to lambda expressions

define (cube x) (* x (* x x )))


function name and parameters

function body – lambda expression


Function Evaluation

• Evaluation process (for normal functions): 1. Parameters are evaluated, in no particular order

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)


Data Structures in Scheme: Box Notation for Lists

first element (car) rest of list (cdr)

1

List manipulation is typically written using ‘car’ and ‘cdr’


Data Structures in Scheme

1 2 3 (1,2,3)

c

da

b((a b) c (d))


Basic List Manipulation

• (car L) – returns the first element of L

• (cdr L) – returns L minus the first element

(car ‘(1 2 3)) = 1

(car ‘((a b)(c d))) = (a b)

(cdr ‘(1 2 3)) = (2 3)

(cdr ‘((a b)(c d))) = ((c d))


Basic List Manipulation

• (list e1 … en) – return the list created from the individual elements

• (cons e L) – returns the list created by adding expression e to the beginning of list L

(list 2 3 4) = (2 3 4)

(list ‘(a b) x ‘(c d) ) = ((a b)x(c d))

(cons 2 ‘(3 4)) = (2 3 4)

(cons ‘((a b)) ‘(c)) = (((a b)) c)


Example Functions

1. member - takes an atom and a simple 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))) ))


Example Functions2. 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 '())))


Example Functions3. equal - takes two general lists as parameters; returns #T if the two

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 '())))


Example Functions

(define (count L)

(if (null? L) 0

(+ 1 (count (cdr L)))

)

)

(count ‘( a b c d)) =

(+ 1 (count ‘(b c d))) =

(+ 1 (+ 1(count ‘(c d))))

(+ 1 (+ 1 (+ 1 (count ‘(d)))))=

(+ 1 (+ 1 (+ 1 (+ 1 (count ‘())))))=

(+ 1 (+ 1 (+ 1 (+ 1 0))))= 4


Scheme Functions

Now define(define (count1 L) ??

)

so that (count1 ‘(a (b c d) e)) = 5


Scheme Functions

This function counts the individual elements:

(define (count1 L)

(cond ( (null? L) 0 )

( (list? (car L))

(+ (count1 (car L))(count1 (cdr L))))

(else (+ 1 (count (cdr L))))

)

)

so that (count1 ‘(a (b c d) e)) = 5


Example Functions

(define (append L M)

(if (null? L) M

(cons (car L)(append(cdr L) M))

)

)

(append ‘(a b) ‘(c d)) = (a b c d)


How does append do its job?


(if (null? L) M

(cons (car L)(append(cdr L) M))))

(append ‘(a b) ‘(c d)) =

(cons a (append ‘(b) ‘(c d))) =




(if (null? L) M




(cons a (cons b (append ‘() ‘(c d)))) =



(define (append L M) (if (null? L) M (cons (car L)(append(cdr L) M))))

(append ‘(a b) ‘(c d)) =(cons a (append ‘(b) ‘(c d))) =

(cons a (cons b (append ‘() ‘(c d)))) =(cons a (cons b ‘(c d))) =




(if (null? L) M





(cons a (cons b ‘(c d))) =

(cons a ‘(b c d)) =




(if (null? L) M





(cons a (cons b ‘(c d))) =

(cons a ‘(b c d)) =

(a b c d)


Reverse

Write a function that takes a list of elements and reverses it:

(reverse ‘(1 2 3 4)) = (4 3 2 1)


Reverse

(define (reverse L)

(if (null? L) ‘()

(append (reverse (cdr L))(list (car L)))

)

)


Selection Sort in Scheme

Let’s define a few useful functions first:(DEFINE (findsmallest lis small) (COND ((NULL? lis) small) ((< (CAR lis) small) (findsmallest (CDR lis) (CAR lis))) (ELSE (findsmallest (CDR lis) small)) ) )



(DEFINE (remove lis item) (COND ((NULL? lis) ‘() ) ((= (CAR lis) item) lis) (ELSE (CONS (CAR lis) (remove (CDR lis)

item))) ))

Cautious programming!

Assuming integers



(DEFINE (selectionsort lis)

(IF (NULL? lis) lis

(LET ((s (findsmallest (CDR lis) (CAR lis))))

(CONS s (selectionsort (remove lis s)) )

)

)


Higher order functions

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

• Mapcar

• Eval


Higher-Order Functions: mapcar

Apply to All - 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)))) ))


Higher-Order Functions: mapcar

• Using mapcar:(mapcar (LAMBDA (num) (* num num num)) ‘(3 4 2 6))

returns (27 64 8 216)


Higher Order Functions: EVAL

• 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


Using EVAL for adding a List of Numbers

Suppose we have a list of numbers that must be added:(DEFINE (adder lis) (COND((NULL? lis) 0) (ELSE (+ (CAR lis) (adder(CDR lis ))))))

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

(adder ‘(3 4 5 6 6))Returns 24


Other Features of Scheme• 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


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


COMMON LISP

• Includes:– records

– arrays

– complex numbers

– character strings

– powerful I/O capabilities

– packages with access control

– imperative features like those of Scheme

– iterative control statements


ML

• 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

• 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


ML

• 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;


Haskell

• 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)


Haskell

• 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


Haskell 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


Haskell Examples

2. Factorial (illustrates guards)

fact n

| n == 0 = 1

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

The special word otherwise can appear as a guard


Haskell Examples3. 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]


Haskell Examples

3. List operations (cont)– Catenation is with ++

e.g., [1, 3] ++ [5, 7] results in

[1, 3, 5, 7]– CONS, CAR, CDR via the colon operator (as in

Prolog)

e.g., 1:[3, 5, 7] results in

[1, 3, 5, 7]


Haskell Examples

• Quicksort:

sort [] = []

sort (a:x) = sort [b | b ← x; b <= a]

++ [a] ++

sort [b | b ← x; b > a]


Applications of Functional Languages

• 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

CS 363 Comparative Programming Languages Functional Languages: Scheme.

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