Functional programming

FUNCTIONAL
PROGRAMMING
REFERENCES:
1. PROGRAMMING LANGUAGES: PRINCIPLES AND PRACTICES. K. C. LAUDEN, SECOND EDITION
2. PROGRAMMING LANGUAGE PRAGMATICS. M. L. SCOTT

AUGUST 31, 2011

Thursday, September 8, 2011

FUNCTIONAL PROGRAMMING

REASONS TO STUDY FP:

RECURSION

ABSTRACTION

HIGHER-ORDER FUNCTIONS

> IMPACT MOST PROGRAMMING LANGUAGES


PROGRAMS AS FUNCTIONS
PROGRAM ≡ DESCRIPTION OF A SPECIFIC COMPUTATION
Y = F(X)
F:X➝Y
IN MATHEMATICS:
*VARIABLES ≡ ACTUAL VALUES
*NO MEMORY LOCATION CONCEPT.

IN IMPERATIVE LANGUAGES:
*VARIABLES ≡ MEMORY LOCATIONS + VALUES


CONSEQUENCES OF NO ASSIGNMENT OPERATION:

*NO LOOPS

*INSTEAD: RECURSION.

IN FUNCTIONAL PROGRAMMING (PURE FUNCTIONAL PROGRAMMING):

*NO VARIABLE
> EXCEPT AS A NAME FOR A VALUE

*NO ASSIGNMENT OPERATION
> X = X + 1 DOES NOT MAKE ANY SENSE IN MATH!

*ONLY CONSTANTS, PARAMETERS, AND VALUES.


EXAMPLE: ITERATIONS VS RECURSION

VOID GCD ( INT U, INT V, INT * X ) INT GCD ( INT U, INT V )
{ {
INT Y, T, Z; IF ( V== 0 )
Z = U; RETURN U;
Y = V; ELSE
WHILE( Y != 0 ) RETURN GCD(V, U % V);
{ }
T = Y;
Y = Z % Y;
Z = T;
}
*X = Z;
}


CONSEQUENCES OF NO VARIABLES AND NO ASSIGNMENT OPERATION:

*NO NOTION OF THE INTERNAL STATE OF A FUNCTION

*THE VALUE OF ANY FUNCTION CANNOT DEPEND ON THE ORDER
OF EVALUATION OF ITS ARGUMENTS

>REFERENTIAL TRANSPARENCY


CONSEQUENCES OF NO VARIABLES AND NO ASSIGNMENT OPERATION:

*NO NOTION OF THE INTERNAL STATE OF A FUNCTION

*THE VALUE OF ANY FUNCTION CANNOT DEPEND ON THE ORDER
OF EVALUATION OF ITS ARGUMENTS

>REFERENTIAL TRANSPARENCY

*THE RUNTIME ENVIRONMENT ASSOCIATES NAMES TO VALUES
ONLY; ONCE A NAME ENTERS THE ENVIRONMENT IT NEVER
CHANGES

>VALUE SEMANTICS


IN FUNCTIONAL PROGRAMMING:

*FUNCTIONS MUST BE GENERAL LANGUAGE OBJECTS

> FIRST-CLASS VALUES

*FUNCTIONS ≡ VALUES ⇒ CAN BE COMPUTED BY OTHER
FUNCTIONS ⇒ WHICH CAN BE PARAMETERS TO FUNCTIONS.

*FUNCTIONS CAN HAVE OTHER FUNCTIONS AS INPUT OR
OUTPUT OR BOTH AS PARAMETERS

> HIGHER-ORDER FUNCTIONS


SUMMARY:

1. ALL PROCEDURES ARE FUNCTIONS AND CLEARLY DISTINGUISH
INCOMING VALUES (PARAMETERS) FROM OUTGOING VALUES
(RESULTS).

2. THERE ARE NO VARIABLES OR ASSIGNMENTS - VARIABLES ARE
REPLACED BY PARAMETERS.

3. THERE ARE NO LOOPS - LOOPS ARE REPLACED BY RECURSIVE
CALLS.

4. THE VALUE OF A FUNCTION DEPENDS ONLY ON THE VALUE OF ITS
INPUT PARAMETERS AND NOT ON THE ORDER OF EVALUATION OR THE
EXECUTION PATH THAT LED TO THE CALL.

5. FUNCTIONS ARE FIRST-CLASS VALUES.


FUNCTIONAL PROGRAMMING IN
AN IMPERATIVE LANGUAGE
BASIC REQUIREMENT:

1. AVAILABILITY OF RECURSION

2. MECHANISM FOR IMPLEMENTING GENERAL FUNCTIONS


EXAMPLE:

CONSIDER A FUNCTION THAT RETURNS THE SUM OF THE INTEGERS
BETWEEN I AND J:

SUM(I, J) = I + (I + 1) + (I + 2) + ... + (J - 1) + J


EXAMPLE:

BETWEEN I AND J:

SUM(I, J) = I + (I + 1) + (I + 2) + ... + (J - 1) + J

INT SUM ( INT I, INT J )
{
INT K, TEMP;
TEMP = 0;
FOR( K = I; K <= J; K++ )
TEMP += K;
RETURN TEMP;
}


EXAMPLE:

BETWEEN I AND J:

SUM(I, J) = I + (I + 1) + (I + 2) + ... + (J - 1) + J

INT SUM ( INT I, INT J ) INT SUM ( INT I, INT J )
{ {
INT K, TEMP; IF ( I > J )
TEMP = 0; RETURN 0;
FOR( K = I; K <= J; K++ ) ELSE
TEMP += K; RETURN I + SUM(I + 1, J);
RETURN TEMP; }
}


TAIL RECURSION:

LAST OPERATION IN A PROCEDURE CALLS ITSELF WITH
DIFFERENT ARGUMENTS.

AUTOMATIC RECURSIVE TO LOOP-CONVERSION:
> REASSIGN PARAMETERS AND START OVER.

INT GCD ( INT U, INT V )
{
IF ( V== 0 )
RETURN U;
ELSE
RETURN GCD(V, U % V);
}


TAIL RECURSION:


AUTOMATIC RECURSIVE TO LOOP-CONVERSION:
> REASSIGN PARAMETERS AND START OVER.

{ INT T1, T2; /* TEMPS INTRODUCED BY TRANSLATOR */
FOR ( ; ; )
{ IF (V == 0 )
{
RETURN U;
IF ( V== 0 )
ELSE
RETURN U;
{
ELSE
T1 = V;
RETURN GCD(V, U % V);
T2 = U % V;
}
U = T1;
V = T2;
}
}
}


TAIL RECURSION:


NOT SO AUTOMATIC RECURSIVE TO LOOP-CONVERSION:
ACCUMULATING PARAMETERS

> USED TO PRECOMPUTE OPERATIONS PERFORMED AFTER
THE RECURSIVE CALL.

EXAMPLE: SUM OF INTEGERS FROM I TO J.

INT SUM ( INT I, INT J )
{
IF ( I > J )
RETURN 0;
ELSE
RETURN I + SUM(I + 1, J);
}


TAIL RECURSION:


NOT SO AUTOMATIC RECURSIVE TO LOOP-CONVERSION:
ACCUMULATING PARAMETERS

> USED TO PRECOMPUTE OPERATIONS PERFORMED AFTER
THE RECURSIVE CALL.

EXAMPLE: SUM OF INTEGERS FROM I TO J.

INT SUM ( INT I, INT J ) INT SUM1 ( INT I, INT J, INT SUMSOFAR )
{ { /* SUM1 IS CALLED A HELPING PROCEDURE */
IF ( I > J ) { IF (I > J )
RETURN 0; RETURN SUMSOFAR;
ELSE ELSE
RETURN I + SUM(I + 1, J); RETURN SUM1( I + 1, J, SUMSOFAR + I );
} }

INT SUM( INT I, INT J )
{ RETURN SUM1( I, J, 0 );}


RESTRICTIONS OF IMPERATIVE LANGUAGES
FOR FUNCTIONAL PROGRAMMING:

1. STRUCTURED VALUES SUCH AS ARRAYS AND RECORDS CANNOT
BE RETURNED VALUES FROM FUNCTIONS.

2. THERE IS NO WAY TO BUILD A VALUE OF A STRUCTURED TYPE
DIRECTLY.

3. FUNCTIONS ARE NOT FIRST-CLASS VALUES, SO HIGHER-ORDER
FUNCTIONS CANNOT BE WRITTEN.


RESTRICTIONS OF IMPERATIVE LANGUAGES
FOR FUNCTIONAL PROGRAMMING:

EXAMPLE: SORTING ARRAY IN C

1. AN ARRAY CANNOT BE RETURNED FROM A FUNCTION VALUE.

2. ANY ARRAY IS AUTOMATICALLY PASSED BY REFERENCE

> SORT IN PLACE

> VIOLATES THE RULE THAT THE INPUT OF A FUNCTION IN FP
SHOULD ALWAYS BE DISTINGUISH FROM THE OUTPUT.

*MOST PROBLEMATIC RESTRICTION: NON-FIRST-CLASSNESS OF
FUNCTIONS IN IMPERATIVE LANGUAGES.


EXAMPLE
C CODE FOR A FUNCTION INT_ARRAY_MAX THAT COMPUTES THE MAXIMUM
VALUE IN A NON-EMPTY ARRAY OF INTEGERS.

INT INT_ARRAY_MAX ( INT A[], INT SIZE )
/* SIZE MUST BE > 0 */
{
INT I, MAX = A[ 0 ];
FOR(I = 1; I < SIZE; I++)
IF (MAX < A[ I ] )
MAX = A[ I ];

RETURN MAX;
}


EXAMPLE
C CODE FOR A FUNCTIONAL VERSION OF THE INT_ARRAY_MAX FUNCTION. IT
USES A HELPING PROCEDURE INT_ARRAY_MAX1 WITH RECURSION AND AN
ACCUMULATING PARAMETER.

INT INTMAX ( INT X, INT Y)
{
RETURN X > Y ? X : Y;
}

INT INT_ARRAY_MAX1 ( INT A[], INT SIZE, INT SOFAR )
{
IF( SIZE == 0 ) RETURN SOFAR;
ELSE
RETURN INT_ARRAY_MAX1( A, SIZE - 1, INTMAX(SOFAR, A[SIZE-1]));
}

INT INT_ARRAY_MAX ( INT A[], INT SIZE )
/* SIZE MUST BE > 0 */
{ RETURN INT_ARRAY_MAX1( A, SIZE - 1, A[ SIZE - 1 ]);
}


SCHEME: A DIALECT OF LISP

LISP [LATE 1950 AND EARLY 1960] CREATED BY JOHN MCCARTHY FROM MIT

LISP = LIST PROCESSOR

BASED ON LAMBDA CALCULUS OF A. CHURCH.

WAS AN INTERPRETER FOR AN IBM 704

FEATURES

UNIFORM REPRESENTATION OF PROGRAMS AS DATA USING A
SINGLE GENERAL DATA STRUCTURE - THE LIST.

THE DEFINITION OF THE LANGUAGE USING AN INTERPRETER
WRITTEN IN THE LANGUAGE ITSELF-CALLED A METACIRCULAR
INTERPRETER.

THE AUTOMATIC MANAGEMENT OF ALL MEMORY BY THE RUNTIME
SYSTEM.


THE ELEMENTS OF SCHEME
THE CFG FOR SCHEME IS:

EXPRESSION ➝ ATOM | LIST
ATOM ➝ NUMBER | STRING | IDENTIFIER | CHARACTER | BOOLEAN
LIST ➝ ‘(‘EXPRESSION-SEQUENCE’)’
EXPRESSION-SEQUENCE ➝ EXPRESSION EXPRESSION-SEQUENCE | EXPRESSION

EXAMPLES OF SCHEME EXPRESSIONS:

47 -A NUMBER
“HELLO” -A STRING
#T -THE BOOLEAN VALUE TRUE
#A -THE CHARACTER ‘A’
(2.1 3.5 7.2 4.2) -A LIST OF NUMBERS
B -AN IDENTIFIER
HELLO -AN IDENTIFIER
(+ 2 4) -A LIST CONSISTING OF THE IDENTIFIER ‘+’ AND TWO NUMBERS
(* (+ 2 4) (/ 6 2)) -A LIST CONSISTING OF AN IDENTIFIER FOLLOWED BY TWO LISTS


THE CFG FOR SCHEME IS:

EXPRESSION ➝ ATOM | LIST
ATOM ➝ NUMBER | STRING | IDENTIFIER | CHARACTER | BOOLEAN
LIST ➝ ‘(‘EXPRESSION-SEQUENCE’)’
EXPRESSION-SEQUENCE ➝ EXPRESSION EXPRESSION-SEQUENCE | EXPRESSION

SCHEME EVALUATION RULES [APPLICATIVE ORDER EVALUATION]

1. CONSTANT ATOMS, SUCH AS NUMBERS AND STRINGS EVALUATE TO THEMSELVES.

2. IDENTIFIERS ARE LOOKED UP IN THE CURRENT ENVIRONMENT AND REPLACED BY THE
VALUE FOUND THERE. [DYNAMIC SYMBOL TABLE]

3. A LIST IS EVALUATED BY RECURSIVELY EVALUATING EACH ELEMENT IN THE LIST AS
AN EXPRESSION; THE FIRST EXPRESSION IN THE LIST MUST EVALUATE TO A FUNCTION.
THIS FUNCTION IS THEN APPLIED TO THE EVALUATED VALUES OF THE REST OF THE LIST.


SCHEME EVALUATION RULES [APPLICATIVE ORDER EVALUATION]

1. CONSTANT ATOMS, SUCH AS NUMBERS AND STRINGS EVALUATE TO THEMSELVES.

2. IDENTIFIERS ARE LOOKED UP IN THE CURRENT ENVIRONMENT AND REPLACED BY THE
VALUE FOUND THERE. [DYNAMIC SYMBOL TABLE]

3. A LIST IS EVALUATED BY RECURSIVELY EVALUATING EACH ELEMENT IN THE LIST AS
AN EXPRESSION; THE FIRST EXPRESSION IN THE LIST MUST EVALUATE TO A FUNCTION.
THIS FUNCTION IS THEN APPLIED TO THE EVALUATED VALUES OF THE REST OF THE LIST.

EXAMPLES: EVALUATE

47
“HELLO”
#T
#A
(2.1 3.5 7.2 4.2)
B
HELLO
(+ 2 4)
(* (+ 2 4) (/ 6 2))


BUILT-IN FUNCTIONS

QUOTE: PREVENTS EVALUATION:

>(2.1 3.4 5.6)
ERROR: THE OBJECT 2.1 IS NOT A PROCEDURE

>(QUOTE (2.1 3.4 5.6))
(2.1 3.4 5.6)
IF: IF-THEN-ELSE CONSTRUCT
(IF (=A 0 ) 0
(/ 1 A))

COND: IF-ELSEIF CONSTRUCT
(COND((= A 0) 0)
((= A 1) 1)
(ELSE (/ 1 A)))

SEMANTICS OF (IF EXP1 EXP2 EXP3):
EXP1 EVALUATES FIRST
IF EXP1 EVALUATES TO FALSE (#F), THEN EXP3 IS EVALUATED; THEN EXP2 IS
EVALUATED AND RETURNED.
IF EXP3 IS ABSENT AND EXP1 EVALUATES TO FALSE, THEN THE VALUE OF THE
EXPRESSION IS UNDEFINED.


SEMANTICS OF (IF EXP1 EXP2 EXP3):
EXP1 EVALUATES FIRST

IF EXP1 EVALUATES TO FALSE (#F), THEN EXP3 IS EVALUATED; THEN EXP2 IS
EVALUATED AND RETURNED.

IF EXP3 IS ABSENT AND EXP1 EVALUATES TO FALSE, THEN THE VALUE OF THE
EXPRESSION IS UNDEFINED.

SEMANTICS OF (COND EXP1 EXP2 ... EXPN):
EACH EXPI MUST BE A PAIR EXPI = (FST SND).

EACH EXPRESSION EXPI IS CONSIDERED IN ORDER, AND THE FIRST PART OF IT IS
EVALUATED.

IF FST EVALUATES TO TRUE, THEN SND IS EVALUATED AND ITS VALUE IS RETURNED
BY THE COND EXPRESSION.

IF NONEN OF THE CONDITIONS EVALUATE TO TRUE, THEN THE EXPRESSION IN THE
ELSE IS EVALUATED AND RETURNED.

DELAYED EVALUATION:



BUILT-IN FUNCTIONS

> (LET ((A 2) (B 3)) (+ A B)) ; BINDING LIST
5


FUNCTION DEFINITION

(DEFINE A 2)
(DEFINE EMPTYLIST ‘())

(DEFINE (GCD U V) ; FUNCTION NAME AND PARAMETERS
(IF (= V 0) ; FUNCTION BODY
U
(GCD V (REMAINDER U V))))


BUILT-IN FUNCTIONS FOR I/O: READ AND DISPLAY

>(READ)

>(DISPLAY “HELLO WORLD”)

>(DISPLAY 234)

SEE PAGE 486 FOR A COMPLETE EXAMPLE WITH I/O


DATA STRUCTURES IN SCHEME
BASIC DATA STRUCTURE: LIST

*EVERYTHING ELSE SHOULD BE PUT IN THAT FORMAT!!!

EXAMPLE: BINARY SEARCH TREE

NODE IS (NAME LEFT RIGHT)

(“HORSE” (“COW” () (“DOG” () ())) (“ZEBRA” (“YAK” () ()) () ))

“HORSE”

“COW” “ZEBRA”

() “DOG” “YAK” ()

() ()


DATA STRUCTURES IN SCHEME
BASIC OPERATIONS ON LIST:

>(CAR L)

>(CONS L)

BINARY TREE:

(DEFINE (LEFTCHILD B) (CAR ( CDR B)))
(DEFINE (RIGHTCHILD B) (CAR (CDR (CDR B))))
(DEFINE (DATA B) (CAR B))

TREE TRAVERSAL:

(DEFINE (PRINT-TREE B)
(COND ((NULL? B) ‘() )
(ELSE (PRINT-TREE (LEFTCHILD B))
(DISPLAY (DATA B))
(NEWLINE)
(PRINT-TREE (RIGHTCHILD B)))))


PROGRAMMING TECHNIQUES IN
SCHEME
EXAMPLE: SQUARING THE NUMBERS IN A LIST

(DEFINE (SQUARE-LIST L)
(IF (NULL? L) ‘()
(CONS (* (CAR L) (CAR L)) (SQUARE-LIST (CDR L)))))

EXAMPLE: PRINTING OUT THE SQUARES OF INTEGERS FROM 1 TO N

(DEFINE (PRINT-SQUARES LOW HIGH)
(COND ((> LOW HIGH) ‘())
(ELSE (DISPLAY(* LOW LOW))
(NEWLINE)
(PRINT-SQUARES (+ 1 LOW) HIGH))))


Delayed Evaluation
An expression subject to lazy evaluation is not evaluated until its value
is required and once is evaluated is never reevaluated.

delay

force

used only in the absence of side effects: the order of evaluation
is unimportant.

benefits: some amount of computation might be avoided altogether
if it is delayed until absolutely required.

stream (infinite lists) construction


Delayed Evaluation

Use language R5RS

(define stream-car
(lambda (s)
(car (force s))))

(define stream-cdr
(lambda (s)
(cdr (force s))))

(define counters
(let next ((n 1))
(delay (cons n (next (+ n 1))))))

(stream-car counters)


Delayed Evaluation

Use Language R5RS from DrRacket

(stream-car (stream-cdr counters))

(define stream-add
(lambda (s1 s2)
(delay (cons
(+ (stream-car s1) (stream-car s2))
(stream-add (stream-cdr s1) (stream-cdr s2))))))

(define even-counters
(stream-add counters counters))

(stream-car even-counters)

(stream-car (stream-cdr even-counters))


Exercises
Write recursive functions in Scheme for:

1. The nth Fibonacci number.

2. The sum of the nth first natural numbers.

3. The factorial of n.

4. Reversing a list of elements.

5. Computing the number of elements in a list.


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