CSC3315 (Spring 2009) 1

CSC 3315CSC 3315Programming Programming ParadigmsParadigmsScheme LanguageScheme Language

Hamid HarroudHamid HarroudSchool of Science and Engineering, Akhawayn School of Science and Engineering, Akhawayn

UniversityUniversityhttp://www.aui.ma/~H.Harroud/csc3315/

Functional Programming & Functional Programming & LispLisp

Designed by John McCarthy at MIT (1956 – 1959) for AI applications

Derived from -calculus theory (allows functions to be values in an expression)

Various dialects: Lisp 1.5 (1960), Scheme (1975), Common Lisp (1985)…[LISP = LISt Processor]

Rich Language: functional, symbolic. Uniform Syntax and Semantics

In Scheme: Symbolic calculus. Basic objets : atoms (words), Atoms groups: lists (sentences).

Atoms + Lists = Symbolic Expressions (S-expr) Scheme manipulates S-exprs: A scheme

program is a S-expr Programs and Data have the same representation.

Functional Programming & Functional Programming & SchemeScheme

Domains: non numerical applications, especially: AI (expert systems, natural languages,...) Automatic reasoning (proof of theorems,

proof of programs,...) Formal calculus Games

Functional Programming & Functional Programming & SchemeScheme

Functional Programming: basic entity = function control structure = recursion

A function is a first class that can be created, assigned to variables, passed as a parameter or returned as a value.

Scheme is implemented as an

interactif loop

(read-eval-print loop).

Functional Programming & Functional Programming & SchemeScheme

Basic ExpressionsBasic Expressions Numbers: integers / floats. A variable is a name associated to a data, for example:

(define pi 3.14159) ; pi is a global variable

A variable has an implicit type, depending on its value. It can have a value of another type:

(set! pi 3.141592)

(set! pi 'alpha)

(set! pi (cons pi '(rho))) A variable can be local to a block: (let ((var1 E1)(var2 E2)...) <expr>)

Composed ExpressionsComposed Expressions

The general format of a list is:(E1 E2 ...... En) where Ei is a S-expression.

A list can be processed as a data:

'((William Shakespeare) (The Tempest))

or as a function call with variables passed by value:(append x y)

Defining a FunctionDefining a Function

Two ways:

>(define (carre x) (* x x))

or:>(define carre (lambda (x) (* x x)))

>(carre 2)

4 Functions may not have names!

>((lambda (x) (* x x)) 3)

9

> (define (f x) (* (+ x 1)(- x 1))) > (define (fact n) ( if (> n 0)

( * n (fact (- n 1))) 1 ) )

>(fact 40)

815915283247897734345611269596115894272000000000

Defining a FunctionDefining a Function

QuoteQuote

quote avoids the evaluation of an argument (expression or atom):

(quote pi) or 'pi If pi est defined as: (define pi 3.141592)

(write 'pi) displays pi symbol (write pi) displays 3.141592 (* 2.0 pi) returns 6.283184 (* 2.0 'pi) invalid parameter

Primitive FunctionsPrimitive Functions

cons, car, cdr are primitives (functions) that allow the construction and access to lists.

lists are defined recusively:empty list: (),non-empty list: (cons )where is a list.

The head and the tail of a list:(car (cons )) returns (cdr (cons )) returns (car '()) and (cdr '()): invalid

parameter

Primitive FunctionsPrimitive Functions

Predicates: functions that return #t or #f. (symbol? x)

#t if x is a symbol,

(number? x)#t if x is a number,

(eq? x y)#t if x and y are equal symbols.

More FunctionsMore Functions

Can be defined using primitive functions:(equal? x y) if x and y are identical objects (not necesary atoms)(null? x) if x is () – empty list.(append x y) concatenate x and y.

(if (f ...) (g ...) "hello")

More Functions: ExampleMore Functions: Example

> ( define ( NumberLists? x ) ( if ( not ( list? x ) ) #f ( if ( null? x ) #t ( if ( not ( number? ( car x ) ) ) #f ( NumberLists? ( cdr x )) ) ) ) )

> ( NumberLists? '( 1 2 3 4 ))#t

> ( define ( numberList? x ) ( cond ( ( not ( list? x ) ) #f ) ( ( null? x ) #t ) ( ( not ( number? ( car x ) ) ) #f ) ( else ( numberList? ( cdr x ) ) )) )> ( numberList? ' ( 1 2 3 4 ) )#t> ( numberList? ' ( 1 2 3 bad 4 ) )#f

More Functions: ExampleMore Functions: Example

More FunctionsMore Functions Elements Access in a list:

(caar x) (car (car x))(cdadr x) (cdr (car (cdr x))))

For example, the following evaluation is made in 4 steps:

(caadar '((p ((q r) s) u) (v)))(caadr '(p ((q r) s) u))(caar '(((q r) s) u))(car '((q r) s))'(q r)

The second element of a list (if it exists):(cadr x)

The third element : (caddr x), (cadddr x), ...

Example 2Example 2(define (same_neighbours? l)

(cond

((null? l) #f)

((null? (cdr l)) #f)

((equal? (car l)(cadr l)) #t)

(else

(same_neighbours? (cdr l)))

) )

> ( define ( eqExpr? x y ) ( cond ( ( symbol? x ) ( eq? x y ) ) ( ( number? x ) ( eq? x y ) ) ; x is a list: ( ( null? x ) ( null? y ) ) ; x is non-empty list: ( ( null? y ) #f ) ( ( eqExpr? ( car x ) ( car y ) ) ( eqExpr? ( cdr x ) ( cdr y ) ) ) ( else #f )) )

Example 3Example 3

>(define (repeatedElems L) (if (list? L) (doRepeatedElems L) ‘list-error))(define (doRepeatedElems L) (cond ((null? L) '()) ((member (car L) (cdr L)) (doRepeatedElems (cdr L)))

(else (cons (car L) (doRepeatedElems (cdr L))))) )

Example 4Example 4

>(define reverse(lambda (x) (if (null? x) ’()

(append (reverse (cdr x)) (list (car x))) )) )

> (reverse '(a b c))(c b a)

More ExamplesMore Examples

CSI2520, Hiver 2007

(define (empty?empty? stack)

(null? stack)

) 

(define (poppop stack)

(if (empty? stack)

()

(cdr stack)

) )

(define (pushpush e stack)

(cons e stack)

)

(define (toptop stack)

(if (empty? stack)

()

(car stack)

) )

Pile en SchemePile en Scheme

(define (min x) (cond ((not (numberlist? x)) (list 'not-number-list x)) ((null? x) x) (else (min-aux (car x) (cdr x))) ) )

(define (min-aux elt x) (cond ((null? x) elt) ((> elt (car x) (min-aux (car x)(cdr x))) (else (min-aux elt (cdr x)))) )

Minimum of a ListMinimum of a List

(define (minL-aux Elt Lst) (if (null? Lst) Elt (let ((v1 (car Lst)) (v2 (cdr Lst))) (if (> Elt v1) (minl-aux v1 v2) (minl-aux Elt v2)

) ) ) )

Minimum of a List: local Minimum of a List: local variablesvariables

(define append

(lambda (x y)

(cond

((null? x) y)

(else (cons (car x) (append (cdr x) y)))

) ) )

> (append '(a b c) '(d e))

(a b c d e)

> (string-append “abc” “123”)

“abc123”

Concatenation of two listsConcatenation of two lists

(define member? (lambda (x y) (cond ((null? y) #f) ((equal? x (car y)) #t) (else (member? x (cdr y)))) ) )

> (member? 'aa '(bb ccc aa eeee rr ttttt))> (member? 'aa '(bb ccc aa eeee rr ttttt))#t> (member? 'aa '(bb ccc (aa) eeee rr ttttt))#f> (member 'aa '(bb ccc aa eeee rr ttttt))(aa eeee rr ttttt)

Member Function Member Function

Some functions can have other functions as parameters.

> (define (combine f1 f2 x) (f1 (f2 x)) )> (combine (lambda (x) (+ 1 x)) (lambda (x) (* 2 x)) 6)13> (combine (lambda (x) (* 2 x)) (lambda (x) (+ 1 x)) 6)14Équivalent to:> ((lambda (x) (+ 1 x)) ((lambda (x) (* 2 x)) 6))> ((lambda (x) (* 2 x)) ((lambda (x) (+ 1 x)) 6))

High-level functions: High-level functions: Functions as ParametersFunctions as Parameters

(define f (lambda (x) (lambda (y) (lambda (z) (+ x y z)) ) ) )

> (((f 1) 2) 3)6

(define g ((f 1) 2))>(g 3)6

High-level functions: High-level functions: Functions as ParametersFunctions as Parameters

(define (map F Lst) (if (null? Lst) Lst (cons (F (car Lst)) (map F (cdr Lst)))) )

map: applies a function to each element of a list:(E1 E2 ... En) ((f E1) (f E2) ... (f En))

For example(map (lambda(x) (+ x 1)) '(1 2 3))returns: (2 3 4)

High-level Functions: mapHigh-level Functions: map

( map car '((a b) (c d) (e f)) )

( map cadr '((a b) (c d) (e f)) )

( map (lambda (x) (cons 3 (list x))) '(a b c d) )

High-level Functions: mapHigh-level Functions: map

Consider a function F with two parameters, and F0 a constant.

We want to have the following transformation

(F E1 (F E2 (F ... (F En F0) ) ...)

Which can be represented using an infix notation:

(E1 E2 ...... En) E1 F E2 F ...... F En F F0

High-level functions: High-level functions: ReducersReducers

Example:

(E1 E2 ...... En) E1 + E2 + ...... + En + 0

(E1 E2 ...... En) E1 * E2 * ...... * En * 1

(define (reduce F F0 L)

(if (null? L)

F0

(F (car L)

(reduce F F0 (cdr L)))

))

High-level functions: High-level functions: ReducersReducers

> (reduce + 0 '(1 2 3 4))

> (reduce * 1 '(1 2 3 4))

> (reduce (lambda (x y) (+ x y 1)) 8 '(1 2 3))

> (reduce cons '() '(1 2 3 4))

> (reduce append '() '((1) (2) (3)))

High-level functions: High-level functions: ReducersReducers

Lists Manipulation

carcdrconsappendlistlengthcaar, cadr, cdar, cddr,caaaar, cddddr, ...

To define Functions

definelambda

Pre-defined FunctionsPre-defined Functions

Logic

not and or #t #f

Control

ifcondelselet

Arithmetic and comparison + < - > * <= / >= max = min

Pre-defined FunctionsPre-defined Functions

Predicatessymbol? number? integer? list? null? eq? equal? procedure?member?

I/O Functions readwrite display print

Pre-defined FunctionsPre-defined Functions

CSI2520, Hiver 2007

Other functions

quote ---> ‘a, ‘(1 2)set! ---> (define a 1) (set! a (+ a 1))load ---> (load “scm1.txt”)map eval ---> (eval ‘(+ 1 2))apply ---> (apply + ‘(1 2))

Pre-defined FunctionsPre-defined Functions