CSC321: Programming Languages14-1 Programming Languages Tucker and Noonan Chapter 14: Functional...

CSC321: Programming Languages 14-1

Programming LanguagesProgramming LanguagesTucker and NoonanTucker and Noonan

Chapter 14: Functional Programming

14.1 Functions and the Lambda Calculus14.2 Scheme

14.2.1 Expressions14.2.2 Expression Evaluation14.2.3 Lists14.2.4 Elementary Values14.2.5 Control Flow14.2.6 Defining Functions14.2.7 Let Expressions14.2.8 Example: Semantics of Clite14.2.9 Example: Symbolic Differentiation14.2.10 Example: Eight Queens


Overview of Functional LanguagesOverview of Functional Languages

• They emerged in the 1960’s with Lisp• Functional programming mirrors mathematical

functions: domain = input, range = output• Variables are mathematical symbols: not

associated with memory locations.• Pure functional programming is state-free: no

assignment• Referential transparency: a function’s result

depends only upon the values of its parameters.


Functions and the Lambda CalculusFunctions and the Lambda Calculus

• The function Square has R (the reals) as domain and range.– Square : R R– Square(n) = n2

• A function is total if it is defined for all values of its domain. Otherwise, it is partial. E.g., Square is total.

• A lambda expression is a particular way to define a function:

LambdaExpression variable | ( M N) | ( variable . M )

M LambdaExpression

N LambdaExpression

• E.g., ( x . x2 ) represents the Square function.


Properties of Lambda ExpressionsProperties of Lambda Expressions

• In ( x . M), x is bound. Other variables in M are free.

• A substitution of N for all occurrences of a variable x in M is written M[x N]. Examples:

• A beta reduction (( x . M)N) of the lambda expression ( x . M) is a substitution of all bound occurrences of x in M by N. E.g.,

• (( x . x2)5) = 52


Function EvaluationFunction Evaluation

• In pure lambda calculus, expressions like (( x . x2)5) = 52 are uninterpreted.

• In a functional language, (( x . x2)5) is interpreted normally (25).

• Lazy evaluation = delaying argument evaluation in a function call until the argument is needed.– Advantage: flexibility

• Eager evaluation = evaluating arguments at the beginning of the call.– Advantage: efficiency


Status of FunctionsStatus of Functions

• In imperative and OO programming, functions have different (lower) status than variables.

• In functional programming, functions have same status as variables; they are first-class entities.– They can be passed as arguments in a call.– They can transform other functions.

• A function that operates on other functions is called a functional form. E.g., we can define

g(f, [x1, x2, … ]) = [f(x1), f(x2), …], so that

g(Square, [2, 3, 5]) = [4, 9, 25]


SchemeScheme

• A derivative of Lisp

• Our subset:– omits assignments– simulates looping via recursion– simulates blocks via functional composition

• Scheme is Turing complete, but

• Scheme programs have a different flavor


ExpressionsExpressions

• Cambridge prefix notation for all Scheme expressions:

(f x1 x2 … xn)

• E.g.,– (+ 2 2) ; evaluates to 4– (+ (* 5 4) (- 6 2)) ; means 5*4 + (6-2)– (define (Square x) (* x x)) ; defines a function– (define f 120) ; defines a global

• Note: Scheme comments begin with ;


Expression EvaluationExpression Evaluation

• Three steps:1.Replace names of symbols by their current bindings.

2.Evaluate lists as function calls in Cambridge prefix.

3.Constants evaluate to themselves.

• E.g.,x ; evaluates to 5

(+ (* x 4) (- 6 2)) ; evaluates to 24

5 ; evaluates to 5

‘red ; evaluates to ‘red


ListsLists• A list is a series of expressions enclosed in

parentheses.– Lists represent both functions and data.– The empty list is written (). – E.g., (0 2 4 6 8) is a list of even numbers. Here’s how

it’s stored:


List Transforming FunctionsList Transforming Functions

• Suppose we define the list evens to be (0 2 4 6 8). I.e., we write (define evens ‘(0 2 4 6 8)). Then:

(car evens) ; gives 0

(cdr evens) ; gives (2 4 6 8)

(cons 1 (cdr evens)) ; gives (1 2 4 6 8)

(null? ‘()) ; gives #t, or true

(equal? 5 ‘(5)) ; gives #f, or false

(append ‘(1 3 5) evens) ; gives (1 3 5 0 2 4 6 8)

(list ‘(1 3 5) evens) ; gives ((1 3 5) (0 2 4 6 8))• Note: the last two lists are different!


Elementary ValuesElementary Values

• Numbers• integers• floats• rationals• Symbols• Characters• Functions• Strings• (list? evens)• (symbol? ‘evens)


Control FlowControl Flow

• Conditional(if (< x 0) (- 0 x)) ; if-then

(if (< x y) x y) ; if-then-else

• Case selection(case month

((sep apr jun nov) 30)

((feb) 28)

(else 31)

)


Defining FunctionsDefining Functions• (define ( name arguments ) function-body )• Examples:

(define (min x y) (if (< x y) x y))

(define (abs x) (if (< x 0) (- 0 x) x))

(define (factorial n)

(if (< n 1) 1 (* n (factorial (- n 1)))

))

• Note: be careful to match all parentheses.


The subst FunctionThe subst Function

(define (subst y x alist)

(if (null? alist) ‘())

(if (equal? x (car alist))

(cons y (subst y x (cdr alist)))

(cons (car alist) (subst y x (cdr alist)))

)))

E.g., (subst ‘x 2 ‘(1 (2 3) 2))

returns (1 (2 3) x)


Let ExpressionsLet Expressions

• Allows simplification of function definitions by defining intermediate expressions. E.g.,

(define (subst y x alist)

(if (null? alist) ‘()

(let ((head (car alist)) (tail (cdr alist)))

(if (equal? x head)

(cons y (subst y x tail))

(cons head (subst y x tail))

)))


Functions as argumentsFunctions as arguments

(define (mapcar fun alist)

(if (null? alist) ‘()

(cons (fun (car alist))

(mapcar fun (cdr alist)))

))

• E.g., if (define (square x) (* x x)) then(mapcar square ‘(2 3 5 7 9)) returns

(4 9 25 49 81)

F


Example: Symbolic DifferentiationExample: Symbolic Differentiation

• Symbolic Differentiation Rules

ddx

(c) 0 c is a constant

d

dx(x) 1

ddx

(u v) dudx

dvdx

u and v are functions of x

d

dx(u v)

du

dxdv

dxddx

(uv) u dvdx

v dudx

d

dx(u /v) v

du

dx udv

dx

/v 2


Scheme EncodingScheme Encoding

1. Uses Cambridge Prefix notationE.g., 2x + 1 is written as (+ (* 2 x) 1)

2. Function diff incorporates these rules.E.g., (diff ‘x ‘(+ (* 2 x) 1))

should give an answer.

3. However, no simplification is performed.E.g. the answer for (diff ‘x ‘(+ (* 2 x) 1)) is

(+ (+ (* 2 1) (* x 0)) 0) which is equivalent to the simplified answer, 2.


Scheme ProgramScheme Program

(define (diff x expr)(if (not (list? Expr))

(if (equal? x expr) 1 0)(let ((u (cadr expr)) (v (caddr expr)))

(case (car expr)((+) (list ‘+ (diff x u) (diff x v)))((-) (list ‘- (diff x u) (diff x v)))((*) (list ‘+ (list ‘* u (diff x v))

(list ‘* v (diff x u))))((/) (list ‘div (list ‘- (list ‘* v (diff x

u)) (list ‘* u (diff x v)))

(list ‘* u v)))))))


Trace of the ExampleTrace of the Example(diff ‘x ‘(+ ‘(* 2 x) 1))

= (list ‘+ (diff ‘x ‘(*2 x)) (diff ‘x 1))

= (list ‘+ (list ‘+ (list ‘* 2 (diff ‘x ‘x))

(list ‘* x (diff ‘x 2)))

(diff ‘x 1))

= (list ‘+ (list ‘+ (list ‘* 2 1) (list ‘* x (diff ‘x 2)))

(diff ‘x 1))

= (list ‘+ (list ‘+ ‘(* 2 1) (list ‘* x (diff ‘x 2)))

(diff ‘x 1))

= (list ‘+ (list ‘+ ‘(* 2 1) (list ‘* x (diff ‘x 2)))

(diff ‘x 1))

= (list ‘+ (list ‘+ ‘(* 2 1) (list ‘* x 0)) 0)

= ‘(+ (+ (* 2 1) (* x 0)) 0)


Example: Eight QueensExample: Eight Queens

• A backtracking algorithm for which each trial move’s:1. Row must not be occupied,

2. Row and column’s SW diagonal must not be occupied, and

3. Row and column’s SE diagonal must not be occupied.

• If a trial move fails any of these tests, the program backtracks and tries another. The process continues until each row has a queen (or until all moves have been tried).


Checking for a Valid MoveChecking for a Valid Move

(define (valid move soln)(let ((col (length (cons move soln))))

(and (not (member move soln))(not (member (seDiag move col) (selist

soln)))(not (member (swDiag move col) (swlist

soln))))))

Note: the and encodes the three rules listed on the previous slide.


Representing the Developing SolutionRepresenting the Developing Solution

• Positions of the queens kept in a list soln whose nth entry gives the row position of the queen in column n, in reverse order. E.g., soln = (5 3 1) represents queens in (row, column) positions (1,1), (3,2), and (5,3); i.e., see previous slide.

• End of the game occurs when soln has N (= 8) entries:• (define (done soln) (>= (length soln) N))

• Continuing the game tests hasmore and generates nextmove: (define (hasmore move) (<= move N))

•(define (nextmove move) (+ move 1)


Generating Trial MovesGenerating Trial Moves

(define (trywh move soln) (if (and (hasmore move) (not (car soln)))

(let ((atry (tryone move (cdr soln))))(if (car atry) atry (trywh (nextmove move)

soln)))soln

))

The try function sets up the first move:

(define (try soln) (trywh 1 (cons #f soln)))


Trying One MoveTrying One Move(define (tryone move soln)

(let ((xsoln (cons move soln))) (if (valid move soln) (if (done xsoln)

(cons #t xsoln)(try xsoln))

(cons #f soln))))

• Note: the #t or #f reveals whether the solution is complete.


SW and SE DiagonalsSW and SE Diagonals(define (swdiag row col) (+ row col))(define (sediag row col) (- row col))(define (swlist alist)

(if (null? Alist) ‘()(cons (swDiag (car alist) (length alist))(swlist (cdr alist)))))

(define (selist alist) (if (null? Alist) ‘()(cons (seDiag (car alist) (length alist))

(selist (cdr alist)))))

CSC321: Programming Languages14-1 Programming Languages Tucker and Noonan Chapter 14: Functional...

Documents

Transcript of CSC321: Programming Languages14-1 Programming Languages Tucker and Noonan Chapter 14: Functional...