1. CSC321: Programming Languages14-1 Programming Languages Tucker and Noonan Chapter 14: Functional Programming 14.1 Functions and the Lambda Calculus 14.2 Scheme 14.2.1 Expressions 14.2.2 Expression Evaluation 14.2.3 Lists 14.2.4 Elementary Values 14.2.5 Control Flow 14.2.6 Defining Functions 14.2.7 Let Expressions 14.2.8 Example: Semantics of Clite 14.2.9 Example: Symbolic Differentiation 14.2.10 Example: Eight Queens

2. CSC321: Programming Languages14-2 Overview 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.

3. CSC321: Programming Languages14-3 Functions and the Lambda Calculus The function Square has R (the reals) as domain and range. –Square : R  R –Square(n) = n 2 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. x 2 ) represents the Square function.

4. CSC321: Programming Languages14-4 Properties 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. x 2 )5) = 5 2

5. CSC321: Programming Languages14-5 Function Evaluation In pure lambda calculus, expressions like (( x. x 2 )5) = 5 2 are uninterpreted. In a functional language, (( x. x 2 )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

6. CSC321: Programming Languages14-6 Status 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]

7. CSC321: Programming Languages14-7 Scheme 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

8. CSC321: Programming Languages14-8 Expressions 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 ;

9. CSC321: Programming Languages14-9 Expression 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

10. CSC321: Programming Languages14-10 Lists 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:

11. CSC321: Programming Languages14-11 List 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!

12. CSC321: Programming Languages14-12 Elementary Values Numbers integers floats rationals Symbols Characters Functions Strings (list? evens) (symbol? ‘evens)

13. CSC321: Programming Languages14-13 Control 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) (else31) )

14. CSC321: Programming Languages14-14 Defining 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.

15. CSC321: Programming Languages14-15 The 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)

16. CSC321: Programming Languages14-16 Let 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)) )))

17. CSC321: Programming Languages14-17 Functions 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

18. CSC321: Programming Languages14-18 Example: Symbolic Differentiation Symbolic Differentiation Rules

19. CSC321: Programming Languages14-19 Scheme Encoding 1.Uses Cambridge Prefix notation E.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.

20. CSC321: Programming Languages14-20 Scheme 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))) ))))

21. CSC321: Programming Languages14-21 Trace 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)

22. CSC321: Programming Languages14-22 Example: 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).

23. CSC321: Programming Languages14-23 Checking 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.

24. CSC321: Programming Languages14-24 Representing 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)

25. CSC321: Programming Languages14-25 Generating 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)))

26. CSC321: Programming Languages14-26 Trying 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.

27. CSC321: Programming Languages14-27 SW 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)))))