1. Copyright © 2006 The McGraw-Hill Companies, Inc. Programming Languages 2nd edition Tucker and Noonan Chapter 14 Functional Programming It is better to have 100 functions operate one one data structure, than 10 functions on 10 data structures. A. Perlis

2. Copyright © 2006 The McGraw-Hill Companies, Inc. Contents 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 14.3 Haskell

3. Copyright © 2006 The McGraw-Hill Companies, Inc. 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.

4. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.1 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.

5. Copyright © 2006 The McGraw-Hill Companies, Inc. 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

6. Copyright © 2006 The McGraw-Hill Companies, Inc. 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

7. Copyright © 2006 The McGraw-Hill Companies, Inc. 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]

8. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2 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

9. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.1 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 ;

10. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.2 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

11. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.3 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:

12. Copyright © 2006 The McGraw-Hill Companies, Inc. 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!

13. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.4 Elementary Values Numbers integers floats rationals Symbols Characters Functions Strings (list? evens) (symbol? ‘evens)

14. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.5 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) )

15. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.6 Defining Functions ( define ( name arguments ) function-body ) (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.

16. Copyright © 2006 The McGraw-Hill Companies, Inc. 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)

17. Copyright © 2006 The McGraw-Hill Companies, Inc. 14.2.7 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)) )))

18. Copyright © 2006 The McGraw-Hill Companies, Inc. 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