Programming Languages 2nd edition Tucker and Noonan

1. Programming Languages2nd editionTucker 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. 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. 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. 14.1 Functions 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.

5. 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 . x2)5) = 52

6. Function 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

7. 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. 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. 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. 14.2.2 Expression Evaluation • Three steps: • Replace names of symbols by their current bindings. • Evaluate lists as function calls in Cambridge prefix. • Constants evaluate to themselves. E.g., x ; evaluates to 5 (+ (* x 4) (- 6 2)) ; evaluates to 24 • ; evaluates to 5 ‘red ; evaluates to ‘red

11. 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. 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. 14.2.4 Elementary Values Numbers integers floats rationals Symbols Characters Functions Strings • (list? evens) • (symbol? ‘evens)

14. 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) (else 31) )

15. 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. 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. 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. Functions as arguments F • (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)