Introduction to Functional Programming in Racket CS 270

Introduction to Functional Programming in Racket CS 270 Math Foundations of CS Jeremy Johnson 1

Objective v To introduce functional programming in racket v Programs are functions and their semantics involve function application. Programs may also produce function by returning functions as values. In pure functional programming, this is it, there are no variables, side effects, nor loops. This simplifies semantics but does not reduce computational power. v We will investigate the style of programming this implies and how to model the semantics of such programs. 2

Outline 1. Syntax and semantics 2. Functional programming 1. Programs are functions – for every input there is a unique output (referential transparency) 2. No variables no assignment and no loops 3. Use recursion for iteration 4. Functions are first class objects 1. Pass as arguments and return as values 3. Racket language and Dr. Racket IDE 3

A Pure Functional Language x 1 = y 1, …, xn=yn f(x 1, …, xn) = f(y 1, …, yn) No side-effects, no assignments, no state, no loops Use recursion instead of iteration Still Turing complete Makes reasoning about programs easier 4

C++ Function with Side-Effects #include <iostream> using namespace std; % g++ count. c int cc() %. /a. out { static int x = 0; return ++x; cc() = 1 cc() = 2 cc() = 3 } int main() { cout << "cc() = " << cc() << endl; } 5

Syntax v Programs and data are lists – delimited by ( and ) or [ and ] and separated by space v S expressions (E 1 … En) v Special forms § § § § Self evaluating: numbers, Booleans, strings, … (quote expr) (if test-expr then-expr else-expr) (cond ([P 1 E 1] … [Pt Et])) (lambda (p 1 … pn) E 1 … Et) (define name E) 6 (let ([b 1 v 1] … [bt vt] E)

Semantics v To evaluate (E 1 E 2. . . En), recursively evaluate E 1, E 2, . . . , En - E 1 should evaluate to a function - and then apply the function value of E 1 to the arguments given by the values of E 2, . . . , En. v In the base case, there are self evaluating expressions (e. g. numbers and symbols). In addition, various special forms such as quote and if must be handled separately. 7

Read-Eval-Print-Loop (REPL) v Dr. Racket IDE (racket-lang. org) Definition Window Click Run to load and run definitions Interaction Window Enter expressions at the prompt (REPL) 8

Example Evaluation

Booleans and Predicates • Boolean constants: #t and #f (= 2 3) => #f (or (= 2 3) (not (= 2 3))) => #t (and #t #t #t) => #t • Predicates are Boolean functions • Convention is name? (equal? 2 3) => #f (eq? 2 3) => #f (number? 2) => #t (boolean? (and #t #f)) => #t

Conditional v (if test-expr then-expr else-expr) § Evaluate test-expr if not #f evaluate and return then-expr else evaluate and return else -expr (if (< 2 3) 0 1) => 0 (if (< 3 2) 0 1) => 1 (if (= 3 (+ 2 1)) 0 1) => 0 (if (or (= 2 3) (= 3 3)) (+2 3) (+ 3 3)) => 5

Conditional v (cond [test-expr 1 then-body 1] [test-exprn then-bodyn] [else then-body]) v Evaluate test-expr 1 if #f then goto next case otherwise return then-body 1. The else case always returns then-body (cond [(= 2 3) 2] [(= 3 4) 3] [else 4]) => 4

List Processing Functions (null? ‘()) => #t (null? ‘(1 2 3)) => #f (car ‘(1 2 3)) => 1 ; same as (first ‘(1 2 3)) (cdr ‘(1 2 3)) => ‘(2 3) ; same as (rest ‘(1 2 3)) (cons 1 ‘()) => ‘(1) (cons 1 ‘(2 3)) => ‘(1 2 3) (cons 1 (cons 2 (cons 3 '()))) => ‘(1 2 3) (cons 1 ‘()) ‘(2 3)) => ‘((1) 2 3)

Lambda Expressions v (lambda (parameters) body) § Evaluates to a function § When applied the actual arguments are substituted for the formal parameters into the body which is then evaluated and returned (lambda (x) (* x x)) => #<procedure> ((lambda (x) (* x x)) 2) => 4 (define sqr (lambda (x) (* x x))) (define (sqr x) (* x x)) ; shorthand for above (sqr 2) => 4

Recursion v In a functional language there are no side effects, hence no assignment and no loops. v All control must be done through recursion (define (fact n) (if (= n 0) 1 (* n (fact (- n 1))))) (fact 3) => 6 (define (ones n) (if (= n 0) '() (cons 1 (ones (- n 1))))) (ones 3) => ‘(1 1 1)

Trace Recursion (define (fact n) (if (= n 0) 1 (* n (fact (- n 1))))) v (fact 0) = 1 v (fact 3) = (* 3 (fact 2)) = (* 3 (* 2 (fact 1))) = (* 3 (* 2 (* 1 (fact 0)))) = (* 3 (* 2 (* 1 1))) = 6 v When n=0 [base case] no recursion v When n>0 [recursive case] recursion occurs

Recursion

Tail Recursion v A tail recursive function is a function where the recursive call is the last operation. Such procedures can easily be converted to loops. (define (fact n) (if (= n 0) 1 (* n (fact (- n 1))))) (define (factt n sofar) (if (= n 0) sofar (factt (- n 1) (* n sofar))))) (fact n) = (factt n 1)

Tail Recursion v An equivalent loop can be constructed, which updates the arguments each iteration of the loop. for (; ; ){ if (n == 0) return sofar; else { t 1 = n - 1; t 2 = sofar * n; n = t 1; sofar = t 2; } }

Testing v Test cases give examples of what a function should compute if implemented properly. They can be used for debugging. v v (fact 3) = 6 (fact 2) = 2 (fact 1) = 1 (fact 0) = 1

Unit Testing in Racket (require rackunit) (require rackunit/text-ui) (define-test-suite fact-suite (check-equal? (fact 0) 1) (check-equal? (fact 1) 1) (check-equal? (fact 2) 2) (check-equal? (fact 3) 6) ) (run-tests fact-suite 'verbose) 4 success(es) 0 failure(s) 0 error(s) 4 test(s) run 0

Higher Order Functions sort: (sort '(4 3 2 1) <) => (1 2 3 4) (sort '("one" "two" "three" "four") string<? ) => '("four" "one" "three" "two") map: (map sqr '(1 2 3 4)) => ‘(1 4 9 16) 22

Higher Order Functions filter: (filter odd? '(1 2 3 4 5)) => ‘(1 3 5) (filter even? ‘(1 2 3 4 5)) => ‘(2 4) fold: (foldr cons '() '(1 2 3 4)) => ‘(1 2 3 4) (foldr list '() '(1 2 3 4)) => '(1 (2 (3 (4 ())))) (foldr + 0 '(1 2 3 4)) => 10 (foldl cons ‘() ‘(1 2 3 4)) => ‘(4 3 2 1) (foldl list '() '(1 2 3 4)) => '(4 (3 (2 (1 ())))) (foldl * 1 ‘(1 2 3 4)) => 24 23

Functions that Return Functions • Make-adder (define (make-adder x) (lambda (y) (+ x y))) (define add 1 (make-adder 1)) (add 1 3) => 4 (define (make-multiplier x) (lambda (y) (* x y))) (define double (make-multiplier 2)) (double 3) => 6 24

Function Composition (define (compose f g) (lambda (x) (f (g x)))) (define add 2 (compose add 1)) (add 2 3) => 5 (define getsecond (compose first rest)) (getsecond ‘(1 2 3 4 5)) => 2 25

Currying (define (curry f a) (lambda (b) (f a b))) (define add 1 (curry + 1)) (add 1 3) => 4 (define double (curry * 2)) (doulble 3) => 6 26