Download Propositional Calculus

Introduction to Functional
Programming in Racket
CS 270 Math Foundations of CS
Jeremy Johnson
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Objective
• To introduce functional programming in racket
• 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.
• We will investigate the style of programming this
implies, and how to model the semantics of such
programs.
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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 control
4. Functions are first class objects
1.
Pass as arguments and return as values
5. Simple semantics [value semantics] (no side
effects, referential transparency)
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A Pure Functional Language
x1 = y1,…,xn=yn  f(x1,…,xn) = f(y1,…,yn)
No side-effects, no assignments, no state, no
loops
Use recursion instead of iteration
Still Turing complete
Makes reasoning about programs easier
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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;
cout << "cc() = " << cc() << endl;
cout << "cc() = " << cc() << endl;
}
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Syntax
• Programs and data are lists – delimited by
( and ) or [ and ] and separated by space
• S expressions (E1 … En)
• Special forms
•
•
•
•
•
•
•
Self evaluating: numbers, Booleans, strings, …
(quote expr)
(if test-expr then-expr else-expr)
(cond ([P1 E1] … [Pt Et]))
(lambda (p1 … pn) E1 … Et)
(define name E)
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(let ([b1 v1] … [bt vt] E)
Semantics
• To evaluate (E1 E2 ... En), recursively evaluate
E1, E2,...,En - E1 should evaluate to a function and then apply the function value of E1 to the
arguments given by the values of E2,...,En.
• 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.
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Read-Eval-Print-Loop (REPL)
• Dr. Racket IDE
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Example Evaluation
 2 => 2
 (/ 4 6) =>







2
3
+ => #<procedure:+>
(+ 2 (* 3 4)) => (+ 2 12) => 14
(max 1 2 3) => 3
(1 2 3) => error
(list 1 2 3) => ‘(1 2 3)
(list 1 (2 3) 4) => error
(list 1 (list 2 3) 4) => ‘(1 (2 3) 4)
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
• (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
• (cond [test-expr1 then-body1]
[test-exprn then-bodyn]
[else then-body])
• Evaluate test-expr1 if #f then goto next
case otherwise return then-body1. 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
(cdr ‘(1 2 3)) => ‘(2 3)
(cons 1 ‘()) => ‘(1)
(cons 1 ‘(2 3)) => ‘(1 2 3)
(cons 1 (cons 2 (cons 3 '()))) => ‘(1 2 3)
(cons (cons 1 ‘()) ‘(2 3)) => ‘((1) 2 3)
Lambda Expressions
• (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))
(sqr 2) => 4
Recursion
• In a functional language there are no side
effects, hence no assignment and no loops.
• 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)
Tail Recursion
• A tail recursive function is a function where
the recursive call is the last operation. Such
procedures can easily be converted to
loops.
 (define (sum n)
(if (zero? n) 0 (+ n (sum (- n 1))))))
 (define (sumt n sofar)
 (if (zero? n) sofar
(sumt (- n 1) (+ n sofar)))))
Tail Recursion
• An equivalent loop can be constructed,
which updates the arguments each
iteration of the loop.
for (;;){
if (n == 0)
return sofar;
else {
t1 = n - 1;
t2 = sofar + n;
n = t1;
sofar = t2; } }
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)
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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
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Functions that Return Functions
• Make-adder
 (define (make-adder x) (lambda (y) (+ x y)))
 (define add1 (make-adder 1))
 (add1 3) => 4
 (define (make-multiplier x) (lambda (y) (* x y)))
 (define double (make-multiplier 2))
 (double 3) => 6
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Function Composition

(define (compose f g) (lambda (x) (f (g x))))


(define add2 (compose add1 add1))
(add2 3) => 5


(define getsecond (compose car cdr))
(getsecond ‘(1 2 3 4 5)) => 2
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Currying
 (define (curry f a) (lambda (b) (f a b)))
 (define add1 (curry + 1))
 (add1 3) => 4
 (define double (curry * 2))
 (doulble 3) => 6
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Tower of Hanoi
 (define (move from to)
(list (string-append from to)))
 (define (hanoi n from to using)
(if (= n 1)
(move from to)
(append (hanoi (- n 1) from using to)
(move from to)
(hanoi (- n 1) using to from))))
 (hanoi 3 "a" "b" "c")
'("ab" "ac" "bc" "ab" "ca" "cb" "ab")
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