Lecture 5

1. Lecture 5 Material in the textbook on pages 50-53 (1.2.6) 56-66 (1.3.1, 1.3.2) Of second edition מבוא מורחב

2. Primality Testing - II (define (prime? n) (= n (find-smallest-divisor n 2))) n is a prime iff its only divisors are 1 and n Iff it has no divisors between 2 and (sqrt n) (define (divides? a b) (= (remainder b a) 0)) (define (find-smallest-divisor n i) (cond ((> i (sqrt n)) n) ((divides? i n) i) (else (find-smallest-divisor n (+ i 1))))) מבוא מורחב

3. (Prime? 47) (= 47 (find-smallest-divisor 47 2)) (= 47 (cond (divides? 2 47) 2) (else (find-smallest-divisor 47 3)))) (= 47 (find-smallest-divisor 47 3)) (= 47 (find-smallest-divisor 47 4)) (= 47 (find-smallest-divisor 47 5)) (= 47 (find-smallest-divisor 47 6)) (= 47 (find-smallest-divisor 47 7)) (= 47 47) #t מבוא מורחב

4. Analysis • Correctness: If n is not a prime, then n=a * b for a,b>1. Then at least one of them is n. So n must have a divisor smaller then n. • Time complexity: • first test - (n) • second test - • (n) . For a number n, we test at most n • numbers to see if they divide n. If n is a 800 digit number, that’s very bad. Absolutely infeasible. מבוא מורחב

5. The Fermat Test: Do 400 times: Pick a random a < n and compute an (mod n) If  a then for sure n is not a prime. If all 400 tests passed, declare that n is a prime. The Fermat Primality Test Fermat’s little theorem: If n is a prime number then: an = a (mod n) for every 0 < a < n, integer מבוא מורחב

6. Computing ab (mod m) fast. (define (expmod a b m) ; computes ab (mod m) (cond ((= b 0) 1) ((even? b) (remainder (expmod (remainder (* a a) m) (/ b 2) m) m)) (else (remainder (* a (expmod a (- b 1) m)) m))))

7. Implementing Fermat test (define (test a n)(= (expmod a n n) a)) (define (one-test n) (test (+ 1 (random (- n 1))) n)) (define (many-tests n t); calls one-test t times (cond ((= t 0) true) ((one-test n) (many-test n (- t 1))) (else false))) מבוא מורחב

8. Time complexity To test if n is a prime. We run 400 tests. Each takes about log(n) multiplcations. T(n) = O(log n) מבוא מורחב

9. Correctness – I (prime numbers) Fermat’s theorem: Every prime will always pass the test. It therefore follows that if n is a prime then for every a, test(a n) is true, hence we always pass the test, And we declare n to be a prime. For a prime n: We are always right. מבוא מורחב

10. Correctness II – Carmichael numbers. If n is a Carmichael number we always pass the test, hence we always declare that n is prime. • Definition: A Carmichael number, is a number • such that • n is Composite, and • n always passes the test. • For every a, an = a (mod n) For a Carmichael number n: We are always wrong. מבוא מורחב

11. Correctness III – any other number A fact: If n is not prime and not a Carmichael number then: for at least half of the choices of a, an <> a (mod n). Hence, if we chose a at random, then with probability half the test fails and we declare that n is composite. The probability all 100 tests fail is at most 2-400 For such n: We are wrong with probability at most 2-400 מבוא מורחב

12. Correctness Suppose we do the test t=400 times. • If n is a prime we are never wrong. • If n is a Carmichael number, we are always wrong • If n is a composite number and not a Carmichael number • we are wrong with probability at most 2-400 . • Error probability smaller than the chance the hardware • is faulty. מבוא מורחב

13. A probabilistic algorithm An algorithm that uses random coins, and for every input gives the right answer with a good probability. Even though Carmichael numbers are very rare Fermat test is not good enough. There are inputs on which it is wrong. There are modifications of Fermat’s test, that for every input give the right answer, with a high probability. מבוא מורחב

14. Types so far • Numbers: 1, 7, 1.2 • Boolean: #t , #f • Strings: “this is a string” • Procedures: (< 2 3), (even? 7), (+ 6 3), (define (f x) (if (< x 0) “x is negative” “x is not negative”)) מבוא מורחב

15. Procedures have types • A procedure • may have requirements regarding the • number of its arguments, • may expect each argument to be of a certain type. The procedure + expects numbers as its arguments. Can not be applied on strings. The procedure < expects at least one argument. Will not accept strings as arguments. (< “abc” “xyz”) מבוא מורחב

16. Procedures have types The type of a procedure is a contract: • If the operands have the specified types,the procedure will result in a value of the specified type • otherwise, its behavior is undefined • maybe an error, maybe random behavior מבוא מורחב

17. number, number number two arguments,both numbers result value of integer-addis a number Example The type of the integer-add procedure is (+ 7 “xx”) - causes an error. מבוא מורחב

18. number, number, number number Boolean string Your turn • The following expressions evaluate to values of what type?(lambda (a b c) (if (> a 0) (+ b c) (- b c)))(lambda (p) (if p "hi" "bye"))(* 3.14 (* 2 5)) number

19. Types (summary) • type: a set of values • every value has a type • procedure types (types which include ) indicate • number of arguments required • type of each argument • type of result of the procedure מבוא מורחב

20. Can procedures get and return procedures? • Can a procedure return a procedure as its value? • Can a procedure get a procedure as an argument? • Can this be useful? מבוא מורחב

21. Consider the following three sums • 1 + 2 + … + 100 = (100 * 101)/2 • 1 + 4 + 9 + … + 1002 = (100 * 101 * 201)/6 • 1 + 1/32 + 1/52 + … + 1/1012 = p2/8 In mathematics they are all captured by the notion of a sum: מבוא מורחב

22. (define (sum-integers a b) (if (> a b) 0 (+ a (sum-integers (+ 1 a) b)))) (define (sum-squares a b) (if (> a b) 0 (+ (square a) (sum-squares (+ 1 a) b)))) (define (pi-sum a b) (if (> a b) 0 (+ (/ 1 (square a)) (pi-sum (+ a 2) b)))) Let’s have a look at the three programs (define (sum term a next b) (if (> a b) 0 (+ (term a) (sum term (next a) next b)))) מבוא מורחב

23. procedure procedure procedure Let’s check this new procedure out! (define (sum term a next b) (if (> a b) 0 (+ (term a) (sum term (next a) next b)))) What is the type of this procedure? (number  number, number, number number, number)  number מבוא מורחב

24. Higher order procedures A higher order procedure: takes a procedure as an argument or returns one as a value Examples: 1. (define (sum-integers1 a b) (sum (lambda (x) x) a (lambda (x) (+ x 1)) b)) 2. (define (sum-squares1 a b) (sum square a (lambda (x) (+ x 1)) b)) 3. (define (pi-sum1 a b) (sum (lambda (x) (/ 1 (square x))) a (lambda (x) (+ x 2)) b)) מבוא מורחב

25. Does it work? (define (sum term a next b) (if (> a b) 0 (+ (term a) (sum term (next a) next b)))) (sum square 1 (lambda (x) (+ x 1)) 100) (+ (square 1) (sum square ((lambda (x) (+ x 1)) 1) (lambda (x) (+ x 1)) 100)) (+ 1 (sum square 2 (lambda (x) (+ x 1)) 100)) (+ 1 (+ (square 2) (sum square 3 (lambda (x) (+ x 1)) 100))) (+ 1 (+ 4 (sum square 3 (lambda (x) (+ x 1)) 100))) (+ 1 (+ 4 (+ 9 (sum square 4 (lambda (x) (+ x 1)) 100))) מבוא מורחב

26. f dx b a Integration as a procedure Integration under a curve f is given roughly by dx (f(a) + f(a + dx) + f(a + 2dx) + … + f(b)) (define (integral f a b) (* (sum f a (lambda (x) (+ x dx)) b) dx)) (define dx 1.0e-3) (define atan (lambda (a) (integral (lambda (x) (/ 1 (+ 1 (square x)))) 0 a)))

27. A moment of reflection It is nice that procedures can be treated as any other value. It can help abstract our thinking as with the sum example. Sometime we would actually send a program rather than execute it. E.g., if the data is not under our control. In fact, it happens quite a lot with web (and other highly distributed) settings. It is nice we can send a mobile agent free to Wonder around and execute somewhere else. What does it cost us? מבוא מורחב

28. The syntactic sugar “Let” Suppose we wish to implement the function f(x,y) = x(1+x*y)2 + y(1-y) + (1+x*y)(1-y) We can also express this as a = 1+x*y b = 1-y f(x,y) = xa2 + yb + ab

29. The syntactic sugar “Let” (define (f x y) ((lambda (a b) (+ (* x (square a)) (* y b) (* a b))) (+ 1 (* x y)) (- 1 y))) (define (f x y) (define (f-helper a b) (+ (* x (square a)) (* y b) (* a b))) (f-helper (+ 1 (* x y)) (- 1 y))) (define (f x y) (let ((a (+ 1 (* x y))) (b (- 1 y))) (+ (* x (square a)) (* y b) (* a b))))

30. The syntactic sugar “Let” (Let ((<var1> <exp1>) (<var2> <exp2>) .. (<varn> <expn>)) <body>) ((lambda (<var1> ….. <varn>) <body>) <exp1> <exp2> … <expn>)