Proofs, Recursion, and Analysis of Algorithms Mathematical Structures for Computer Science Chapter 2 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesProofs, Recursion, and Analysis of Algorithms

Section 2.3More on Proof of Correctness1 Loop Rule Suppose that s i is a loop statement in the form: while condition B do P end while where B is a condition that is either true or false and P is a program segment. The precondition Q holds before the loop is entered and after it terminates. The Loop Rule of inference states that we can derive {Q} s i {Q Λ B} from {Q Λ B} P {Q} Because it may not be known exactly when the loop will terminate, Q must remain true after each iteration through the loop. Q represents a predicate, or relation, among the values of the program variables. The loop invariant is the relation among these variables unaffected by the action of the loop iteration.

Section 2.3More on Proof of Correctness2 Euclidean Algorithm The Euclidean algorithm finds the greatest common divisor of two positive integers a and b. The greatest common divisor of a and b, denoted by gcd(a, b), is the largest integer n such that n|a and n|b. For example, gcd(12, 18) = 6 and gcd(420, 66) = 6. The Euclidean algorithm works by a succession of divisions. To find gcd(a, b), assuming that a >= b, you first divide a by b, getting a quotient and a remainder. Next, you divide the divisor, b, by the remainder and keep doing this until the remainder is 0, at which point the greatest common divisor is the last divisor used.

Section 2.3More on Proof of Correctness3 Euclidean Algorithm ALGORITHM: Euclidean algorithm GCD(positive integer a; positive integer b) //a  b Local variables: integers i, j i = a j = b while j != 0 do compute i = qj + r, 0  r < j i = j j = r end while //i now has the value gcd(a, b) return i; end function GCD

Section 2.3More on Proof of Correctness4 Euclidean Algorithm Proof To prove the correctness of this function, we need one additional fact: (  integers a, b, q, r)[(a = qb + r)  (gcd(a, b) = gcd(b, r))] Using function GCD, we will prove the loop invariant Q: gcd(i, j) = gcd(a, b) and evaluate Q when the loop terminates. We use induction to prove: Q(n): gcd(i n, j n ) = gcd(a, b) for all n  0. Q(0) is gcd(i 0, j 0 ) = gcd(a, b) is true because when we first get to the loop statement, i and j have the values a and b. Assume Q(k): gcd(i k, j k ) = gcd(a, b). Show Q(k + 1): gcd(i k + 1, j k + 1 ) = gcd(a, b).

Section 2.3More on Proof of Correctness5 Euclidean Algorithm Proof By the assignment statements within the loop body, we know that i k + 1 = j k j k + 1 = r k Then, by the additional fact on the previous slide: gcd(i k + 1, j k + 1 ) = gcd(j k, r k ) = gcd(i k, j k ) By the inductive hypothesis, the above is equal to gcd(a, b) Q is therefore a loop invariant. At loop termination, gcd(i, j) = gcd(a, b) and j = 0, so gcd(i, 0) = gcd(a, b). But gcd(i, 0) is i, so i = gcd(a, b). Therefore, function GCD is correct.