Transcript Proofs, Recursion and Analysis of Algorithms

Proofs, Recursion, and Analysis of
Algorithms
Mathematical Structures
for Computer Science
Chapter 2
Copyright © 2006 W.H. Freeman & Co.
MSCS Slides
Proofs, Recursion, and Analysis of
Loop Rule
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Suppose that si is a loop statement in the form:
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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} si {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.
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Section 2.3
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Euclidean Algorithm
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Section 2.3
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.
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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 2.3
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Euclidean Algorithm Proof
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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(in, jn) = gcd(a, b) for all n  0.
Q(0) is gcd(i0, j0) = 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(ik, jk) = gcd(a, b).
Show Q(k + 1): gcd(ik + 1, jk + 1) = gcd(a, b).
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Section 2.3
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Euclidean Algorithm Proof
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By the assignment statements within the loop body, we
know that
ik + 1 = jk
j k + 1 = rk
Then, by the additional fact on the previous slide:
gcd(ik + 1, jk + 1) = gcd(jk, rk) = gcd(ik, jk)
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.
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Section 2.3
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