1. Loop Invariants
CSC 171 FALL 2001
LECTURE 6

2. History: Herman Hollerith
Herman is said to have been a bright and able child at school, but had an inability to learn spelling easily. His determined teacher made his life miserable to the extent that he used to avoid school whenever possible and run away when his teacher showed renewed effort to improve his spelling.

3. History: Herman Tabulating
Herman Hollerith won the competition for the processing equipment to assist in the 1890 US Census
The Hollerith Tabulating Company, eventually became the Calculating-Tabulating-Recording (C-T-R) company in 1914, and eventually was renamed IBM in 1924.

4. Loop invariants
In order to verify loops we often establish an assertion (boolean expression) that is true each time we reach a specific point in the loop.
We call this assertion, a loop invariant

5. Assertions
When ever the program reaches the top of the while loop, the assertion is true
INIT
BODY
INVARIANT
TEST

6. Example Write a method to compute an
public static int power(int a , int n)
You have 5 minutes

7. Possible solution public static double power(int a, int n) {
double r = 1; double b = a; int i = n ;
while (i>0){
if (i%2 == 0) { b = b * b; i = i / 2;}
else { r = r * b; i--; }
return r; }

8. Does it work? b i r a 100 1 a2 50 a4 25 24 a8 12 a16 6 a32 3 2 a36 a64
a100
Does it work?
SURE! TRUST ME!
Well, look at a100 if you don’t believe me!
Note, less loops!
Can you “prove” that it works?

9. What is the loop invariant?
At the top of the while loop, it is true that
r*bi = an
It is?
Well, at the top of the first loop
r==1
b==a
i==n

10. So, if it’s true at the start
Even case
rnew= rold
bnew == (bold)2
inew==(iold)/2
Therefore,
rnew * (bnew)i-new == rold * ((bold)2)i-old/2
== rold * ((bold)2)i-old
== an

11. So, if it’s true at the start II
Odd case
rnew= rold*bold
bnew == bold
inew==iold-1
Therefore,
rnew * (bnew)i-new == rold *bold* (bold)i-old-1
== rold * (bold)i-old
== an

12. r*bi = an So, If it’s true at the start
And every time in the loop, it remains true
Then, it is true at the end
r*bi = an
And, i == 0 ( the loop ended)
What do we know?

13. Correctness Proofs Proof are more valuable than testing
Tests demonstrate limited correctness
Proofs demonstrate correctness for all inputs
For some time, people hoped that all formal logic would replace programming
The naïve idea that “programming is a form of math” proved to be an oversimplification

14. Correctness Proofs
Unfortunately, in practice, these methods never worked very well.
Instead of buggy programs,
people wrote buggy logic
Nonetheless, the approach is useful for program analysis

15. The take away message?
In the end, engineering and (process) management are at least as important as mathematics and logic for the successful completion of large software projects