finding the loop invariant for an algorithm

Question

I am having some issues finding the invariant for the algorithm below.Also,I have to follow all the steps to proof how i find the specific invariant and I don't know how I can demonstrate that. I saw that this algorithm is a multiplication by adding.

The algorithm is:

alg1(integer a,b)
 x<-a
 y<-b
 z<-0
 while y>0 do
   z<-z+x
   y<-y-1
 end while
 return z

I hope someone can help share some light on this for me, as the similar cases I've found in here, have not been sufficient.

Thanks alot in advance for your time.

There is not *the* loop invariant. There are many. But only few (or maybe even only one) that is useful for a correctness prove. — MrSmith42, Nov 17 '20 at 14:45

score 0 · Answer 1 · answered Nov 17 '20 at 15:05

0

Hint:

The body of the loop shows that z grows from zero by increments x, while y decreases to zero by units. Hence, z + x y is probably constant...

answered Nov 17 '20 at 15:05

amol goel · Answer 2 · 2020-12-20T07:53:30.013

0

Loop invariant: At the start of kth iteration, value of z is k-1 multiplied by x.

Proving correctness:
Initialization
At beginning, k=1, so

z= (1-1) * x
z= 0 *z=0

Maintenance
If at start of kth iteration, z =(k-1)x where k-1< n then within loop z gets multiplied by z and the end of iteration z =kx. This shows loop invariant holds true for k+1 th iteration.

Termination:
At termination,

k=y+1: 
z=(y+1)-1 * x
z=yx

z becomes product of y and x

edited Dec 20 '20 at 07:53

answered Dec 20 '20 at 06:45

amol goel

149
3

a little desciption needed, i didnt nderstand – Abilogos Dec 20 '20 at 07:04

finding the loop invariant for an algorithm

2 Answers2