GATE CSE 1991 | Question: 1,vi

Question

Consider the following PASCAL program segment:

if i mod 2 = 0 then
    while i >= 0 do 
    begin
        i := i div 2;
        if i mod 2 < > 0  then i := i - 1;
        else i := i – 2;
    end;

An appropriate loop-invariant for the while-loop is ________

Comments

yes in syllabus .. syllabus is not defined as such .so you need to read this topic.

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thank u sir

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is it in GATE 2027 CS syllabus?

Answer 1

Loop invariant is some condition which holds at the end of each iteration of the loop. i.e. it is "invariant" => does not vary (or change). It might change inside one iteration, but it will be true at the end of every iteration.

We often use loop invariants to prove that our algorithm works correctly.

In the given program, a loop invariant is:

i mod 2 = 0

i.e. i is even after every iteration.

One can verify this as follows:

• Before the execution of first iteration the loop invariant is true, because of this line of code: 

  if i mod 2 = 0 then

• In every iteration, we divide i by $2$, so now i will be either odd or even. 
  • If odd, we subtract 1 from i 

    if i mod 2 < > 0  then i := i - 1; 

  • so it's now even.
  • otherwise, if even, we subtract 2 from i 

    else i := i – 2;

  • so, it remains even
• So, at the end of every iteration i remains even.

https://stackoverflow.com/questions/3221577/what-is-a-loop-invariant

Comments

https://m.youtube.com/watch?v=ysk1Ef9kxpo

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this link is not working

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meaning of symbol 

< >

is not equal to 

Answer 2

A loop invariant is a condition that is always be same before the loop starts , while in the loop and after the loop ends for each iteration.

Here $i \mod2 = 0$ is the loop invariant.

Comments

loop invariants are true even if the loop condition is false ...

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Recently I studied that loop invariant may be or may not be true during iteration of the loop. Is this statement correct?

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@puja

Yes, that will happen after last iteration.

Answer 3

A loop invariant is a condition [among program variables] that is necessarily true immediately before and immediately after each iteration of a loop. (Note that this says nothing about its truth or falsity part way through an iteration.)

Source: http://www.cs.uofs.edu/~mccloske/courses/cmps144/invariants_lec.html

In simple words, a loop invariant is some predicate (condition) that holds for every iteration of the loop.

• For example, let's look at a simple for loop that looks like this:

                int j = 9;
                for(int i=0; i<10; i++)  
                    j--;

In this example it is true (for every iteration) that i + j == 9. A weaker invariant that is also true is that i >= 0 && i < 10 (because this is the continuation condition!) or that j <= 9 && j >= 0.

• The loop invariants will be true on entry into a loop and following each iteration, so that on exit from the loop both the loop invariants and the loop termination condition can be guaranteed.

Solution : An appropriate loop-invariant for the while-loop is i mod 2 = 0
                (i.e. ' i ' must be even, else the loop breaks when ' i ' is odd)

Comments

No after termination it need not be true :

int j = 9;
for(int i=0; i<10; i++)  
  j--;

Loop invariant here is i+j= 9. at the end of termination i+j= 10.

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okay so how do we relate loop invariants to looping conditions??

+ i think in the code 

int j = 9; 
for(int i=0; i<10; i++)  
 j--; 

After the loop terminates i=10 and j=-1 i+j=10+(-1)=9. 

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$\textbf{i >= 0}$ is not a loop invariant. However, a $\underline{weaker}$ invariant can be $\textbf{i >= -2}$.

Answer 4

'i' is even and decrease in every iteration.