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I am currently in an introductory programming class and was asked to interpret the final state of the given tracing table featuring a loop invariant. The blue numbers in the bottom box are the given answers.

As you can see, low is updated but I don't know how this number was achieved given the information in the contract.

Tracing table

Edit: Is a correct method to solving this finding the "boundary" values where the expressions in the @maintains clause hold true?

Since n doesn't change, high at the least can be 81, and low at the greatest is 9?

itscomcast
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1 Answers1

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So this would be confusing if you didn't understand the full definition of the loop invariant, like I did.

A loop invariant is a conditional statement that is true at the beginning of the method, before and after each iteration, and at the end of the method. Otherwise, it is not a valid loop invariant.

Knowing variables low and high are updated, we know their values have a chance of being changed in the method. n, on the other hand, will remain the same.

The loop invariants are low * low <= high and low * low <= n <= high. These must hold true at the end of the method body.

The loop only runs when low * low < high, i.e., the loop stops once low * low >= high. But, low * low > high cannot happen due to our loop invariant (equals-to symbol omitted for clarity)!

So following the rule that the loop invariant must hold, and knowing when the while loop stops, it's concluded that the final values must be:

low * low = high.

Factor in n and we have low * low = n = high. Again, we know low and high are the only values that have a chance of being changed.

Finally, we have low * low = 81 = high.

low = 9, n = 81, high = 81.

itscomcast
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