Question

Prove the correctness of the following algorithm ​​Use this structure.

1. State the loop invariant.
2. Prove the loop invariant.
3. Use the loop invariant to prove postcondition s satisfied.
4. Show that the algorithm terminates.

Answer 1

To prove an algorithm's correctness, one must establish and verify a loop invariant, demonstrate its maintenance, show that the loop's termination ensures the postcondition, and confirm that the algorithm indeed terminates.

Step-by-step explanation:

The question is asking to prove the correctness of an algorithm using loop invariants. To do this, you'd typically follow a structured approach:

1. State the loop invariant: Identify a property that holds before and after each iteration of the loop.
2. Prove the loop invariant: Demonstrate that the property is true before the loop runs for the first time (initialization), and that if it's true before an iteration of the loop, it remains true after the iteration (maintenance).
3. Use the loop invariant to prove postcondition is satisfied: Show that, when the loop terminates, the invariant provides a proof that the algorithm has fulfilled its specifications (termination).
4. Show that the algorithm terminates: Demonstrate that the loop cannot go on indefinitely, usually by showing that some value is getting closer to a boundary with each iteration.

To provide a concrete example, let’s consider a simple algorithm that finds the maximum value in an array:

max = A[0]
for i from 1 to n-1 :
if A[i] > max:
max = A[i]

For this algorithm:

1. The loop invariant could be "At the start of each iteration, max contains the largest value of the subarray A[0..i-1]."
2. To prove the loop invariant, we show it holds true before the first iteration (initialization) when max is the first element and there's no other element to compare. It's maintained because at each step, if the current element is greater than max, max is updated (maintenance).
3. The loop invariant, along with the fact that the algorithm goes through the full range of elements from 0 to n-1, ensures that at the end, the maximum value in the entire array is stored in max (termination).
4. This algorithm clearly terminates because the loop counter i is incremented each time, and the loop will cease when i reaches n-1.