What is meant by loop invariant?

In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration. It is a logical assertion, sometimes checked within the code by an assertion call.

What is an invariant in induction?

Induction step In the induction step, we know the invariant holds after t iterations, and want to show it still holds after the next iteration. We start by stating all the things we know: 4 Page 5 1. The invariant holds for the values of the variables at the start of the next iteration.

What is loop invariant proof?

Loop invariant proofs provide you with a very structured way of translating your intuition into something solid. Let us start with a very simple example. Consider the following computational problem: given an array A (of size n) of numbers, output the sum of the numbers in A.

What is a loop invariant explain with an example?

Loop invariant condition is a condition about the relationship between the variables of our program which is definitely true immediately before and immediately after each iteration of the loop. For example: Consider an array A{7, 5, 3, 10, 2, 6} with 6 elements and we have to find maximum element max in the array.

When does the invariant hold in a loop?

Termination: When the loop terminates, the invariant gives us a useful property that helps show that the algorithm is correct. When the first two properties hold, the loop invariant is true prior to every iteration of the loop.

Is the loop invariant the same as the inductive hypothesis?

Because of the similarity of loops and recursive programs, proving partial correctness of loops with invariants is very similar to proving correctness of recursive programs via induction. In fact, the loop invariant is often the same as the inductive hypothesis to be proved for a recursive program equivalent to a given loop.

Can a loop invariant prove the correctness of an algorithm?

(Note that this says nothing about its truth or falsity part way through an iteration.) By itself, a loop invariant doesn’t do much. However, given an appropriate invariant, it can be used to help prove the correctness of an algorithm. The simple example in CLRS probably has to do with sorting.

How to use induction to prove correctness?

2 Loop invariants Induction is the method of choice for analyzing properties of algorithms with loops. A typical such algorithm: 1 Initialization The algorithm initializes some variables based on the inputs. Loop Then the loop starts. Each iteration of the loop changes variables ac- cording to some loop instructions until a guard condition fails.