Many times when I've heard an instructor describe the loop invariant, it has simply been 'the loop condition', or something similar. Your example shows that the invariant can be much more. CLRS uses it to proove the correctness of a sorting algorithm.

This again can be weakened to give us the invariant. Let's summarize our analysis: If c is not present in s, then -1 is returned. This deduces the function's postcondition. For this reason, we must pause our study of programming logic for a study of symbolic logic, that is, the algebra of logical assertions.

The phrase reads better like this: The specification of the previous example now looks like this: To understand how it might be used, consider this variation of the previous example. For string, s, s[: If c is not in s, a copy of s is returned, unchanged.

Invariant discovery is an art form; there cannot exist a mechanical algorithm to do this. (This is a key result of computability theory, the study of what problems are mechanically solvable.) So, we now study how to discover loop invariants. Examples of invariant discovery A key intellectual task in programming is stating a loop's invariant. Invariant discovery is an art form; there cannot exist a mechanical algorithm to do this. (This is a key result of computability theory, the study of what problems are mechanically solvable.) So, we now study how to discover loop invariants. Examples of invariant discovery A key intellectual task in programming is stating a loop's invariant. loop invariant for simple algorithms. Ask Question. discrete-mathematics algorithms invariance. share | cite | improve this question. [i-1]) as the loop invariant, positioned immediately after the increment to i (last line of while loop). To make it even easier to prove that the invariant holds on each pass through the loop, you can.

You are welcome to deduce that the program meets its postcondition. Reuse the loop invariant from find in the previous exercise. But there is a technical problem variable index is a variable local to the function's body and is not part of the precondition nor is it part of the answer returned by delete it makes no sense to include it in the postcondition.

Worse yet, its presence can lead to false deductions at the point where the function is called! We do it like this: No answer is returned because the precondition is violated and the loop is unable to terminate. What if we ignored the function's precondition the proof of the loop remains the same?!

The deduction law for loops guarantees, if the loop terminates, then the postcondition must hold true. There can be silly applications of the loop law. Consider this faulty program: But the loop body preserves the invariant only because its body, pass, is too timid to make any progress at all towards the goal.

So, the loop never terminates. Now, if the loop would terminate, then the proof shows we will achieve the goal. But, for every argument but 0, the loop will not terminate. Because of this limitation of the loop law, it is called a partial correctness law.

To ensure total correctness, that is, to prove the loop must terminate and satisfies its goal, we must use additional reasoning. The value of the measure must always compute to a nonnegative integer, and after each iteration of the loop, the measure must decrease by at least 1.

This means, after some finite number of iterations, the measure hits 0 and the loop stops. We won't develop this further It is called mathematical induction. We will introduce mathematical induction by means of examples from programming.

Testing and induction Go back to the first section in this chapter. A typical computer programmer would pretend that the definition of!

For example, we can test 5! Before we grind out 4!

There is an important principle hiding in here, and once we understand it we will realize there are only two test cases that matter: This will save us a lot of time testing the definition of!.Michael Ernst.

Reasoning about loops A loop represents an unknown number of paths Case analysis is problematic Write loop invariants and decrementing functions when you are unsure about a loop If a loop is not working: Add invariant and decrementing function if missing. Invariant discovery is an art form; there cannot exist a mechanical algorithm to do this.

(This is a key result of computability theory, the study of what problems are mechanically solvable.) So, we now study how to discover loop invariants.

Examples of invariant discovery A key intellectual task in programming is stating a loop's invariant. • Loop invariants are useful for: – code specification – debugging continued kaja-net.com • A loop invariant is typically written as an inductive statement S(n), where n .

2) Also, what is the purpose of the loop invariant? Is it to show that $1$ runs all the way in the loop or what?

Any comment is appreciated. Thank you for your kind help. So you basically just change the upper and lower bound of the loop Since hloop from the next element until k iterates over the reqiured elements (if h==k, the loop .

The boolean formula in this rule is known as the loop invariant. The loop invariant plays an important role in the intuitive argument for soundness of the Floyd-Hoare rule for while loops. The loop invariant has to be true before each iteration of the loop body, and also after each iteration of the loop body.

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discrete mathematics - loop invariant for simple algorithms - Mathematics Stack Exchange