Loop Invariants Overview A loop invariant is a condition 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. In what follows, we assume that evaluation of B has no "side effects" i. The flow chart is annotated to indicate the "locations" at which we assume that the precondition Q holds, we want the postcondition R to hold, and we want the loop invariant P to hold.

This again can be weakened to give us the invariant. If c is not present in s, then -1 is returned. 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. You are welcome to deduce that the program meets its postcondition. Reuse the loop invariant from find in the previous exercise. 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. 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. 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!The loop's invariant is exactly the precondition for executing the loop's body, and it is exactly the postcondition of what is generated by executing the loop's body.

Even if you forget all about algebra and proofs, whenever you write a loop, document the loop with its invariant stated in words.

Welcome to part 8 on loop invariants. Please refer to the first part (introduction) here. Today, we are going to discuss insertion sort algorithm. Let us get started. Problem definition. Given an array (A) of (n) numbers.

Assume the array starts at position (1). Develop an algorithm to sort the array in increasing order using insertion sort. A loop invariant is a formal statement about the relationship between variables in your program which holds true just before the loop is ever run (establishing the invariant) and is true again at the bottom of the loop, each time through the loop (maintaining the invariant).

Then in the inner loop, we still have the above invariant, but then we also have All entries in a[i..j-1] are greater than or equal to a[best]. When the inner loop terminates, that last invariant tells us that the variable best indexes a minimum element of the unsorted array, so we can swap it with the item at index i.

Loop Invariant Condition with Examples of Sorting Algorithms Loop Invariant Condition: Loop invariant condition is a condition about the relationship between the variables of our program which is definitely true immediately before and .

A loop invariant is a condition 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.).

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Loop Invariants