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. Knowing its invariant(s) is essential in understanding the effect of a loop.

In formal program verification, particularly the Floyd-Hoare approach, loop invariants are expressed by formal predicate logic and used to prove properties of loops and by extension algorithms that employ loops (usually correctness properties). The loop invariants will be true on entry into a loop and following each iteration, so that on exit from the loop both the loop invariants and the loop termination condition can be guaranteed.

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.

Loop-invariant code motion, which involves moving code out of the loop if that does not change the effect of the program, is not directly related to loop invariants, which are properties of the loop overall.

The following C subroutine max() returns the maximum value in its argument array a[], provided its length n is at least 1. Comments are provided at lines 3, 6, 9, 11, and 13. Each comment makes an assertion about the values of one or more variables at that stage of the function. The highlighted assertions within the loop body, at the beginning and end of the loop (lines 6 and 11), are exactly the same. They thus describe an invariant property of the loop. When line 13 is reached, this invariant still holds, and it is known that the loop condition i!=n from line 5 has become false. Both properties together imply that m equals the maximum value in a[0...n-1], that is, that the correct value is returned from line 14.

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