Loop Variant

In computer science, a loop variant is a mathematical function defined on the state space of a computer program whose value is monotonically decreased with respect to a (strict) well-founded relation by the iteration of a while loop under some invariant conditions, thereby ensuring its termination. A loop variant whose range is restricted to the non-negative integers is also known as a bound function, because in this case it provides a trivial upper bound on the number of iterations of a loop before it terminates. However, a loop variant may be transfinite, and thus is not necessarily restricted to integer values.

A well-founded relation is characterized by the existence of a minimal element of every non-empty subset of its domain. The existence of a variant proves the termination of a while loop in a computer program by well-founded descent. A basic property of a well-founded relation is that it has no infinite descending chains. Therefore a loop possessing a variant will terminate after a finite number of iterations, as long as its body terminates each time.

A while loop, or, more generally, a computer program that may contain while loops, is said to be totally correct if it is partially correct and it terminates.