Formal methods

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. (Wikipedia).

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Related pages

Tractable problem | Infinite descending chain | Primitive recursive function | Μ operator | Gödel's incompleteness theorems | Transfinite induction | Pseudocode | Transfinite number | Well-order | Large countable ordinal | Partial correctness | Order type | Correctness (computer science) | Function (mathematics) | Integer | First uncountable ordinal | Turing machine | Loop invariant | Halting problem | Predicate transformer semantics | Computational complexity theory | Universal quantification | Well-founded relation | Termination analysis | State (computer science) | Recursion (computer science)