terminability

Terminability, in general, is the property of a process, procedure, or system to come to an end after a finite sequence of steps. The term is related to termination and is often used to emphasize the ability to reach a final state, as opposed to running indefinitely. The concept appears across disciplines such as computer science, mathematics, and logic, where ensuring termination is important for predictability and correctness.

In computer science, a program or algorithm is terminating if it halts after a finite number of steps for every valid input. Terminating programs contrast with non-terminating ones, which can loop forever. Terminability is a key concern in program verification and formal methods. Techniques for establishing termination include designing algorithms with structural recursion (where each recursive call operates on a strictly smaller input), using a well-founded measure or ranking function that decreases with each step, and applying lexicographic or other well-founded orderings. Termination proofs may be automatic or manual and are central to proving total correctness, alongside soundness and partial correctness.

In term rewriting and formal logic, a system is terminating if every sequence of rule applications eventually ends in a form with no applicable rules. This is also called strong normalization in some contexts. Termination guarantees that computations produce a final result, which is essential for the consistency of certain logical systems and for enabling automated reasoning.

Termination is not guaranteed in general; the Halting Problem shows that no universal procedure can decide termination for all programs. Nevertheless, termination analysis is feasible for many practical languages and systems, and remains an active area of research in ensuring reliable, predictable computation.