nonhalting

Nonhalting is a term used in computer science to describe a computation or process that does not reach a termination state. In theoretical contexts, a nonhalting computation runs indefinitely on a given input, such as a Turing machine that enters an infinite sequence of steps or a program that executes an infinite loop.

The concept is central to the halting problem, a result proven by Alan Turing. The halting problem asks whether there exists a general algorithm that can determine, for any program and input, whether the program will halt. Turing showed that such an algorithm cannot exist for all possible programs, making nonhalting undecidable in general. In practice, engineers can sometimes prove nontermination for specific programs and inputs by identifying a recurrent state or an invariant that prevents the program from exiting its loop, but no universal method exists to decide nonhalting across all cases.

Nonhalting also appears in practical programming patterns. Some programs are designed to run indefinitely, such as servers, event loops, or daemons, where continuous operation is expected. In languages with lazy evaluation or infinite data structures, nontermination can be intentional, generating endless streams of data or results. Conversely, many analyses aim to prove termination to ensure programs complete after a finite amount of work.

In summary, nonhalting denotes infinite execution, a concept tightly linked to undecidability through the halting problem and relevant to both theoretical and practical aspects of computing.