Nonterminating

Nonterminating describes a state in which a process, computation, or sequence does not come to an end in finite time. In this sense, something nonterminating continues indefinitely or remains unfinished.

In computer science, nonterminating programs or processes do not halt for a given input. Examples include infinite loops, servers designed to run forever, and reactive or embedded systems that must operate continuously. The concept is closely related to the halting problem, which asks whether a given program will terminate on a given input; many questions about termination cannot be decided algorithmically in general.

In mathematics, nonterminating can refer to decimal expansions or series that do not terminate. A nonterminating decimal has an infinite number of digits after the decimal point, such as 0.41421356… for the square root of 2, or 0.333… for 1/3. By contrast, terminating decimals end after a finite number of digits. Nonterminating decimals can be either repeating (such as 0.333… or 0.142857142857…) or non-repeating (as with many irrational numbers).

In formal languages and logic, nonterminating derivations or computations describe procedures that do not reach a final, completed form. This is relevant for reasoning about program behavior, language recognition, and the limits of algorithmic analysis.

Termination analysis is a field focused on proving that certain programs always halt or identifying inputs that cause nontermination. Understanding nontermination helps assess reliability, resource usage, and the theoretical limits of computation.