Nontermination

Nontermination is a property of a computation in which a program or process continues running without reaching a stopping condition. A program that halts is terminating; one that does not halt is nonterminating. Most programs are intended to terminate after producing a result, but some systems are designed to run indefinitely, such as servers or background services, where progress is defined in terms of responsiveness or ongoing maintenance rather than immediate output.

Causes include infinite loops (for example, a loop with a condition that never becomes false), unbounded or missing base cases in recursion, and waiting for events or inputs that may never arrive. In concurrent or distributed systems, nontermination can also arise from deadlock, livelock, or unbounded waiting, where components make no further progress.

In theory, nontermination relates to the halting problem. Alan Turing proved that there is no general algorithm that can decide for every program-input pair whether it will terminate. Nevertheless, for many restricted programming languages and specific programs, termination can be proven, using techniques such as well-founded induction, structural recursion, or termination metrics. A program that halts for all inputs defines a total function; if it may fail to halt for some inputs, it defines a partial function.

Practically, nontermination can be addressed with safeguards such as time or resource limits, timeouts, watchdog processes, and termination analysis tools. Static analysis and termination-checking techniques aim to certify that programs will terminate under specified conditions, while language features like explicit recursion schemes help ensure termination for well-founded problems.