Divergence (computer science)

"Terminating" redirects here. For other uses, see Termination.

In computer science, a computation is said to diverge if it does not terminate or terminates in an (unobservable) exceptional state. Otherwise it is said to converge. In domains where computations are expected to be infinite, such as process calculi, a computation is said to diverge if it fails to be productive (always produces an action within a finite amount of time.)

Definitions

Various subfields of computer science use varying, but mathematically precise, definitions of what it means for a computation to converge or diverge.

Rewriting

In abstract rewriting, an abstract rewriting system is called convergent if it is both confluent and terminating.^[1]

The notation t ↓ n means that t reduces to normal form n in zero or more reductions, t↓ means t reduces to some normal form in zero or more reductions, and t↑ means t does not reduce to a normal form; the latter is impossible in a terminating rewriting system.

In the lambda calculus an expression is divergent if it has no normal form.^[2]

Denotational semantics

In denotational semantics an object function f : A → B can be modelled as a mathematical function f : A ∪ {⊥} → B ∪ {⊥} where ⊥ (bottom) indicates that the object function or its argument diverges.

Concurrency theory

In the calculus of communicating sequential processes, divergence is a drastic situation where a process performs an endless series of hidden actions. For example, consider the following process, defined by CSP notation:

Clock = tick \rightarrow Clock

The traces of this process are defined as:

\operatorname{traces}(Clock) = \{\langle\rangle, \langle tick \rangle, \langle tick,tick \rangle, \cdots \} = \{ tick \}^*

Now, consider the following process, which conceals the tick event of the Clock process:

P= Clock \backslash tick

By definition, P is called a divergent process.

Notes

↑ Baader & Nipkow 1998, p. 9.
↑ Pierce 2002, p. 65.

References

Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press.
Pierce, Benjamin C. (2002). Types and Programming Languages. MIT Press.
J. M. R. Martin and S. A. Jassim (1997). "How to Design Deadlock-Free Networks Using CSP and Verification Tools: A Tutorial Introduction" in Proceedings of the WoTUG-20.

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