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 exceptional state.^[1]: 377 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 (i.e. to continue producing 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.^[2]

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.^[3]

Denotational semantics

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

Concurrency theory

In the calculus of communicating sequential processes (CSP), 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:

${\displaystyle Clock=tick\rightarrow Clock}$ 

The traces of this process are defined as:

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

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

${\displaystyle P=Clock\backslash tick}$ 

By definition, P is called a divergent process.

See also

• Infinite loop
• Termination analysis

Notes

1. C.A.R. Hoare (Oct 1969). "An Axiomatic Basis for Computer Programming" (PDF). Communications of the ACM. 12 (10): 576–583. doi:10.1145/363235.363259. S2CID 207726175.
2. Baader & Nipkow 1998, p. 9.
3. Pierce 2002, p. 65.

References

• Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press. ISBN 9780521779203.
• 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.