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 occurs when a process performs an endless series of hidden actions.[4] For example, consider the following process, defined by CSP notation: $${\displaystyle Clock=tick\rightarrow Clock}$$ The traces of this process are defined as: $${\displaystyle \operatorname {traces} (Clock)=\{\langle \rangle ,\langle tick\rangle ,\langle tick,tick\rangle ,\ldots \}=\{tick\}^{*}}$$ Now, consider the following process, which hides the tick event of the Clock process: $${\displaystyle P=Clock\setminus tick}$$ As ${\displaystyle P}$ cannot do anything other than perform hidden actions forever, it is equivalent to the process that does nothing but diverge, denoted ${\displaystyle \mathbf {div} }$. One semantic model of CSP is the failures-divergences models, which refines the stable failures model by distinguishes processes based on the sets of traces after which they can diverge.

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.
4. Roscoe, A.W. (2010). Understanding Concurrent Systems. Texts in Computer Science. doi:10.1007/978-1-84882-258-0. ISBN 978-1-84882-257-3.

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.