In computer science, a computation is said to diverge if it does not terminate or terminates in an exceptional state.: 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 f : A ∪ { ⊥ } → B ∪ { ⊥ } {\displaystyle f:A\cup \{\perp \}\rightarrow B\cup \{\perp \}} where ⊥ (bottom) indicates that the object function or its argument diverges.
Concurrency theory
See also: Communicating sequential processes § Failures/divergences model
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: C l o c k = t i c k → C l o c k {\displaystyle Clock=tick\rightarrow Clock} The traces of this process are defined as: traces ( C l o c k ) = { ⟨ ⟩ , ⟨ t i c k ⟩ , ⟨ t i c k , t i c k ⟩ , … } = { t i c k } ∗ {\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: P = C l o c k ∖ t i c k {\displaystyle P=Clock\setminus tick} As P {\displaystyle P} cannot do anything other than perform hidden actions forever, it is equivalent to the process that does nothing but diverge, denoted d i v {\displaystyle \mathbf {div} } . One semantic model of CSP is the failures-divergences model, which refines the stable failures model by distinguishing processes based on the sets of traces after which they can diverge.
See also
Notes
- 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.
References
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. http://extras.springer.com/2002/978-3-642-63970-8/DVD3/rom/pdf/Hoare_hist.pdf ↩
Baader & Nipkow 1998, p. 9. - Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press. ISBN 9780521779203. https://books.google.com/books?id=N7BvXVUCQk8C&q=Divergent ↩
Pierce 2002, p. 65. - Pierce, Benjamin C. (2002). Types and Programming Languages. MIT Press. ↩
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. 978-1-84882-257-3 ↩