Divergence (computer science)

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 $f:A\cup \{\perp \}\rightarrow B\cup \{\perp \}$ 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: $Clock=tick\rightarrow Clock$ The traces of this process are defined as: $\operatorname {traces} (Clock)=\{\langle \rangle ,\langle tick\rangle ,\langle tick,tick\rangle ,\ldots \}=\{tick\}^{*}$ Now, consider the following process, which hides the tick event of the Clock process: $P=Clock\setminus tick$ As $P$ cannot do anything other than perform hidden actions forever, it is equivalent to the process that does nothing but diverge, denoted $\mathbf {div}$ . 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.