# 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 ${\displaystyle f:A\cup \{\perp \}\rightarrow B\cup \{\perp \}}$ where (bottom) indicates that the object function or its argument diverges.

### Concurrency theory

In the calculus of communicating sequential processes, 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.