computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

, a computation is said to diverge if it does not terminate or terminates in an exceptional

state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * '' Our ...

. Otherwise it is said to converge. In domains where computations are expected to be infinite, such as

process calculi In computer science, the process calculi (or process algebras) are a diverse family of related approaches for formally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions, communications, and ...

, 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

abstract rewriting In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting s ...

, an

abstract rewriting system In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting s ...

is called convergent if it is both confluent and

terminating In computer science, a computation is said to diverge if it does not terminate or terminates in an exceptional state. Otherwise it is said to converge. In domains where computations are expected to be infinite, such as process calculi, a computatio ...

. 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 Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...

an expression is divergent if it has no normal form.

Denotational semantics

In denotational semantics an object function ''f'' : ''A'' → ''B'' can be modelled as a

mathematical function In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the func ...

f : A \cup\ \rightarrow B \cup\

where ⊥ ( bottom) indicates that the object function or its

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

diverges.

Concurrency theory

In the calculus of

communicating sequential processes In computer science, communicating sequential processes (CSP) is a formal language for describing patterns of interaction in concurrent systems. It is a member of the family of mathematical theories of concurrency known as process algebras, or ...

, divergence is a drastic situation where a process performs an endless series of hidden actions. For example, consider the following process, defined by

CSP CSP may refer to: Education * College Student Personnel, an academic discipline * Commonwealth Supported Place, a category in Australian education * Concordia University (Saint Paul, Minnesota), US Organizations * Caledonian Steam Packet Compa ...

notation: :

Clock = tick \rightarrow Clock

The traces of this process are defined as: :

\operatorname(Clock) = \ = \^*

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

P= Clock \backslash tick

By definition, ''P'' is called a divergent process.

Notes

References