Menzerath's law: Difference between revisions

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In [[computer science]], a computation is said to '''diverge''' if it does not terminate or terminates in an (unobservable) [[exceptional state]]. 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 [[productivity (computer science)|productive]] (always produces 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]] a [[reduction (abstract rewriting)|reduction]] is called convergent if and only if it is both [[confluent (abstract rewriting)|confluent]] and [[terminating (abstract rewriting)|terminating]].{{sfn|Baader|Nipkow|1998|p=9}} The notation ''t'' ↓ ''n'' means term ''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.


In the [[lambda calculus]] an expression is divergent if it has no [[normal form (abstract rewriting)|normal form]].{{sfn|Pierce|2002|p=65}}
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=== Denotational semantics ===
In [[denotational semantics]] an [[function (computer science)|object function]] ''f'' : ''A'' &rarr; ''B'' can be modelled as a [[function (mathematics)|mathematical function]] ''f'' : ''A'' &cup; {&perp;} &rarr; ''B'' &cup; {&perp;} where &perp; ([[bottom element|bottom]]) indicates that the object function or its [[argument (computer science)|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 [[Communicating sequential processes|CSP]] notation:
:<math>Clock = tick \rightarrow Clock</math>
The traces of this process are defined as:
:<math>\operatorname{traces}(Clock) = \{\langle\rangle, \langle tick \rangle, \langle tick,tick \rangle, \cdots \} = \{ tick \}^*</math>
Now, consider the following process, which conceals the ''tick'' event of the ''Clock'' process:
:<math>P= Clock \backslash tick</math>
By definition, ''P'' is called a divergent process.
 
== See also ==
* [[Infinite loop]]
 
== Notes ==
{{reflist|2}}
 
== References ==
* {{cite book|first1=Franz|last1=Baader|authorlink1=Franz Baader|first2=Tobias|last2=Nipkow|authorlink2=Tobias Nipkow|title=[[Term Rewriting and All That]]|year=1998|publisher=Cambridge University Press|ref=harv}}
* {{cite book|first=Benjamin C.|last=Pierce|authorlink=Benjamin C. Pierce|title=[[Types and Programming Languages]]|year=2002|publisher=MIT Press|ref=harv}}
* 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''.
 
{{comp-sci-stub}}
 
[[Category:Programming language theory]]
[[Category:Process (computing)]]
[[Category:Rewriting systems]]
[[Category:Lambda calculus]]
[[Category:Denotational semantics]]

Latest revision as of 13:12, 26 August 2014


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