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 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 is called convergent if and only if it is both confluent and terminating.Template:Sfn 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.Template:Sfn

Denotational semantics

In denotational semantics an object function f : A → B can be modelled as a mathematical function f : A ∪ {⊥} → B ∪ {⊥} 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:

C l o c k = t i c k \to C l o c k

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 ⟩, \dots} = {t i c k}^{*}

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

P = C l o c k ∖ t i c k

By definition, P is called a divergent process.

Notes

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References

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

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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'' &darr; ''n'' means term ''t'' reduces to normal form ''n'' in zero or more reductions, ''t''&darr; means ''t'' reduces to some normal form in zero or more reductions, and ''t''&uarr; 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}}
+=== 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]]