Event segment: Difference between revisions

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A ''segment'' or ''trajectory'' which is association between an element of an arbitrary set <math>Z </math> and a time of time base <math> \mathbb{T} </math> [[Event_Segment#References|[Zeigler76]]] and [[Event_Segment#References|[ZPK00]]]. As timed sequences of ''events'', event segments are a special class of the general segment. Event segments are used to define [[Timed Event System|Timed Event Systems]] such as [[DEVS]], timed automata, and timed petri nets.

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{{Short description|Computational modeling concept}}

A '''segment''' of a system variable in [[computing]] shows a homogenous status of [[system dynamics]] over a time period. Here, a homogenous status of a variable is a state which can be described by a set of coefficients of a formula. For example, of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96&nbsp;km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by a real interval, to the set <math>Z</math> [[Event_Segment#References|[Zeigler76]]], [[Event_Segment#References|[ZPK00]]], [[Event_Segment#References|[Hwang13]]]. A '''trajectory''' of a system variable is a sequence of segments concatenated. We call a trajectory constant (respectively linear) if its concatenating segments are constant (respectively linear).

An '''event segment''' is a special class of the constant segment with a constraint in which the constant segment is either one of a timed event or a null-segment. The event segments are used to define [[Timed Event System]]s such as [[DEVS]], [[timed automaton|timed automata]], and [[timed petri nets]].

== Event Segments ==

=== Event and Null Event ===

An ''event'' is a label that abstracts a change. Given an event set <math> Z</math>, the ''null event'' denoted by <math> \epsilon \not \in Z</math> stands for nothing change.

=== Time Base ===

The ''time base'' of our concerning systems is denoted by <math> \mathbb{T} </math>, and defined

== Event Segmentssegments ==

<center><math> \mathbb{T}=[0,\infty) </math> </center>

=== Time Basebase ===

The ''time base'' of ourthe concerning systems is denoted by <math> \mathbb{T} </math>, and defined

<{{center>|1=<math> \mathbb{T}=[0,\infty) </math> </center>}}

as the set of non-negative real numbers.

=== Timed Event and null event ===

AAn ''timed event'' <math>is (z,t)a </math>label overthat abstracts a change. Given an event set <math> Z </math> and, the time''null baseevent'' <math>denoted \mathbb{T}</math> denotes that an eventby <math> z\epsilon \not \in Z</math> occursstands atfor timenothing <math> t \in \mathbb{T}</math>change.

=== NullTimed Event Segmentevent ===

TheA ''nulltimed event segment'' overis timea intervalpair <math> [t_l(t,z) t_u]</math> where <math>t \subseteqin \mathbb{T} </math> is denoted byand <math> z \epsilon_{[t_l,in Z t_u]}</math> which meansdenotes that therean isevent no<math> eventz over\in Z</math> [t_l,occurs t_u]at time <math> t \in \mathbb{T}</math>.

=== UnitNull Event Segmentsegment ===

The ''null segment'' over time interval <math> [t_l, t_u] \subset \mathbb{T} </math> is denoted by <math> \epsilon_{[t_l, t_u]}</math> which means nothing in <math>Z</math> occurs over <math> [t_l, t_u] </math>.

An ''unit event segment'' is either a [[Event Segment#Null Event Segment|null event segment]] or a [[Event Segment#Timed Event|timed event]].

=== ConcatenationUnit ofevent Event Segmentssegment ===

GivenA an''unit event set <math>Z</math>, ''concatenationsegment'' ofis twoeither a [[Event Segment#UnitNull Eventevent Segmentsegment|unitnull event segmentssegment]] or <math>\omega</math>a over <math>[t_1,[Event t_2]</math>Segment#Timed andevent|timed <math>\omega'</math> over <math>[t_3,event]].

=== Concatenation ===

Given an event set <math>Z</math>, ''concatenation'' of two [[Event Segment#Unit event segment|unit event segments]] <math>\omega</math> over <math>[t_1, t_2]</math> and <math>\omega'</math> over <math>[t_3,

t_4]</math> is denoted by <math>\omega\omega'</math> whose time interval is <math>[t_1,

t_4]</math>, and implies <math>t_2 = t_3</math>.

=== Multi-Event Segmenttrajectory ===

AAn ''multi-event segmenttrajectory''

<math>(z_1,t_1,z_1)(z_2,t_2,z_2) \cdots (z_n,t_n,z_n)</math> over an event set <math> Z </math> and a time interval <math>[t_l, t_u] \subseteqsubset \mathbb{T} </math> is concatenationsconcatenation of [[Event Segment#Unit Eventevent Segmentsegment|unit event segments]] <math>\epsilon_{[t_l,t_1]},(z_1,t_1,z_1), \epsilon_{[t_1,t_2]},(z_2,t_2,z_2),\ldots, (z_n,t_n,z_n),</math> and <math>\epsilon_{[t_n,t_u]}</math> where

<math>t_l\le t_1 \le t_2 \le \cdots \le t_{n-1} \le t_n \le t_u</math>.

Mathematically, an event trajectory is a mapping <math>\omega</math> a time period <math>[t_l,t_u] \subseteq \mathbb{T} </math> to an event set <math>Z</math>. So we can write it in a function form :

== Timed Language ==

The ''universal timed language'' over an event set <math>Z</math> and a time interval <math>[t_l, t_u] \subseteq \mathbb{T}</math>, is denoted by

<math>\Omega_{Z,[t_l, t_u]}</math>, and is defined as the set of all possible

event segments. Formally,

<center><math>

\Omega_{Z,[t_l,t_u]}=\{(z,t)^*| z \in Z \cup \{\epsilon\}, t \in [t_l, t_u] \}

</math> </center>

where <math>^*</math> denotes a none or multiple concatenation(s) of timed

events. Notice that the number of events in a string <math>\omega \in

\Omega_{Z,[t_l, t_u]}</math> can be either of zero, finite or infinite.

Infinite many events in a string <math>\omega \in \Omega_{Z,[t_l,

t_u]}</math> implies that <math>t_u - t_l \rightarrow \infty</math>, however <math>t_u - t_l \rightarrow

\infty</math> does not imply infinite many events in it.

{{center|1=<math> \omega:[t_l,t_u] \rightarrow Z^* .</math>}}

== Timed Languagelanguage ==

A ''timed language'' over an event set <math>Z</math> and a timed interval

The ''universal timed language'' <math>\Omega_{Z,[t_l, t_u]}</math> isover an event set <math>Z</math> and ''a time interval <math>[t_l, t_u] \subset \mathbb{T}</math>, is the set of all event segments''trajectories over <math>Z</math> and <math>[t_l,t_u]</math>.

t_u]</math>. If <math>L</math> is a language over <math>Z</math> and <math>[t_l, t_u]</math>, then <math>L

TheA ''universal timed language'' <math>L</math> over an event set <math>Z</math> and a timetimed interval <math>[t_l, t_u] \subseteq \mathbb{T}</math>, is denoted by

t_u]</math>.[t_l, If <math>Lt_u]</math> is ''a languageset of event trajectories'' over <math>Z</math> and <math>[t_l, t_u]</math>, then <math>L

t_u]</math> if <math>L

\subseteq \Omega_{Z, [t_l, t_u]}</math>.

== See also ==

* [[Outline of computing]]

== References ==

* [Zeigler76] {{cite book|author = Bernard Zeigler | year = 1976| title = Theory of Modeling and Simulation| publisher = Wiley Interscience, New York | id = |edition=first}}

* [ZKP00] {{cite book|author author1= Bernard Zeigler, |author2=Tag Gon Kim, |author3=Herbert Praehofer | year = 2000| title = Theory of Modeling and Simulation| publisher = Academic Press, New York | idisbn= ISBN 978-01277845570-12-778455-7 |edition=second}}

* [Giambiasi01] Giambiasi N., Escude B. Ghosh S. “Generalized Discrete Event Simulation of Dynamic Systems”, in: Issue 4 of SCS Transactions: Recent Advances in DEVS Methodology-part II, Vol. 18, pp.&nbsp;216–229, dec 2001

* [Hwang13] M.H. Hwang, ``Revisit of system variable trajectories``, ''Proceedings of the Symposium on Theory of Modeling & Simulation - DEVS Integrative M&S Symposium '', San Diego, CA, USA, April 7–10, 2013

[[Category:Automata theory(computation)]]

[[Category:Formal specification languages]]