Event segment: Difference between revisions

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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 onea coefficientset of acoefficients simpleof a formula. For example, of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96km96&nbsp;km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by ana real interval, to the set <math>Z</math> [[Event_Segment#References|[Zeigler76]]] and, [[Event_Segment#References|[ZPK00]]], [[Event_Segment#References|[Hwang13]]]. A '''trajectory''' of a system variable is a concatenationsequence of segments concatenated. We call it a trajectory of constant-segments (respectively linear-segments) if its concatenating segments are constant (respectively linear).

An '''event segment''' is a special class of the constant segment thatwith 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]].

A '''segment''' of a system variable 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 one coefficient of a simple formula. For example of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by an real interval, to the set <math>Z</math> [[Event_Segment#References|[Zeigler76]]] and [[Event_Segment#References|[ZPK00]]]. A '''trajectory''' of a system variable is a concatenation of segments. We call it a trajectory of constant-segments (respectively linear-segments) if its concatenating segments are constant (respectively linear).

An event segment is a special class of the constant segment that is 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 the 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.

=== 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.

=== Timed event ===

A ''timed event'' is a pair <math> (z,t,z) </math> overwhere an event set <math>t Z\in \mathbb{T}</math> and the time base <math> z \mathbb{T}in Z </math> denotes that an event <math> z \in Z</math> occurs at time <math> t \in \mathbb{T}</math>.

=== Null event segment ===

The ''null event 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 thatnothing therein is no<math>Z</math> eventoccurs over <math> [t_l, t_u] </math>.

=== Unit event segment ===

Line 30 ⟶ 35:

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

=== Multi-eventEvent 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] \subset \mathbb{T} </math> is concatenation of [[Event Segment#Unit event segment|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 :

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

== Timed language ==

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

<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 an event segment <math>\omega \in

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

Infinitely many events in an event segment <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.

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

<math>[t_l, t_u]</math> is ''a set of event segmentstrajectories'' over <math>Z</math> and <math>[t_l,

t_u]</math> if <math>L

t_u]</math>. If <math>L</math> is a language over <math>Z</math> and <math>[t_l, t_u]</math>, then <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 | isbn= 978-0-12-778455-7 |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 - 107–10, 2013

[[Category:Automata theory]]

[[Category:Automata theory(computation)]]

[[Category:Formal specification languages]]