Event segment: Difference between revisions

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A '''segment''' or ''trajectory'' is a relation 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]].

== Event Segmentssegments ==

=== Event and Nullnull Eventevent ===

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 Basebase ===

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

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

as the set of non-negative real numbers.

=== Timed Eventevent ===

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

=== Null Eventevent Segmentsegment ===

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 that there is no event over <math> [t_l, t_u] </math>.

=== Unit Eventevent Segmentsegment ===

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

=== Concatenation of Event Segments ===

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-Eventevent Segmentsegment ===

A ''multi-event segment''

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

== Timed Languagelanguage ==

The ''universal timed language'' over an event set <math>Z</math> and a time interval <math>[t_l, t_u] \subset \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,

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 either one of zero, finite or infinite.

events. Notice that the numberInfinitely ofmany events in an event segment <math>\omega \in \Omega_{Z,[t_l,

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

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