Krull–Schmidt theorem: 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]]s such as [[DEVS]], [[timed automaton|timed automata]], and [[timed petri nets]].


== Event segments ==
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=== 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><math> \mathbb{T}=[0,\infty) </math> </center>
as the set of non-negative real numbers.
 
=== Timed event ===
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 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 that there is no event over <math> [t_l, t_u]  </math>.
 
=== Unit event segment ===
A ''unit event segment'' is either a [[Event Segment#Null event segment|null event segment]] or a [[Event Segment#Timed event|timed 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 segment ===
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 concatenation 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 language ==
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,
<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'' over an event set <math>Z</math> and a timed interval
<math>[t_l, t_u]</math> is ''a set of event segments'' over <math>Z</math> and <math>[t_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>.
 
== 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 = Bernard Zeigler, Tag Gon Kim, Herbert Praehofer| year = 2000| title = Theory of Modeling and Simulation| publisher = Academic Press, New York  | isbn= 978-0-12-778455-7 |edition=second}}
 
[[Category:Automata theory]]
[[Category:Formal specification languages]]

Latest revision as of 09:00, 11 August 2014


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