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 == | |||
=== 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]] |
Revision as of 10:03, 28 November 2013
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A segment or trajectory is a relation between an element of an arbitrary set and a time of time base [Zeigler76] and [ZPK00]. As timed sequences of events, event segments are a special class of the general segment. Event segments are used to define Timed Event Systems such as DEVS, 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 , the null event denoted by stands for nothing change.
Time base
The time base of the concerning systems is denoted by , and defined
as the set of non-negative real numbers.
Timed event
A timed event over an event set and the time base denotes that an event occurs at time .
Null event segment
The null event segment over time interval is denoted by which means that there is no event over .
Unit event segment
A unit event segment is either a null event segment or a timed event.
Concatenation
Given an event set , concatenation of two unit event segments over and over is denoted by whose time interval is , and implies .
Multi-event segment
A multi-event segment over an event set and a time interval is concatenation of unit event segments and where .
Timed language
The universal timed language over an event set and a time interval , is denoted by , and is defined as the set of all possible event segments. Formally,
where denotes a none or multiple concatenation(s) of timed events. Notice that the number of events in an event segment can be one of zero, finite or infinite. Infinitely many events in an event segment implies that , however does not imply infinite many events in it.
A timed language over an event set and a timed interval is a set of event segments over and . If is a language over and , then .
References
- [Zeigler76] 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 - [ZKP00] 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