Krull–Schmidt theorem

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A segment or trajectory is a relation between an element of an arbitrary set $Z$ and a time of time base $\mathbb{𝕋}$ [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 $Z$, the null event denoted by $ϵ\in ̸Z$ stands for nothing change.

Time base

The time base of the concerning systems is denoted by $\mathbb{𝕋}$, and defined

$\mathbb{𝕋}=[0,\mathrm{\infty })$

as the set of non-negative real numbers.

Timed event

A timed event $(z,t)$ over an event set $Z$ and the time base $\mathbb{𝕋}$ denotes that an event $z\in Z$ occurs at time $t\in \mathbb{𝕋}$.

Null event segment

The null event segment over time interval $[t_l,t_u]\subset \mathbb{𝕋}$ is denoted by ${ϵ}_{[t_l,t_u]}$ which means that there is no event over $[t_l,t_u]$.

Unit event segment

A unit event segment is either a null event segment or a timed event.

Concatenation

Given an event set $Z$, concatenation of two unit event segments $\omega$ over $[t_1,t_2]$ and ${\omega }^{\prime }$ over $[t_3,t_4]$ is denoted by $\omega {\omega }^{\prime }$ whose time interval is $[t_1,t_4]$, and implies $t_2=t_3$.

Multi-event segment

A multi-event segment $(z_1,t_1)(z_2,t_2)\cdots (z_n,t_n)$ over an event set $Z$ and a time interval $[t_l,t_u]\subset \mathbb{𝕋}$ is concatenation of unit event segments ${ϵ}_{[t_l,t_1]},(z_1,t_1),{ϵ}_{[t_1,t_2]},(z_2,t_2),\dots ,(z_n,t_n),$ and ${ϵ}_{[t_n,t_u]}$ where $t_l\le t_1\le t_2\le \cdots \le t_{n-1}\le t_n\le t_u$.

Timed language

The universal timed language over an event set $Z$ and a time interval $[t_l,t_u]\subset \mathbb{𝕋}$, is denoted by ${\mathrm{\Omega }}_{Z,[t_l,t_u]}$, and is defined as the set of all possible event segments. Formally,

${\mathrm{\Omega }}_{Z,[t_l,t_u]}=\{(z,t{)}^{*}|z\in Z\cup \{ϵ\},t\in [t_l,t_u]\}$

where ${*}^{}$ denotes a none or multiple concatenation(s) of timed events. Notice that the number of events in an event segment $\omega \in {\mathrm{\Omega }}_{Z,[t_l,t_u]}$ can be one of zero, finite or infinite. Infinitely many events in an event segment $\omega \in {\mathrm{\Omega }}_{Z,[t_l,t_u]}$ implies that $t_u-t_l\to \mathrm{\infty }$, however $t_u-t_l\to \mathrm{\infty }$ does not imply infinite many events in it.

A timed language over an event set $Z$ and a timed interval $[t_l,t_u]$ is a set of event segments over $Z$ and $[t_l,t_u]$. If $L$ is a language over $Z$ and $[t_l,t_u]$, then $L\subseteq {\mathrm{\Omega }}_{Z,[t_l,t_u]}$.

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.
  
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• [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