Event segment

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 a set of coefficients of a formula. For example, of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96 km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by a real interval, to the set $$Z$$ [Zeigler76], [ZPK00], [Hwang13]. A trajectory of a system variable is a sequence of segments concatenated. We call a trajectory constant (respectively linear) if its concatenating segments are constant (respectively linear).

An event segment is a special class of the constant segment with 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 Systems such as DEVS, timed automata, and timed petri nets.

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

$ \mathbb{T}=[0,\infty) $ 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 $$ Z$$, the null event denoted by $$ \epsilon \not \in Z$$ stands for nothing change.

Timed event
A timed event is a pair $$ (t,z) $$ where $$t \in \mathbb{T}$$ and $$ z \in Z $$ denotes that an event $$ z \in Z$$ occurs at time $$ t \in \mathbb{T}$$.

Null segment
The null segment over time interval $$ [t_l, t_u] \subset \mathbb{T} $$ is denoted by $$ \epsilon_{[t_l, t_u]}$$ which means nothing in $$Z$$ occurs 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 $$\omega'$$ over $$[t_3, t_4]$$ is denoted by $$\omega\omega'$$ whose time interval is $$[t_1, t_4]$$, and implies $$t_2 = t_3$$.

Event trajectory
An event trajectory $$(t_1,z_1)(t_2,z_2) \cdots (t_n,z_n)$$ over an event set $$ Z $$ and a time interval $$[t_l, t_u] \subset \mathbb{T} $$ is concatenation of unit event segments $$\epsilon_{[t_l,t_1]},(t_1,z_1), \epsilon_{[t_1,t_2]},(t_2,z_2),\ldots, (t_n,z_n),$$ and $$\epsilon_{[t_n,t_u]}$$ where $$t_l\le t_1 \le t_2 \le \cdots \le t_{n-1} \le t_n \le t_u$$.

Mathematically, an event trajectory is a mapping $$\omega$$ a time period $$[t_l,t_u] \subseteq \mathbb{T} $$ to an event set $$Z$$. So we can write it in a function form :

$ \omega:[t_l,t_u] \rightarrow Z^* .$

Timed language
The universal timed language $$\Omega_{Z,[t_l, t_u]}$$ over an event set $$Z$$ and a time interval $$[t_l, t_u] \subset \mathbb{T}$$, is the set of all event trajectories over $$Z$$ and $$[t_l,t_u]$$.

A timed language $$L$$ over an event set $$Z$$ and a timed interval $$[t_l, t_u]$$ is a set of event trajectories over $$Z$$ and $$[t_l, t_u]$$ if $$L \subseteq \Omega_{Z, [t_l, t_u]}$$.