# Event segment

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A segment of a system variable 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.

## Event segments

### 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 $t_{l}\leq t_{1}\leq t_{2}\leq \cdots \leq t_{n-1}\leq t_{n}\leq t_{u}$ .

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 $L\subseteq \Omega _{Z,[t_{l},t_{u}]}$ .