Event segment

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

${\displaystyle \mathbb {T} =[0,\infty )}$

as the set of non-negative real numbers.

An event is a label that abstracts a change. Given an event set ${\displaystyle Z}$, the null event denoted by ${\displaystyle \epsilon \not \in Z}$ stands for nothing change.

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

The null segment over time interval ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$ is denoted by ${\displaystyle \epsilon _{[t_{l},t_{u}]}}$ which means nothing in ${\displaystyle Z}$ occurs over ${\displaystyle [t_{l},t_{u}]}$.

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

Given an event set ${\displaystyle Z}$, concatenation of two unit event segments ${\displaystyle \omega }$ over ${\displaystyle [t_{1},t_{2}]}$ and ${\displaystyle \omega '}$ over ${\displaystyle [t_{3},t_{4}]}$ is denoted by ${\displaystyle \omega \omega '}$ whose time interval is ${\displaystyle [t_{1},t_{4}]}$, and implies ${\displaystyle t_{2}=t_{3}}$.

An event trajectory ${\displaystyle (t_{1},z_{1})(t_{2},z_{2})\cdots (t_{n},z_{n})}$ over an event set ${\displaystyle Z}$ and a time interval ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$ is concatenation of unit event segments ${\displaystyle \epsilon _{[t_{l},t_{1}]},(t_{1},z_{1}),\epsilon _{[t_{1},t_{2}]},(t_{2},z_{2}),\ldots ,(t_{n},z_{n}),}$ and ${\displaystyle \epsilon _{[t_{n},t_{u}]}}$ where ${\displaystyle t_{l}\leq t_{1}\leq t_{2}\leq \cdots \leq t_{n-1}\leq t_{n}\leq t_{u}}$.

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

${\displaystyle \omega$ :[t_{l},t_{u}]\rightarrow Z^{*}.}