Event segment

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 formula_1 [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 formula_2, and defined
 formula_3 
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 formula_4, the "null event" denoted by formula_5 stands for nothing change.
Timed event.
A "timed event" is a pair formula_6 where formula_7 and formula_8 denotes that an event formula_9 occurs at time formula_10.
Null segment.
The "null segment" over time interval formula_11 is denoted by formula_12 which means nothing in formula_1 occurs over formula_14.
Unit event segment.
A "unit event segment" is either a null event segment or a timed event.
Concatenation.
Given an event set formula_1, "concatenation" of two unit event segments formula_16 over formula_17 and formula_18 over formula_19 is denoted by formula_20 whose time interval is formula_21, and implies formula_22.
Event trajectory.
An "event trajectory"
formula_23 over an event set formula_24 and a time interval formula_25 is concatenation of unit event segments formula_26 and formula_27 where
formula_28.
Mathematically, an event trajectory is a mapping formula_16 a time period formula_30 to an event set formula_1. So we can write it in a function form :
formula_32
Timed language.
The "universal timed language" formula_33 over an event set formula_1 and a time interval formula_35, is the set of all event trajectories over formula_1 and formula_37.
A "timed language" formula_38 over an event set formula_1 and a timed interval
formula_40 is "a set of event trajectories" over formula_1 and formula_42 if formula_43.