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Timed event system
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<h2 id="timed-event-systems">Timed Event Systems</h2> <p>A timed event system is a structure </p> G =< Z , Q , Q 0 , Q A , Δ > {\displaystyle {\mathcal {G}}=<Z,Q,Q_{0},Q_{A},\Delta >} <p>where </p> <ul><li> Z {\displaystyle \,Z} is <i>the set of events</i>;</li> <li> Q {\displaystyle \,Q} is <i>the set of states</i>;</li> <li> Q 0 ⊆ Q {\displaystyle \,Q_{0}\subseteq Q} is <i>the set of initial states</i>;</li> <li> Q A ⊆ Q {\displaystyle Q_{A}\subseteq Q} is <i>the set of accepting states</i>;</li> <li> Δ ⊆ Q × Ω Z , [ t l , t u ] × Q {\displaystyle \Delta \subseteq Q\times \Omega _{Z,[t_{l},t_{u}]}\times Q} is <i>the set of state trajectories</i> in which ( q , ω , q ′ ) ∈ Δ {\displaystyle (q,\omega ,q')\in \Delta } indicates that a state q ∈ Q {\displaystyle q\in Q} can change into q ′ ∈ Q {\displaystyle q'\in Q} along with an <a href="/facts/Event_Segment/kdrhDIu1">event segment</a> ω ∈ Ω Z , [ t l , t u ] {\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}} . If two state trajectories ( q 1 , ω 1 , q 2 ) {\displaystyle (q_{1},\omega _{1},q_{2})} and ( q 3 , ω 2 , q 4 ) ∈ Δ {\displaystyle (q_{3},\omega _{2},q_{4})\in \Delta } are called contiguous if q 2 = q 3 {\displaystyle q_{2}=q_{3}} , and two event trajectories ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} are contiguous. Two contiguous state trajectories ( q , ω 1 , p ) {\displaystyle (q,\omega _{1},p)} and ( p , ω 2 , q ′ ) ∈ Δ {\displaystyle (p,\omega _{2},q')\in \Delta } implies ( q , ω 1 ω 2 , q ′ ) ∈ Δ {\displaystyle (q,\omega _{1}\omega _{2},q')\in \Delta } .</li></ul> <h2 id="behaviors-and-languages-of-timed-event-system">Behaviors and Languages of Timed Event System</h2> <p>Given a timed event system G =< Z , Q , Q 0 , Q A , Δ > {\displaystyle {\mathcal {G}}=<Z,Q,Q_{0},Q_{A},\Delta >} , <i>the set of its behaviors</i> is called its <i>language</i> depending on the observation time length. Let t {\displaystyle t} be the observation time length. If 0 ≤ t < ∞ {\displaystyle 0\leq t<\infty } , <i> t {\displaystyle t} -length observation language of</i> G {\displaystyle {\mathcal {G}}} is denoted by L ( G , t ) {\displaystyle L({\mathcal {G}},t)} , and defined as </p> L ( G , t ) = { ω ∈ Ω Z , [ 0 , t ] : ∃ ( q 0 , ω , q ) ∈ Δ , q 0 ∈ Q 0 , q ∈ Q A } . {\displaystyle L({\mathcal {G}},t)=\{\omega \in \Omega _{Z,[0,t]}:\exists (q_{0},\omega ,q)\in \Delta ,q_{0}\in Q_{0},q\in Q_{A}\}.} <p>We call an event segment ω ∈ Ω Z , [ 0 , t ] {\displaystyle \omega \in \Omega _{Z,[0,t]}} a t {\displaystyle t} -length behavior of G {\displaystyle {\mathcal {G}}} , if ω ∈ L ( G , t ) {\displaystyle \omega \in L({\mathcal {G}},t)} . </p><p>By sending the observation time length t {\displaystyle t} to infinity, we define <i>infinite length observation language of</i> G {\displaystyle {\mathcal {G}}} is denoted by L ( G , ∞ ) {\displaystyle L({\mathcal {G}},\infty )} , and defined as </p> L ( G , ∞ ) = { ω ∈ lim t → ∞ Ω Z , [ 0 , t ] : ∃ { q : ( q 0 , ω , q ) ∈ Δ , q 0 ∈ Q 0 } ⊆ Q A } . {\displaystyle L({\mathcal {G}},\infty )=\{\omega \in {\underset {t\rightarrow \infty }{\lim }}\Omega _{Z,[0,t]}:\exists \{q:(q_{0},\omega ,q)\in \Delta ,q_{0}\in Q_{0}\}\subseteq Q_{A}\}.} <p>We call an event segment ω ∈ lim t → ∞ Ω Z , [ 0 , t ] {\displaystyle \omega \in {\underset {t\rightarrow \infty }{\lim }}\Omega _{Z,[0,t]}} an infinite-length behavior of G {\displaystyle {\mathcal {G}}} , if ω ∈ L ( G , ∞ ) {\displaystyle \omega \in L({\mathcal {G}},\infty )} . </p> <h2 id="see-also">See also</h2> <p><a href="/facts/State_transition_system/AABVmNzn">State Transition System</a> </p> <ul><li>[Zeigler76] Bernard Zeigler (1976). <i>Theory of Modeling and Simulation</i> (first ed.). Wiley Interscience, New York.</li> <li>[ZKP00] Bernard Zeigler; Tag Gon Kim; Herbert Praehofer (2000). <i>Theory of Modeling and Simulation</i> (second ed.). Academic Press, New York. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-778455-7.</li> <li>[Hwang2012] Moon H. Hwang. "Qualitative Verification of Finite and Real-Time DEVS Networks". <i>Proceedings of 2012 TMS/DEVS</i>. Orlando, FL, USA. pp. 43:1–43:8. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-61839-786-7.</li></ul>