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Timed Event Systems

A timed event system is a structure

G =< Z , Q , Q 0 , Q A , Δ > {\displaystyle {\mathcal {G}}=<Z,Q,Q_{0},Q_{A},\Delta >}

where

• Z {\displaystyle \,Z} is the set of events;
• Q {\displaystyle \,Q} is the set of states;
• Q 0 ⊆ Q {\displaystyle \,Q_{0}\subseteq Q} is the set of initial states;
• Q A ⊆ Q {\displaystyle Q_{A}\subseteq Q} is the set of accepting states;
• Δ ⊆ Q × Ω Z , [ t l , t u ] × Q {\displaystyle \Delta \subseteq Q\times \Omega _{Z,[t_{l},t_{u}]}\times Q} is the set of state trajectories 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 event segment 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 } .

Behaviors and Languages of Timed Event System

Given a timed event system G =< Z , Q , Q 0 , Q A , Δ > {\displaystyle {\mathcal {G}}=<Z,Q,Q_{0},Q_{A},\Delta >} , the set of its behaviors is called its language depending on the observation time length. Let t {\displaystyle t} be the observation time length. If 0 ≤ t < ∞ {\displaystyle 0\leq t<\infty } , t {\displaystyle t} -length observation language of G {\displaystyle {\mathcal {G}}} is denoted by L ( G , t ) {\displaystyle L({\mathcal {G}},t)} , and defined as

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}\}.}

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)} .

By sending the observation time length t {\displaystyle t} to infinity, we define infinite length observation language of G {\displaystyle {\mathcal {G}}} is denoted by L ( G , ∞ ) {\displaystyle L({\mathcal {G}},\infty )} , and defined as

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}\}.}

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 )} .

See also

State Transition System

• [Zeigler76] Bernard Zeigler (1976). Theory of Modeling and Simulation (first ed.). Wiley Interscience, New York.
• [ZKP00] Bernard Zeigler; Tag Gon Kim; Herbert Praehofer (2000). Theory of Modeling and Simulation (second ed.). Academic Press, New York. ISBN 978-0-12-778455-7.
• [Hwang2012] Moon H. Hwang. "Qualitative Verification of Finite and Real-Time DEVS Networks". Proceedings of 2012 TMS/DEVS. Orlando, FL, USA. pp. 43:1–43:8. ISBN 978-1-61839-786-7.