Refinements and Abstractions of Signal-Event (Timed) Languages

level Concrete level refinement −−−−−−→ ConnectToServer Details used to establish the connection abstraction ←−−−−−− Formalisation of abstraction Let σ : A → P(B) be a substitution. Abstract level Concrete level σ(L) = {w ∈ A | σ(w) ∩ L '= ∅} abstraction ←−−−−−− L ⊆ B 5/29 Adding time to the picture Timed refinement refinement −−−−−−→ Abstract level Concrete level abstraction ←−−−−−− ConnectToServer Req · Wait · Ack ConnectToServer4.5 Req · Wait · Nack · Wait · Retry · Wait · Ack An abstract action a with duration d should be replaced by a concrete execution (word) w with the same duration ‖w‖ = d. 6/29 Outline Introduction 2 Signal-Event (Timed) Words and Automata Signal-Event (Timed) Substitutions Closure under substitutions Closure under inverse substitutions Conclusion 7/29 Signal-Event (Timed) Words Asarin Caspi Maler 2002 ! Σe finite set of (instantaneous) events ! Σs finite set of signals ! T time domain, T = T ∪ {∞} ! Σ = Σe ∪ (Σs × T) ! Notation: ad for (a, d) ∈ Σs × T ! Σ set of signal-event (timed) words Example: a3ffgb1.5a2f ! Signal stuttering: aa ≈ a, a = aaa · · · 8/29 Signal-Event (Timed) Words Unobservable signal τ ! Useful to hide signals: Signal-event word hiding signals −−−−−−−→ Classical timed words afbgfaf τfτgfτf = (f, 3)(g, 4)(f, 4)(f, 6) ! τ ≈ ε : an hidden signal with zero duration is not observable. a '≈ ε : a signal, even of zero duration, is observable. τ '≈ ε : we still observe a time delay but the actual signal has been hidden. Example : aτafτgτfbbb · · · ≈ afgτfb ! Signal-event words SE (Σ) = Σ/ ≈ 9/29 Signal-Event (Timed) automata ! States emit signals ! Transitions emit (instantaneous) events