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Introduction to Uppaal ITV Multiprogramming & Real-Time Systems Anders P. Ravn Aalborg University May 2009

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Networks of Timed Automata

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X = {x 1, … x n } - clock variables (V, E) – control graph init: V pred(X) inv: V pred(X) jump: E pred(X X’) event: E Timed Automaton - Syntax x’ = x-1

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Q – states, e.g. (v=”Off”,x = 17.5) Q 0 – initial states, Q 0 Q A – labels – transition relation, A Q Q Labelled Transition Systems a post a (R) = { q’ | q R and q q’} pre a (R) = { q | q’ R and q q’} a a

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Transition Semantics of TA X = {x 1, … x n } - clock variables (V, E) – control graph init: V pred(X) inv: V pred(X) jump: E pred(X X’) event: E Q - states – {(v,x) | v V and inv(v)[X := x]}. x’ = x-1 Q 0 – initial states - {(v,x) Q | init(v)[X := x]} A - labels - R 0 { (v,x) – (v’,x’) | e E(v,v’) and event(e) = and jump(e) [X := x]} { (v,x) – (v,x+ ) | R 0}.

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Q - states, {(v,x) | v V and inv(v)[X := x]} Q 0 – initial states, … A - labels, … - transition relation, A Q Q Trace Semantics a Trajectory: = where q 0 Q 0 and q i –a i q i+1, i 0 Live Transition System: (S, L = { | infinite from S}) Machine Closed: finite from S, prefix(L) Duration of is sum of time labels. S is non-Zeno: duration of L diverges, Machine closed

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Q - states Q 0 – initial states, … A - labels, … - transition relation, A Q Q Composition of Transition Systems a S = S1 || S2 with : A1 A2 A Q = Q1 Q2 Q 0 = Q1 0 Q2 0 (q1,q2) –a (q1’,q2’) iff (qi –ai qi’, i=1,2 and a = a1 a2 Remark p 7

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Verification Tasks Reachability of (v,x) – finitary, time-abstract trace inclusion Emptiness – time-abstract trace inclusion Trace (finitary) inclusion Time-abstract (finitary) trace inclusion

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CTL

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