# Temporal Verification of Reactive Systems: Safety

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CS256/Winter 2007 — Lecture #3 Zohar Manna Announcements Homework 1 due NOW Homework 2 out today (check website), due Tue next week 3-1 TEMPORAL LOGIC(S) Languages that can specify the behavior of a reactive program. Two views: (1) the program generates a set of sequences of states the models of temporal logic are infinite sequences of states LTL (linear time temporal logic ) [Manna, Pnueli] approach x x x x x x x 3-2 (2) the program generates a tree, where the branching points represent nondeterminism in the program the models of temporal logic are infinite trees CTL (computation tree logic ) [Clarke, Emerson] at CMU Also CTL * . x @ @ @ x x x H H H x H H H x x x X X X x x x X X X x x x x x x x x 3-3 Temporal logic: underlying assertion language Assertion language L : first-order language over interpreted typed symbols (functions and relations over concrete domains) Example: x > 0 x + 1 > y x, y Z + formulas in L called: state formulas or assertions 3-4 Temporal logic: underlying assertion language (Con’t) A state formula is evaluated over a single state to yield a truth value. For state s and state formula p s q p if s [ p ] = t We say: p holds at s s satisfies p s is a p -state Example: For state s : { x : 4 , y : 1 } s q x = 0 y = 1 s q / x = 0 y = 1 s q z. x = z 2 3-5 Temporal logic: underlying assertion language (Con’t) p is state-satisfiable if s q p for some state s p is state-valid if s q p for all states s p and q are state-equivalent if s q p iff s q q for all states s Example: ( x, y : integer) state-valid: x y x +1 > y state-equivalent: x = 0 y = 1 and x 6 = 0 y = 1 3-6 TEMPORAL LOGIC (TL) A formalism for specifying sequences of states TL = assertions + temporal operators assertions (state formulas ): First-order formulas describing the properties of a single state temporal operators Fig 0.15 3-7 Future Temporal Operators 0 p Henceforth p 1 p Eventually p p U q p Until q p W q p Waiting-for (Unless) q 2 p Next p Past Temporal Operators p So-far p Q p Once p p S q p Since q p B q p Back-to q « p Previously p 2 p Before p Fig. 0.15. The temporal operators 3-8 future temporal operators ←- past -→|←- future -→-→-→ 0 present 1 q Eventually q q 0 0 p Henceforth p p p p p ······ 0 p U q p  • • • 