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**Unformatted text preview: **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 > → 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 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 operatorsFig....

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