Example: Proving a Congruence

Temporal Verification of Reactive Systems: Safety

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CS256/Winter 2007 — Lecture #4 Zohar Manna Announcements HW1 solutions and grades available. 4-1 Example: Proving a Congruence For temporal formulas ϕ and ψ , show 1 0 ϕ 1 0 ψ 1 ( 0 ϕ 0 ψ ) We have to show 1 0 ϕ 1 0 q 1 ( 0 ϕ 0 ψ and 1 ( 0 ϕ 0 ψ ) 1 0 ϕ 1 0 ψ The left-to-right entailment is valid: Consider arbitrary σ and j such that ( σ, j ) q 1 0 ϕ 1 0 ψ. Thus k 1 j. ( σ, k 1 ) q 0 ϕ and k 2 j. ( σ, k 2 ) q 0 ψ 4-2 Example: Proving a Congruence (Cont’d) 1 0 ϕ 1 0 ψ 1 ( 0 ϕ 0 ψ ) Unraveling the definition of 0 , we get k 1 j. k 0 1 k 1 . ( σ, k 0 1 ) q ϕ and k 2 j. k 0 2 k 2 . ( σ, k 0 2 ) q ψ. This implies that k = max { k 1 ,k 2 } z }| { k j. k 0 k. ( σ, k 0 ) q ϕ and ( σ, k 0 ) q ψ. So k j. ( σ, k ) q ( 0 ϕ 0 ψ ) . That is, ( σ, j ) q 1 ( 0 ϕ 0 ψ ) . The right-to-left entailment is valid. All implications in the first part hold in reverse, so the entailment is valid. 4-3 Example: Proving an Equivalence / Disproving a Congruence For temporal logic formulas ϕ and ψ , show 1 ϕ 1 Q ϕ 1 ϕ 6≈ 1 Q ϕ We shall prove: (1) 1 ϕ 1 Q ϕ is valid; Thus ϕ 1 Q ϕ is valid. (2) 1 Q ϕ 1 ϕ is valid. (3) 1 Q ϕ 1 ϕ is not valid. 4-4
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(1) 1 ϕ 1 Q ϕ is valid: Consider arbitrary σ and j such that ( σ, j ) q 1 ϕ . Then i j. ( σ, i ) q ϕ . Hence i j. k : 0 k i. | {z } k = i ( σ, k ) q ϕ . By def. i j. ( σ, i ) q Q ϕ . Therefore ( σ, j ) q 1 Q ϕ . (2) 1 Q ϕ 1 ϕ is valid: Consider arbitrary σ such that ( σ, 0) q 1 Q ϕ . Then i 0 . ( σ, i ) q Q ϕ . Hence i 0 . k : 0 k i. ( σ, k ) q ϕ . Hence k 0 . ( σ, k ) q ϕ . Therefore ( σ, 0) q 1 ϕ . (3) 1 Q ϕ 1 ϕ is not valid. Counterexample: Take ϕ : p (propositional symbol) σ = h s 0 : p, s 1 : ¬ p, s 2 : ¬ p, s 3 : ¬ p, . . . i and j = 1 Then ( σ, 1) q 1 Q p , but ( σ, 1) q / 1 p . 4-5 Rigid and Flexible Variables Variables in the vocabulary are partitioned into: Rigid Variables : Rigid variable has the same value in all states of a sequence σ Flexible Variables : The values of a flexible variable may be different in different states of a sequence σ .
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  • Logic, Quantification, First-order logic, u., Temporal logic, U. U. U. U

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