Particle Tableau

# Temporal Verification of Reactive Systems: Safety

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CS256/Winter 2007 — Lecture #15 Zohar Manna Particle Tableau 15-1 Particle Tableau: Motivation Consider ϕ : 2 2 p The closure Φ ϕ has three basic formulas: p , 2 p , 2 2 p . Thus, it has eight atoms. The atom tableau T 2 2 p is ? { p , 2 p, 2 2 p } @ @ @ R p , 2 p, 2 2 p } ? - ? { p , ¬ 2 p, 2 2 p } @ @ @ I ? ? p , ¬ 2 p, 2 2 p } 6 { p , 2 p, ¬ 2 2 p } ? p , 2 p, ¬ 2 2 p } 6 @ @ @ R { p , ¬ 2 p, ¬ 2 2 p } p , ¬ 2 p, ¬ 2 2 p } @ @ @ I 15-2 Particle Tableau: Motivation The ω -automaton A 2 2 p : ? n 1 : p @ @ @ R n 2 : ¬ p ? - ? n 3 : p @ @ @ I ? ? n 4 : ¬ p 6 n 5 : p ? n 6 : ¬ p 6 @ @ @ R n 7 : p n 8 : ¬ p @ @ @ I F M = { all SCS’s } F S = {} Note : No promising formulas. 15-3 Particle Tableau: Motivation Because of the atom construction rule : for every ψ Φ ϕ , ψ A iff ¬ ψ 6∈ A , every atom makes a commitment about every formula in the closure. Clearly, some of these commitments are irrelevant in determining the satisfiability of the formula. 15-4

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Particle Tableau: Motivation (Cont’d) Intuitively, the tableau below should suffice to determine satisfiability. The truth value of p at the first two posi- tions is irrelevant: e T 2 2 p : ? n 1 : 2 2 p ? n 2 : 2 p ? n 3 : p ? n 4 : t e A 2 2 p : ? n 1 : t ? n 2 : t ? n 3 : p ? n 4 : t If we change the offending rule to if ψ A then ¬ ψ 6∈ A we get the particle tableau, which is usually considerably smaller than the atom tableau. 15-5 Particles The idea of a particle is to assert what needs to be true, not what needs to be false, except for state formulas. Thus, if ψ A , ψ needs to be true; if ψ 6∈ A , ψ can be true or false. Step 0 : Push negations inside ϕ We push all negations inside the formula such that negations only appear at the state level. This can be done with the help of the following congru- ences: ¬ 1 p 0 ¬ p ¬ 2 p 2 ¬ p ¬ 0 p 1 ¬ p ¬ ( p U q ) ( ¬ q ) W ( ¬ p ∧ ¬ q ) ¬ ( p W q ) ( ¬ q ) U ( ¬ p ∧ ¬ q ) Thus, the closure only needs to contain positive formulas and the negation of state formulas.
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