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**Unformatted text preview: **CS256/Winter 2007 — Lecture #15 Zohar Manna Particle Tableau 15-1 Particle Tableau: Motivation Consider ϕ : 22 p The closure Φ ϕ has three basic formulas: p , 2 p , 22 p . Thus, it has eight atoms. The atom tableau T 22 p is ? { p , 2 p, 22 p } @ @ @ R {¬ p , 2 p, 22 p } ?- ? { p , ¬ 2 p, 22 p } @ @ @ I ? ? {¬ p , ¬ 2 p, 22 p } 6 { p , 2 p, ¬ 22 p } ? {¬ p , 2 p, ¬ 22 p } 6 @ @ @ R { p , ¬ 2 p, ¬ 22 p } {¬ p , ¬ 2 p, ¬ 22 p } @ @ @ I 15-2 Particle Tableau: Motivation The ω-automaton A 22 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 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 22 p : ? n 1 : 22 p ? n 2 : 2 p ? n 3 : p ? n 4 : t e A 22 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 ≈ ¬ p ¬ 2 p ≈ 2 ¬ p ¬ 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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