Satisfiability over a finite-state program

Temporal Verification of Reactive Systems: Safety

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CS256/Winter 2007 — Lecture #14 Zohar Manna 14-1
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Satisfiability over a finite-state program P -validity problem (of ϕ ) Given a finite-state program P and formula ϕ , is ϕ P -valid? i.e. do all P -computations satisfy ϕ ? P -satisfiability problem (of ϕ ) Given a finite-state program P and formula ϕ is ϕ P -satisfiable? i.e., does there exist a P -computation which satisfies ϕ ? To determine whether ϕ is P -valid, it suffices to apply an algorithm for deciding if there is a P -computation that satisfies ¬ ϕ . 14-2
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The Idea To check P -satisfiability of ϕ , we combine the tableau T ϕ and the transition graph G P into one product graph, called the behavior graph B ( P,ϕ ) , and search for paths ( s 0 , A 0 ) , ( s 1 , A 1 ) , ( s 2 , A 2 ) , . . . that satisfy the two requirements: σ q ϕ : there exists a fulfilling path π : A 0 , A 1 , . . . in the tableau T ϕ such that ϕ A 0 . σ is a P -computation: there exists a fair path σ : s 0 , s 1 , . . . in the transition graph G P . 14-3
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State transition graph G P : Construction Place as nodes in G P all initial states s ( s q Θ ) Repeat for some s G P , τ ∈ T , add all its τ -successors s 0 to G P if not already there, and add edges between s and s 0 . Until no new states or edges can be added. If this procedure terminates, the system is finite-state. 14-4
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Example: Program mux-pet1 (Fig. 3.4) (Peterson’s Algorithm for mutual exclusion) local y 1 , y 2 : boolean where y 1 = f , y 2 = f s : integer where s = 1 P 1 :: 0 : loop forever do 1 : noncritical 2 : ( y 1 , s ) := ( t , 1) 3 : await ( ¬ y 2 ) ( s 6 = 1) 4 : critical 5 : y 1 := f P 2 :: m 0 : loop forever do m 1 : noncritical m 2 : ( y 2 , s ) := ( t , 2) m 3 : await ( ¬ y 1 ) ( s 6 = 2) m 4 : critical m 5 : y 2 := f 14-5
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Abstract state-transition graph for mux-pet1 m 2 m 3 m 5 5 2 3 s = 1 9 s = 1 2 6 1 s = 2 3 s = 2 1 2 s = 2 s = 2 s = 2 9 s = 1 s = 1 3 at - m 0 .. 2 at - m 3 at - m 4 , 5 at - 0 .. 2 at - 3 at - 4 , 5 s = 1 6 We use y 1 at - 3 .. 5 y 2 at - m 3 .. 5 14-6
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Some states have been lumped together: a super-state labeled by i represents i states mux-pet1 has 42 reachable states.
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