l5-6 - CS 267 Automated Verification Lectures 5 and...

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CS 267: Automated Verification Lectures 5 and 6: μ -calculus, symbolic model checking Instructor: Tevfik Bultan

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μ -Calculus μ -Calculus is a temporal logic which consist of the following: Atomic properties AP Boolean connectives: ¬ , , Precondition operator: EX Least and greatest fixpoint operators: μ y . F y and ν y. F y F must be syntactically monotone in y meaning that all occurrences of y in within F fall under an even number of negations
μ -Calculus 2200 μ -calculus is a powerful logic Any CTL* property can be expressed in μ -calculus So, if you build a model checker for μ -calculus you would handle all the temporal logics we discussed: LTL, CTL, CTL* One can write a μ -calculus model checker using the basic ideas about fixpoint computations that we discussed However, there is one complication Nested fixpoints!

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Mu-calculus Model Checking Algorithm eval(f : mu-calculus formula) : a set of states case: f AP return {s | L(s,f)=true}; case: f ¬ p return S - eval(p); case: f p q return eval(p) eval(q); case: f p q return eval(p) eval(q); case: f EX p return EX(eval(p));
Mu-calculus Model Checking Algorithm eval(f) case: f μ y . g(y) y := False; repeat { y old := y; y := eval(g(y)); } until y = y old return y;

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Mu-calculus Model Checking Algorithm eval(f) case: f ν y . g(y) y := True; repeat { y old := y; y := eval(g(y)); } until y = y old return y;
Nested Fixpoints Here is a CTL property EG EF p = ν y . ( μ z . p EX z) EX y The fixpoints are not nested. Inner fixpoint is computed only once and then the outer fixpoint is computed Fixpoint characterizations of CTL properties do not have nested fixpoints Here is a CTL* property EGF p = ν y . μ z . ((p EX z) EX y) The fixpoints are nested. Inner fixpoint is recomputed for each iteration of the outer fixpoint

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Nested Fixpoint Example 1 0 2 p 0 |= EG EF p EF p EF p EG EF p = ν y . ( μ z . p EX z) EX y EGF p = ν y . μ z . ((p EX z) EX y) 0 |= EGF p F 1 F 2 F 3 F 1 ( ) = {1} F 1 2 ( ) = {0,1} F 1 3 ( ) = {0,1} S={0,1,2} F 2 (S) = {0,1} F 2 2 (S) = {0} F 2 3 (S) = {0} EG EF p = {0} F 3 y z 0,0 {0,1,2} 0,1 {1} 0,2 {0,1} 0,3 {0,1} 1,0 {0,1} 1,1 2,0 2,1 3,0 EGF p = EF p fixpoint EG {0,1} fixpoint nested fixpoint
Symbolic Model Checking [McMillan et al. LICS 90] Basic idea: Represent sets of states and the transition relation as Boolean logic formulas Fixpoint computation becomes formula manipulation pre-condition (EX) computation: Existential variable elimination conjunction (intersection), disjunction (union) and

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