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CS 267: Automated Verification Lectures 4: -calculus Instructor: Tevfik Bultan

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-Calculus -Calculus is a temporal logic which consist of the following: Atomic properties AP Boolean connectives: , , Precondition operator: EX Least and greatest fixpoint operators: y. F y and 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

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-Calculus -calculus is a powerful logic –Any CTL* property can be expressed in -calculus So, if you build a model checker for -calculus you would handle all the temporal logics we discussed: LTL, CTL, CTL* One can write a -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 APreturn {s | L(s,f)=true}; case: f preturn S - eval(p); case: f p q return eval(p) eval(q); case: f p q return eval(p) eval(q); case: f EX preturn EX(eval(p));

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Mu-calculus Model Checking Algorithm eval(f) … case: f 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 y. g(y) y := True; repeat { y old := y; y := eval(g(y)); } until y = y old return y;

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Nested Fixpoints Here is a CTL property EG EF p = y. ( 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 = y. 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 102 p 0 |= EG EF p EF p EG EF p = y. ( z. p EX z) EX y EGF p = y. z. ((p EX z) EX y) 0 |= EGF p F1F1 F2F2 F3F3 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 yz 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 fixpointEG {0,1} fixpoint nested fixpoint

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