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

1. CS 267: Automated VerificationLectures 4: -calculusInstructor: Tevfik Bultan

2. -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

3. -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!

4. 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));

5. Mu-calculus Model Checking Algorithm eval(f) … case: f  y . g(y) y := False; repeat { yold := y; y := eval(g(y)); } until y = yold return y;

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

7. 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

8. Nested Fixpoint Example 0 |= EGF p 0 1 2 EGF p =  y .  z . ((p  EX z)  EX y) p EF p EF p F3 F2 0 |= EG EF p nested fixpoint F3y 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  EG EF p =  y . ( z . p  EX z)  EX y F1 EF p fixpoint EG {0,1} fixpoint  F1() = {1} F12() = {0,1} F13() = {0,1} S={0,1,2} F2(S) = {0,1} F22(S) = {0} F23(S) = {0} EG EF p = {0} EGF p = 