1. CS 267: Automated Verification Lectures 4:  -calculus Instructor: 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  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));

5. 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;

6. 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;

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 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