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

This preview shows pages 1–10. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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;

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/27/2011 for the course CMPSC 267 taught by Professor Bultan during the Fall '09 term at UCSB.

### Page1 / 29

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online