Download presentation

Presentation is loading. Please wait.

Published byThomasine Franklin Modified over 2 years ago

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

Similar presentations

OK

© 2004 Pearson Addison-Wesley. All rights reserved February 20, 2006 ‘do’ and ‘for’ loops ComS 207: Programming I (in Java) Iowa State University, SPRING.

© 2004 Pearson Addison-Wesley. All rights reserved February 20, 2006 ‘do’ and ‘for’ loops ComS 207: Programming I (in Java) Iowa State University, SPRING.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Free ppt on brain machine interface insect Free ppt on different types of houses Ppt on european central bank Ppt on current account deficit capital account Esi tof ms ppt online Ppt on vitamin b12 deficiency Keynote opening ppt on ipad Resource based view ppt on mac Ppt on knowledge management Ppt on phonetic transcription chart