Model Checking for CTL

Marks the states of K by subformulas of P

s is marked by a subformula Q if Q holds

at TK,s

The algorithm proceeds from simple formulasto more complex formulas for all states

simultaneously.

Algorithm

For atomic formulas – immediately

For Boolean connectives – easy

s is marked by P1& P2 if ….

For modal connectives: P1 U P2 : if from s there is a P1 path to a P2 node.

For modal connectives: P1 U P2 ……

CTL*

Modalities: E( a formula of TL(U))

A ( a formula of TL(U))

Semantics: T,s|= E C if there is a path from s

which has a property C.

Model Checking for CTL*

How to check E (‘ property of a path’)

Construct an automaton A for the property.

Take the product with the Kripke Structure.

Equation for P1 U P2

X - the set that satisfy P1 U P2

X= P2 (X& P1 )

X=H(X) where H = λ Y. P2 (Y & P1 )

How many solution Z=H(Z) has?

Characterization of P1 U P2

P1 U P2 is the minimal solution of

Z= P2 (Z & P1 )

X0= P2

Xn+1= P2 (Xn & P1 )

s in Xn iff there is a P1 path of length≤ n+1 from s to P2

X= Xn X=H(X) and H monotonic

Mu-calculus

E := At| ¬ At| X| E1 &E2| E1E2| E | A E| μ X. E| νX.E

Semantics: μ least fixed point; ν greatest fixed point.

[| E |]ρ the set of states that satisfies E in the enviroment ρ: Var-> States.

EGp

EGp = νX.p& X

From mu-calculus to MLO

Theorem: for every mu-formula c(X1,…,Xn)there is an MLO formula b(t, X1,…Xn) whichis equivalent to c over trees.

Theorem: for every future MLO formula b(t,X1,…Xn) which is invariant under counting there is an equivalent (over trees) mu formula c.

Symbolic Model Checking

Explicit Model Checking:

Input a finite state K and a formula c

Task Find the states of K that satisfy c.

Symbolic model checking

Input a description of K and a formula c

Task Find a description of the states of K that satisfy c.

A description of Kripke structures by formulas

• s(x1,…,xn) describes a set of states

• t(x1,…xn,x1’,…xn’) describes transitions

• For every label p a formula lp(x1,…xn) that describes the states labeled by p.

BDT, and OBDD

• Binary decision trees

• Ordered Binary Decision Diagrams.