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Model Checking for CTL and CTL*: Algorithms, Modalities, and Semantics Explained

This paper delves into the algorithms and semantics of Model Checking for Computation Tree Logic (CTL) and CTL*. We outline a systematic approach to marking states with subformulas and detail the algorithm's operation from simple to complex formulas across all states. The discussion extends to atomic and Boolean connectives, modal connectives, and the semantics surrounding path properties. Additionally, we introduce symbolic and explicit model checking, the significance of Kripke structures, and representation through formulas, trees, and automata, exemplifying the theoretical foundations necessary for practitioners in the field.

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Model Checking for CTL and CTL*: Algorithms, Modalities, and Semantics Explained

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  1. 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 formulas to more complex formulas for all states simultaneously.

  2. 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 ……

  3. 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.

  4. 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.

  5. 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?

  6. 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

  7. Mu-calculus E := At| ¬ At| X| E1 &E2| E1E2|  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.

  8. EGp EGp = νX.p&  X

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

  10. 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.

  11. 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.

  12. BDT, and OBDD • Binary decision trees • Ordered Binary Decision Diagrams.

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