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[Solved]: Given a set of LTL formulas, on which states does the Kripke structure hold?

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Problem Detail: 

I'm currently learning about LTL and CTL formulas and to get a better understanding I try to manually interpret the formulas over a given Kripke structure. Since I'm not 100% sure if my results are correct I would appreciate if anyone can verify them.

Task:

I showed on which states the given LTL formular hold.

Some LTL notation notes:

$X$ equals $\bigcirc$

$G$ equals $\Box$

$F$ equals $\diamond$

Given Kripke structure

  1. $Fc = \{\}$

    My interpretation: $Fc$ means that on all paths $c$ holds sometimes the future. Since all paths come along $t4$ it doesn't hold for any state.

  2. $G(b \vee c) = \{\}$

    My interpretation: For all paths holds globally b or c.

  3. $G(Fb) = \{t0, t1, t2, t3, t4, t5, t6\}$

    My interpretation: For all paths holds globally that eventually b will be true.

  4. $G(b \Rightarrow (Xa \Rightarrow Xb)) = \{t0, t1, t2, t3, t4, t5, t6\}$

    My interpretation: Since $Xa \Rightarrow Xb$ is true for every state the implication $b \Rightarrow (Xa \Rightarrow Xb)$ must hold for all states too sinde $? \Rightarrow true$ is always true.

  5. $a U (b U c) = \{t1, t3, t4, t5\}$

    My interpretation: Following paths are valid: aaaaabbbc, bbbbc, c, ccc. Therefore the states $t1, t3, t4, t5$ are valid.

So can anybody confirm my results?

Asked By : Mad A.

Answered By : Klaus Draeger

A couple of comments:

  1. Note that "in the future" is not strict, i.e. $Fb$ is also satisfied whenever $b$ itself holds.
  2. Looks fine to me.
  3. Also ok, though I'm not sure if your interpretation would be very enlightening in a more complicated Kripke structure.
  4. "$Xa\Rightarrow Xb$ is true for every state" is a CTL-ism. What you want to say is that for every state along every path, if $a$ holds in this state and $b$ holds in the next state, then $a$ also holds in the next state.Is this the case?
  5. Here you want some finite number (possibly $0$) of $a$s, followed by some finite number (possibly $0$) of $b$s, followed by a $c$. Is this what you see along every path starting in the states you give?
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Question Source : http://cs.stackexchange.com/questions/41704

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