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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?

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  • 3
    $\begingroup$ Your question already includes a complete answer to the original problem but no question about this answer. Thus, only "yes/no" answers may remain, helping neither you nor future visitors. Please read related meta discussions here and here and adjust your question accordingly, e.g. by formulating a specific question about a single element of your answer you are uncertain about. If you just want general feedback, you are welcome to visit us in Computer Science Chat. $\endgroup$ – Raphael Apr 22 '15 at 15:18
  • $\begingroup$ I calculated which state holds for which LTL formula. But I'm not certain if my results are correct, therefore I wanted someone to confirm them. I'll join the chat, if you think my question is more appropriate there. $\endgroup$ – Mad A. Apr 22 '15 at 15:30
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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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  • $\begingroup$ Hello Klaus, thanks for your input. AD1: So t1 and t3 are states that satisfy $Fc$, is that correct? AD4: I feel like I didn't understand the Ne$X$t operator correctly. Let's only look at $Xa \Rightarrow Xb$, that LTL formula would be satisfied for every state $s0,...,s6$ right? AD5: Yes exactly. There must be at least one state where $c$ is true. Befor that the state must be a set of a's (i.e. aaaaac) a set of b's (bbc) or a set of a's followed by b's followed by at least one c (aaabbbc). Also do you think the states for #4 and #5 are correct disregarding from the interpretation? $\endgroup$ – Mad A. Apr 22 '15 at 19:43
  • $\begingroup$ Ad 4: You may be thinking of the CTL formula $EXa\Rightarrow EXb$. It is important to remember that LTL semantics is defined in terms of traces; $Xa\Rightarrow Xb$ means that if the second state on a trace satisfies $a$, then it also satisfies $b$. Ad 5: Nothing forces you to ever leave $t4$. $\endgroup$ – Klaus Draeger Apr 23 '15 at 11:38

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