# Kripke model and LTL vs CTL formulae interpretation [closed]

I have this Kripke model $M$:

$$\begin{array}{ccccccc} \to & (p, q) & \to & (\neg p, \neg q) & \to & (p, \neg q) \\ & \circlearrowright & & & & \circlearrowright & \\ \end{array}$$

where $(p, \neg q)$ means “$p$ and not $q$” and $\circlearrowright$ is a self loop.

Now I cannot wrap my mind as to why:

• $M \vDash \mathop{\mathbf{A}}\mathop{\mathbf{F}}\mathop{\mathbf{A}}\mathop{\mathbf{G}}p$ is false in CTL;
• $M \vDash \mathop{\mathbf{A}}\mathop{\mathbf{F}}\mathop{\mathbf{G}}p$ is true in CTL*.

If you have a reasonable explanation for the above I might post a second analogous example which might disprove your intuition.

• What are AFAG and AFG? $\:$ What is the * for? $\;\;$ – user6973 Jun 29 '12 at 8:54
• AFAG is AF(AG(p)) where AF stand for All Finally and AG stands for All Globally, CTL* is a temporal language. If you don't understand the above I think you should first read a book on the subject. – dendini Jun 29 '12 at 15:30
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• @Kaveh Feel free to migrate to Computer Science – Gilles 'SO- stop being evil' Jun 30 '12 at 0:53

Intuitively, what happens here is that for $AFGp$, you check each individual path for whether after some point, $p$ will always be true - no matter what other choices are available in a given state.

In particular, for the path which always stays in the first state, this is true even though a $\neg p$-state is reachable. On all other paths it is true because they eventually reach the third state.

In the case of $AFAGp$, the second path quantifier "overwrites" the first in a sense: here you have to check whether for all paths, you eventually reach a state such that all paths from that state - not just the one you were originally following - always satisfy $p$. For the path staying in the first state, this is not true, because there is always a branch going to the second state.

• what is AFPp ? When you say "In particular, for the path which always stays in the first state, this is true even though a ¬p-state is reachable.", this sounds like a contradiction to me without any further explanation! – dendini Jun 29 '12 at 15:39
• That was a typo - was supposed to be $AFGp$, thanks for catching it. As for the apparent contradiction: The point really is that for this formula, each path is checked individually for whether $FGp$ holds along it. That the $\neg p$-state is reachable from the first one is irrelevant for the path which does not in fact reach it. – Klaus Draeger Jun 29 '12 at 17:35

Lets name the states $s_1,s_2,s_3$, from left to right.

First, we show why the first formula is false: In order to see this, we first need to check which states satisfy the sub-formula $AG(p)$.

$s_3$ satisfies $AG(p)$ because there is only one infinite computation starting from $s_3$ which is $s_3^\omega$. However, $s_1$ does not satisfy $AG(p)$ since there is a computation, namely $s_1 s_2 s_3^\omega$, starting from $s_1$ that does not satisfy $AG(p)$.

Next, we check if $s_1$ satisfies $AFAG(p)$. By the above, we need to check, informally, if $s_1$ satisfies $AF(s_3)$. Clearly, it doesn't because of the computation $s_1^\omega$.

Next, we show that the structure satisfies $AFG(p)$: The computations starting from $s_1$ have one of two formats: either $s_1^\omega$ or $s_1^* \cdot s_2 \cdot s_3^\omega$ (i.e., a finite prefix of $s_1$ and visit to $s_2$ and an infinite suffix of $s_3$). A computation of this form satisfies $FG(p)$, because it either ends in $s_3$ or $s_1$. Since these are all the computations starting from $s_1$ we have that $s_1$ satisfies $AFG(p)$.

I hope this helps..

• You jumped the part which interests me, that is AFG(p) looping on state s1, why don't I consider the fact that it can always in a future state jump to s2 while in CTL I condider that??... moreover you considered FG(p), shouldn't AF be one unbreakable construct and G(p) another one? – dendini Jun 29 '12 at 15:42
• AFGp is an LTL formula, which I find easier to understand. By definition, s1 satisfies AFGp iff all paths starting from s1 satisfy the path formula FGp. So that's what I showed in the answer - I went over all the options of paths starting from s1 and showed they satisfy FGp. – Guy Jun 29 '12 at 16:31
• Why doesn't this work for AFAGp: This is a CTL formula and formally it talks about trees. Intuitively, it means: all paths eventually reach a point from which all paths satisfy Gp. You can think of it as a tree: every branch in the tree corresponds to a computation of the Kripke struct. So every node in the tree is labeled with a subset of AP. The tree satisfies the formula iff every path from the root reaches a subtree in which every node has p. – Guy Jun 29 '12 at 16:35