I am trying to learn about the following temporal operators: $X,F,G$ where $X$ is for neXt (sometimes denoted by $N$), $F$ is Future and $G$ stands for Globally.
I believe I understand what each one means, but I am having a problem understanding the diagrams at Wikipedia.
For $X$:
I am not sure about which color denotes True (they use white and red), but $Xp$ should be true at the next state, hwo did they say this in the diagram ?
Also (and this is a question for all $3$ diagrams): $Xp$ does not depends on if $p$ is currently true or not, so why do they sketch the truth values of $p$ above $Xp$ ?
For $F$ -
again, I don't understand how the diagram tells us what $Fp$ means.
For $G$:
This one is the most confusing, I would of expected $Gp$ to be in the same color (the one that denotes True), why some parts are white while the others are red ?
I would appriciate any help in understanding those diagrams!
Your $Xp$ is their $Np$, for "Next, $p$." The red indicates truth along that fixed $y$-coordinate. So for instance, the semantics for $Np$ state "$Np$ is true just in case $p$ is true at the next time interval." In your first diagram, $Np$ is true at 1-2, so at the next time interval (2-3), $p$ must be true. If, for example, $p$ were true at intervals 3-5 and 7-8, we'd have $Np$ being true at times 2-4 and 6-7.
$Fp$ is true just in case $p$ is true at some future time. So $p$ since $p$ is true at intervals 2-3 and 4-5, $Fp$ will be true up to 5. If $p$ is not true at another other intervals, then $Fp$ will not be true past 5.
$Gp$ is true just in case $p$ will be true at all future times. In this case, since the intervals only go up to 6, if $p$ is true from 4-6, then $Gp$ is true from 4-6. But at intervals 1-4, $p$ isn't always true at all future times, so $Gp$ isn't true.