Temporal Logic - LTL - 2

In the previous post (LTL - 1), we started talking about Linear Temporal Logic, a logic that allows expressing temporal properties about every path of a model.
In particular we have observed the temporal operators (or modalities):
G (Globally)
F (Eventually)
There is a relation between these two operators: Writing that globally something (property) holds, is equivalent to saying that the negation of the property will never hold. In terms of the operators this is expressed as:
G property = not F not property
I take this opportunity to present syntax notations I will use in future posts:
not = \( \neg \)
or = \( \lor \)
and = \( \land \)
Now, other temporal operators of interest are:
X formula (at the next state formula holds)
formula1 U formula2 (formula1 holds until formula2 holds)
For example,

In the given state machine, p q and r are atomic propositions. For example, p could represent x = 1 , q represent y > 0, and r represent that z is within some range of values. These are just examples, for each system, the set of atomic propositions is defined, and with these, the state machine representing the system is built. We will consider that if an atomic proposition does not appear in a state, then that atomic proposition does not hold at that state. For example the first state satisfies \( p \land \neg q \land \neg r \).

When standing at the first state of the model in the example (let's call it M), we can see that the next state satisfies q. That is, \( M \vDash \textbf{X}\ q \). When writing that the model satisfies an LTL formula, we are saying that every path starting from the initial states of the model satisfies the formula.

We can also observe that \(M \vDash  \textbf{X}\ \neg p \) ,\( M \vDash   \neg \textbf{X}\ r \), \( M \vDash  \textbf{X} \textbf{X}\ q \), and \( M \vDash  \textbf{X} \textbf{X}\textbf{X}\ p \). Question to the reader: why?


Thus, X expresses at the next state (from the state we are currently at). For example although the model does not satisfy that at the next state of every path starting from the initial states p will hold, there exists a future state such that its next state will satify p. That is \( M \vDash \textbf{F}\textbf{X}\ p \). And this way the different temporal operators can be combined.

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The remaining operator for this post is U: The idea is that a formula "formula1 U formula2" is satisfied when formula1 holds at every state until some state where formula2 holds. If formula2 holds, then the formula with until holds. If formula2 never holds, then the until formula is not satisifed. In the next blog we will give the more formal definitions of the temporal operators (or modalities) and understand why the express the general idea we have seen in these two posts.