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I've read that in general for a modal formula P, a world w and a Kripke frame ⟨W,R⟩

w⊨□P if and only if for every u∈W, if wRu then u⊢P

In case of LTL, being a modal logic, I assumed that the worlds are different moments in time and the relation R is the chain of "arrows" between those moments. However, in this case the semantic of □ in LTL would not correspond to the definition above.

What did I misunderstood?

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2 Answers 2

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In LTL, you don't have $\diamond$ and $\Box$, but you have $X$ and $U$, meaning next and until, respectively.

In LTL, every world has a unique successor.

Next is defined as $w \models X \phi$ if $w' \models \phi$, where $w'$ is the successor of $w$.

Until is defined as follows: $w \models \phi U \psi$ if there exists $w''$ such that $w'' \models \psi$ and for every $w'$ between $w$ and $w''$, we have that $w' \models \phi$.

Now, from these modalities, we can construct other modalities, such as globally, $G \phi$, and finally, $F \phi$. It is common to denote

  • $X \phi$ as $\bigcirc \phi$
  • $F \phi$ as $\diamond \phi$
  • $G \phi$ as $\Box \phi$.
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  • $\begingroup$ you start with "In LTL, you don't have ⋄ and ◻" and finish with "It is common to denote 𝐹𝜙 as ⋄𝜙 and 𝐺𝜙 as ◻𝜙". Isn't it up to personal preference? $\endgroup$
    – Ayrat
    Mar 7, 2023 at 9:41
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    $\begingroup$ I would rather call it convention than personal preference, and in temporal logics, the convention is using the alphabetic modalities, e.g. $F$, $X$, $G$, and $AF$, $EF$, $AG$, $EG$, $A (\phi U \psi)$ and $E (\phi U \psi)$. Meaning that CTL, CTL*, ATL and LTL are defined on the language using $A,E$ (when applicable), and $F,G,X$, and $U$. Then we throw in the boxes and diamonds as syntactic sugar, if you will. $\endgroup$
    – Pål GD
    Mar 7, 2023 at 12:05
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    $\begingroup$ I'd remove the first sentence. It's an opinion at best and not supported by facts. Manna and Pnueli use $\bigcirc$, $\square$, $\diamond$ in their articles and reference books. Lamport's TLA doesn't have a next operator but he uses $\square$ and $\diamond$. $\endgroup$
    – Kai
    Mar 7, 2023 at 21:02
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tl;dr The accessibility relation needs to be the reflexive transitive closure of what you had in mind.

Details: Let $P$ be a set of atomic proposition. Write $\Sigma$ for $P$'s powerset. Write $T$ for $\Sigma^{\mathbb{N}}$ (infinite traces).

A Kripke frame $\langle W, R\rangle$ where the meaning of formulas agrees with that of the fragment of LTL using just $P$, boolean connectives, $\square$ and $\diamond$ would consist of $W = T\times\mathbb{N}$ (ie infinite traces with an index) and $R = \{~((\sigma,i),(\sigma,k))~|~\sigma\in T\wedge i\leq k\in\mathbb{N}~\}$.

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