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?


2 Answers 2


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 $\Box \phi$.
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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 $R = \{~((\sigma,i),(\sigma,k))~|~\sigma\in T\wedge i\leq k\in\mathbb{N}~\}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.