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@@ -117,6 +117,7 @@ we include an arrow that points to the state for which the formula holds.
\end{figure}
\camil
+\label{pltl:dual-modalities}
Dual modalities, like the LTL $\square\lozenge$ \enquote{infinitely often} and \enquote{eventually forever}, are less useful in PLTL.
$\square^{-1}\lozenge^{-1}\phi$ intuitively holds when at every point in the past, $\phi$ held or there was a previous moment at which $\phi$ held.
This is satisfied precisely when $\phi$ held at the first moment in time.