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\subsubsection{Semantics}
\erin
The semantics of LTL and PLTL are defined in a very similar way.
However, we must make some modifications to the definitions.
\camil
In particular, the $\vDash$ operator must be redefined.
Recall that for LTL we wrote $\sigma \vDash \phi$ when $\sigma$ satisfies $\phi$ and $\sigma[i\dots]$ for the suffix of $\sigma$ starting in the $(j+1)$st symbol.
The latter notation effectively loses information about the prefix.
In the case of PLTL, we cannot lose this information.
For this reason, we use a satisfaction relation at a specific index.
We will write this as $\sigma \vDash_i \phi$, read as \enquote{$\sigma$ satisfies $\phi$ at $i$}.%
\footnote{%
The notation used in literature varies.
\citet{Lichtenstein1985} use $(\sigma,i) \vDash \phi$; \citet{Markey2003} uses $\sigma,i \vDash \phi$.
Although the difference is minor, we find $\vDash_i$ more intuitive because it shows that the object being checked is the same in $\sigma\vDash_i\phi$ and $\sigma\vDash_j\phi$.}
We use $\sigma \vDash \phi$ as a shorthand for $\sigma \vDash_0 \phi$.
\erin
\begin{definition}[Semantics of PLTL (Interpretation over Words)]
Let $\phi$ be a PLTL formula over $AP$. The LT property induced by $\phi$ is
\[Words(\phi) = \{\sigma \in \left(2^{AP}\right)^\omega \mid \sigma \vDash \phi\}\]
where $\vDash\ \subseteq \left(2^{AP}\right)^\omega \times \mathbb{N} \times PLTL$ is the smallest relation with the properties in \cref{fig:PLTL-semantics}.
\end{definition}
\camil
Note that we must redefine the satisfaction relation for the LTL operators, because $\vDash$ is now a ternary relation.
The difference with LTL formulae is well illustrated by the $\bigcirc$ operator.
In the LTL case, $\sigma=A_0A_1A_2\dots\vDash\bigcirc\phi$ iff $\sigma[1\dots]=A_1A_2\dots\vDash\phi$.
Thus, whether $\bigcirc\phi$ holds for $\sigma$ cannot depend on $A_0$.
In the PLTL case, $\sigma\vDash_i\bigcirc\phi$ iff $\sigma\vDash_{i+1}\phi$.
The information about $A_0$ is not lost, so whether $\bigcirc\phi$ holds for $\sigma$ may depend on $A_0$,
as is trivially the case with $\bigcirc\bigcirc^{-1}a$.
\begin{figure}
\centering
\begin{mdframed}
\[
\begin{matrix*}[l]
\sigma &\vDash_i & \top\\
\sigma &\vDash_i & a &\text{iff} & a\in A_i\\
\sigma &\vDash_i & \phi_1\land\phi_2 &\text{iff} & \text{$\sigma\vDash_i\phi_1$ and $\sigma\vDash_i\phi_2$}\\
\sigma &\vDash_i & \lnot\phi &\text{iff} & \sigma \nvDash_i \phi\\
\sigma &\vDash_i & \bigcirc\phi &\text{iff} & \sigma \vDash_{i+1} \phi\\
\sigma &\vDash_i & \phi_1\Uop\phi_2 &\text{iff} & \exists_{j \ge 0}.\text{$\sigma \vDash_j \phi_2$ and $\sigma \vDash_i \phi_1$ for all $0 \le i < j$}\\
\sigma &\vDash_i & \phi_1\Sop\phi_2 &\text{iff} & \exists_{0 \le j \le i}.\text{$\sigma \vDash_j \phi_2$ and $\sigma \vDash_k \phi_1$ for all $j < k \le i$}\\
\sigma &\vDash_i & \Pop\phi &\text{iff} & \text{$i \geq 1$ and $\sigma \vDash_{i-1} \phi$}
\end{matrix*}
\]
\end{mdframed}
\caption{PLTL semantics for infinite words $\sigma=A_0A_1A_2\dots \in \left(2^{AP}\right)^\omega$.}
\label{fig:PLTL-semantics}
\end{figure}
\erin
Given these definitions, it is possible to derive the formal semantics of the $\Fop^{-1}$ and $\Gop^{-1}$ operators as well:
\[
\begin{matrix*}[l]
\sigma &\vDash_i &\Fop^{-1}\phi &\text{iff} &\exists_{0 \leq j \leq i}[\sigma \vDash_j \phi]\\
\sigma &\vDash_i &\Gop^{-1}\phi &\text{iff} &\forall_{0 \leq j \leq i}[\sigma \vDash_j \phi]
\end{matrix*}
\]
\camil
The first follows from the rule for $\Sop$ and the fact that $\sigma \vDash_i \top$ for all $i\in\mathbb N$.
The second follows from:
\begin{align*}
\sigma \vDash_i \square^{-1}\phi
&\quad\Leftrightarrow\quad \sigma \vDash_i \lnot\lozenge^{-1}\lnot\phi\\
&\quad\Leftrightarrow\quad \sigma \nvDash_i \lozenge^{-1}\lnot\phi\\
&\quad\Leftrightarrow\quad \nexists_{0\le j\le i} [\sigma \vDash_j \lnot\phi]\\
&\quad\Leftrightarrow\quad \forall_{0\le j\le i} [\sigma \vDash_j \phi].
\end{align*}
With this result we can also derive the semantics of the dual modalities $\lozenge^{-1}\square^{-1}$ and $\square^{-1}\lozenge^{-1}$:
\[
\sigma \vDash_i \lozenge^{-1}\square^{-1}\phi
\quad\Leftrightarrow\quad
\sigma \vDash_i \square^{-1}\lozenge^{-1}\phi
\quad\Leftrightarrow\quad
\sigma \vDash_0 \phi
\]
This is done in \cref{ex:dual-modalities}.
\erin
\begin{definition}[Semantics of PLTL over Paths and States]
Let $TS = (S, Act, \rightarrow, I, AP, L)$ be a transition system without terminal states, and let $\phi$ be an PLTL-formula over AP.
\begin{itemize}
\item For infinite path fragments $\pi$ of $TS$, the satisfaction relation is defined by
\[\pi \vDash \phi \Leftrightarrow trace(\pi) \vDash \phi\]
\item For state $s \in S$, the satisfaction relation $\vDash$ is defined by
\[s \vDash \phi \Leftrightarrow \forall_{\pi \in Paths(s)}[\pi \vDash \phi]\]
\item $TS$ satisfies $\phi$, denoted by $TS \vDash \phi$ if $Traces \vDash \phi$, if $Traces(TS) \subseteq Words(\phi)$
\qedhere
\end{itemize}
\end{definition}
In order to make these definitions suitable for use with our PLTL operators,
we must provide their semantics.
\Cref{fig:PLTL-semantics} shows the formal semantics of the operators defined in the grammar.
Note that we need to add a index $i$, since we must also be able to look in the past.
\cbend
|
