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Diffstat (limited to 'Assignment1/syntax.tex')
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diff --git a/Assignment1/syntax.tex b/Assignment1/syntax.tex index 8543669..708d3b8 100644 --- a/Assignment1/syntax.tex +++ b/Assignment1/syntax.tex @@ -61,6 +61,7 @@ In discourse, it can be helpful to distinguish different kinds of PLTL formulas. Note that the classes \wffn, \wffp\ and \wfff\ are mutually disjoint. Furthermore, there are \wff\ which are neither, like $\Xop a \land \Xop^{-1} a$. +However, as we will see in \cref{pltl:to-ltl:syntactic}, every PLTL formula can be written as a boolean combination of $\wfff \cup \wffp \cup \wffn$. As with LTL, we can also derive additional operators in PLTL. They can be seen as counterparts of the derived LTL modalities $\square$ and $\lozenge$: @@ -70,11 +71,15 @@ They can be seen as counterparts of the derived LTL modalities $\square$ and $\l \erin \begin{definition}[Derived PLTL operators] Given $\phi \in PLTL$, then: - \begin{equation*} - \Fop^{-1} \phi \enspace\defeq\enspace \top \Sop \phi \qquad - \Gop^{-1} \phi \enspace\defeq\enspace \neg \Fop^{-1} \neg \phi + \[ + \begin{array}{rlrl} + \Fop \phi &\defeq \top \Uop \phi \qquad + & \Gop \phi &\defeq \neg \Fop \neg \phi \\ + \Fop^{-1} \phi &\defeq \top \Sop \phi + & \Gop^{-1} \phi &\defeq \neg \Fop^{-1} \neg \phi + \end{array} \qedhere - \end{equation*} + \] \end{definition} Their intuitive meaning is as follows. @@ -145,14 +150,13 @@ we include an arrow that points to the state for which the formula holds. \camil \label{pltl:dual-modalities} -Dual modalities, like the LTL $\square\lozenge$ \enquote{infinitely often} and \enquote{eventually forever}, are less useful in PLTL. +Dual modalities, like the LTL $\square\lozenge$ \enquote{infinitely often} and \enquote{eventually forever}, are less useful with past modalities. $\square^{-1}\lozenge^{-1}\phi$ 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. Interestingly, $\lozenge^{-1}\square^{-1}\phi$ means the same: it holds when $\phi$ held from the beginning until some moment in the past. -The reason that dual modalities in PLTL are less useful is that we still consider traces with a fixed starting point. +The reason that dual past modalities are less useful is that we still consider traces with a fixed starting point. Thus, while with future modalities it is possible to look infinitely far in the future, it is not possible to look infinitely far in the past. -%TODO: Give examples of semantics along the lines of subsection 5.1.1. Before turning to the formal semantics in the next subsection, we provide some examples of PLTL formulae and their uses. \begin{example}[Properties for a Traffic Light] @@ -164,7 +168,7 @@ Before turning to the formal semantics in the next subsection, we provide some e \[\square\left(\textsl{green} \rightarrow \lnot \Pop\textsl{red}\right) \qedhere\] \end{example} -\begin{example}[Properties for an Authentication System] +\begin{example}[A property for an Authentication System] \label{ex:pltl:authentication-system} \citet{FiterauBrostean2017} use past modalities to describe properties of the Secure Shell (SSH) protocol. One property says that if a channel is opened, there must have been some successful authentication attempt in the past~\citep[p.~149]{FiterauBrostean2017}. @@ -177,4 +181,14 @@ Before turning to the formal semantics in the next subsection, we provide some e The LTL formula is derived algorithmically in \cref{ex:pltl-to-ltl:authentication}. The LTL formula is slightly larger than the PLTL formula and is slightly less understandable. \end{example} + +\begin{remark}[Other Notations] + % TODO: I don't find this place logical for this remark, but it is the same place as Remark 5.16 in the book. + Like for LTL, many different notations are used in literature for PLTL. + These include + $\mathbf X^{-1}, \mathbf G^{-1}, \mathbf F^{-1}$~\citep[e.g.]{Markey2003}, + but also \raisebox{-1pt}{\tikz\draw[black,fill=black](0,0)circle(.4em);}$,\blacksquare,\blacklozenge$~\citep[e.g.]{Gabbay1989} + or $\stackinset{c}{}{c}{}{$\cdot$}{$\bigcirc$},\boxdot,\stackinset{c}{}{c}{}{$\cdot$}{$\lozenge$}$~\citep[e.g.]{Havelund2002}. + % It is always fun to come up with new versions and watch people struggling to reproduce them in \LaTeX. +\end{remark} \cbend |