* doc/tl/tl.tex: More text for the temporal hierarchy.

doc/tl/tl.tex

@@ -16,6 +16,13 @@
 \usepackage{tabulary}
 \usepackage[numbers]{natbib}
 \usepackage{rotating}
+\usepackage{tikz}
+\usetikzlibrary{backgrounds}
+\usetikzlibrary{shadows}
+\usetikzlibrary{arrows}
+\pgfdeclarelayer{background}
+\pgfdeclarelayer{foreground}
+\pgfsetlayers{background,main,foreground}
 
 % TODO
 \usepackage{todonotes}
@@ -115,6 +122,10 @@
 \makeindex
 \newcommand{\Index}[1]{\index{#1}#1}
 
+% Tikz
+\tikzstyle{annote}=[overlay,thick,<-,red,>=stealth']
+
+
 % Make sure we can compile this file even without use the version
 % supplied by the Makefile.
 \ifdefined\SpotVersion\else
@@ -959,6 +970,7 @@ properties can be queried from any \texttt{spot::ltl::formula}
 instance using the following methods:
 
 \noindent
+\label{property-methods}
 \begin{tabulary}{\textwidth}{lL}
 \texttt{is\_boolean()}& Whether the formula uses only Boolean
   operators.
@@ -1057,43 +1069,100 @@ rules:
            \mid \0       \R \varphi
            \mid \varphi_U\W \varphi_U
            \mid \varphi  \W \0
-           \mid \varphi_U\M \varphi_U\\
+           \mid \varphi_U\M \varphi_U
 \end{align*}
 
 \section{Syntactic Hierarchy Classes}
 
+\begin{figure}[tbp]
+  \centering
+ \begin{tikzpicture}
+    \draw[drop shadow,fill=white] (0,0) rectangle (6,7);
+    \path[fill=yellow!20] (0,6) -- (3,4.5) -- (0,3);
+    \path[fill=red!20] (6,6) -- (3,4.5) -- (6,3);
+    \path[fill=orange!20] (0,3) -- (3,4.5) -- (6,3) -- (3,1);
+    \path[fill=blue!40,fill opacity=.5] (0,0) -- (0,3) -- (4.5,0);
+    \path[fill=green!40,fill opacity=.5] (6,0) -- (1.5,0) -- (6,3);
+    \draw (0,0) rectangle (6,7);
+
+    \node[align=center] (rea) at (3,6)  {Reactivity\\ $\bigwedge\G\F p_i\lor \F\G q_i$};
+    \node[align=center] (rec) at (1,4.5) {Recurrence\\ $\G\F p$};
+    \node[align=center] (per) at (5,4.5) {Persistence\\ $\F\G p$};
+    \node[align=center] (obl) at (3,2.85) {Obligation\\ $\bigwedge\G p_i\lor \F q_i$};
+    \node[align=center] (saf) at (1,1) {Safety\\ $\G p$};
+    \node[align=center] (gua) at (5,1) {Guarantee\\ $\F p$};
+
+    \node[align=center,below left] (det) at (-.2,6.7) {Deterministic\\Büchi Automata};
+    \node[align=center,below right](weak) at (6.2,6.7) {Weak Büchi\\Automata};
+    \node[align=center,right](wdba) at (6.2,4.3) {Weak Deteterministic\\Büchi Automata};
+    \node[align=center,above right](ter) at (6.2,2) {Terminal\\Büchi Automata};
+    \node[align=center,above left](occ) at (-.2,2) {Terminal\\co-Büchi Automata};
+
+    \node[above right] (buc) at (3.35,7) {General Büchi Automata};
+
+    \draw[annote] (rec) -- (det);
+    \draw[annote] (per) -- (weak);
+    \draw[annote] (obl.east) -- (wdba);
+    \draw[annote] (gua) -- (ter);
+    \draw[annote] (saf) -- (occ);
+    \draw[annote] (rea.north) .. controls +(north:5mm) and +(west:5mm) .. (buc.west);
+  \end{tikzpicture}
+  \caption{The temporal hierarchy of \citet{manna.87.podc} with their
+    associated classes of automata~\citep{cerna.03.mfcs}.  The
+    formul\ae{} associated to each class are more than canonical
+    examples: they show the normal forms under which any LTL formula of
+    the class can be rewritten, assuming that $p,p_i,q,q_i$ denote
+    subformul\ae{} involving only Boolean operators, $\X$, and past
+    temporal operators (Spot does not support the
+    latter). \label{fig:hierarchy}}
+\end{figure}
+
 The hierarchy of linear temporal properties was introduced
-by~\citet{manna.87.podc}.  A first syntactic characterization of the
+by~\citet{manna.87.podc} and is illustrated on
-classes was presented by~\citet{chang.92.icalp}, but other
+Fig.~\ref{fig:hierarchy}.  In the case of the LTL subset of the
-presentations have been done including negation~\citep{cerna.03.mfcs}
+hierarchy, a first syntactic characterization of the classes was
-and weak until~\citep{schneider.01.lpar}.
+presented by~\citet{chang.92.icalp}, but other presentations have been
+done including negation~\citep{cerna.03.mfcs} and weak
+until~\citep{schneider.01.lpar}.
 
-In the following grammar rules, the symbols $\varphi_G$, $\varphi_S$,
+The following grammar rules describes extend the aforementioned
-$\varphi_O$, $\varphi_P$, $\varphi_R$ designate any formula belonging
+work slightly by dealing with PSL operators.  These are the
-respectively to the Guarantee, Safety, Obligation, Persistence, or
+rules used by Spot to decide upon
-Recurrence classes.  $v$ designates any variable, $r$ any SERE, $r_F$
+construction to which class a formula belongs (see the methods
-a bounded SERE (no loops), and $r_I$ an unbounded SERE.
+\texttt{is\_syntactic\_safety()}, \texttt{is\_syntactic\_guaranty()},
+\texttt{is\_syntactic\_obligation()},
+\texttt{is\_syntactic\_recurrence()}, and
+\texttt{is\_syntactic\_persistence()} listed on
+page~\pageref{property-methods}).
 
+The symbols $\varphi_G$, $\varphi_S$, $\varphi_O$, $\varphi_P$,
+$\varphi_R$ designate any formula belonging respectively to the
+Guarantee, Safety, Obligation, Persistence, or Recurrence classes.
+$v$ denotes any variable, $r$ any SERE, $r_F$ any bounded SERE (no
+loops), and $r_I$ any unbounded SERE.
 
 \begin{align*}
   \varphi_G ::=&\, \0\mid\1\mid v\mid \NOT\varphi_S\mid
                    \varphi_G\AND \varphi_G\mid (\varphi_G\OR \varphi_G)\mid
                    \X\varphi_G \mid \F\varphi_G\mid
                    \varphi_G\U\varphi_G\mid \varphi_G\M\varphi_G\\
-           \mid&\, \sere{r}\Esuffix \varphi_G\mid
+           \mid&\, \sere{r_F}\mid
-                   \sere{r_F}\Asuffix \varphi_G\\
+                   \sere{r}\Esuffix \varphi_G\mid
+                   \sere{r_F}\Asuffix \varphi_G \\
   \varphi_S ::=&\, \0\mid\1\mid v\mid \NOT\varphi_G\mid
                    \varphi_S\AND \varphi_S\mid (\varphi_S\OR \varphi_S)\mid
                    \X\varphi_S \mid \G\varphi_S\mid
                    \varphi_S\R\varphi_S\mid \varphi_S\W\varphi_S\\
-           \mid&\, \sere{r_F}\Esuffix \varphi_S\mid
+           \mid&\, \nsere{r_F}\mid
+                   \sere{r_F}\Esuffix \varphi_S\mid
                    \sere{r}\Asuffix \varphi_S\\
   \varphi_O ::=&\, \varphi_G \mid \varphi_S\mid \NOT\varphi_O\mid
                    \varphi_O\AND \varphi_O\mid (\varphi_O\OR \varphi_O)\mid
                    \X\varphi_O \mid
                    \varphi_O\U\varphi_G\mid\varphi_O\R\varphi_S \mid
                    \varphi_S\W\varphi_O\mid \varphi_G\M\varphi_O\\
-           \mid&\, \sere{r_F}\Esuffix \varphi_O \mid \sere{r_I}\Esuffix \varphi_G\mid
+           \mid&\, \sere{r} \mid \nsere{r}\mid
+                   \sere{r_F}\Esuffix \varphi_O \mid \sere{r_I}\Esuffix \varphi_G\mid
                    \sere{r_F}\Asuffix \varphi_O\mid
                    \sere{r_I}\Asuffix \varphi_S\\
   \varphi_P ::=&\, \varphi_O \mid \NOT\varphi_R\mid
@@ -1112,6 +1181,13 @@ a bounded SERE (no loops), and $r_I$ an unbounded SERE.
            \mid&\, \sere{r_F}\Esuffix \varphi_R \mid \sere{r_I}\Esuffix \varphi_G\mid \sere{r}\Asuffix \varphi_P\\
 \end{align*}
 
+It should be noted that a formula can belong to a class of the
+temporal hierarchy even if it does not syntactically appears so.  For
+instance the formula $(\G(q\OR \F\G p)\AND \G(r\OR \F\G\NOT p))\OR\G
+q\OR \G r$ is not syntactically safe, yet it is a safety formula
+equivalent to $\G q\OR \G r$.  Such a formula is usually said
+\emph{pathologically safe}.
+
 \chapter{Rewritings}
 
 The LTL rewritings described in this section are all implemented in