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* doc/tl/tl.tex: Here.
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Alexandre Duret-Lutz 2012-04-28 21:02:43 +02:00
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doc/tl/tl.tex
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@ -11,6 +11,8 @@
\usepackage{xspace}
\usepackage{dsfont}
\usepackage{mathabx} % vDash
\usepackage{wasysym}
\usepackage{stmaryrd}
\usepackage{showlabels}
\usepackage{tabulary}
\usepackage[numbers]{natbib}
@ -36,6 +38,8 @@
\newcommand{\spottodo}[2][]{\stodo[color=green!40,caption={#2},#1]{#2}}
\newcommand{\ltltodo}[2][]{\stodo[color=red!40,caption={#2},#1]{#2}}
\newcommand{\uni}[1]{\texttt{\small U+#1}}
%% ---------------------- %%
%% Mathematical symbols. %%
@ -359,16 +363,16 @@ Two temporal formul\ae{} $f$ and $g$ can be combined using the
following Boolean operators:
\begin{center}
\begin{tabular}{ccccc}
& preferred & \multicolumn{2}{c}{other supported} \\
operation & syntax & \multicolumn{2}{c}{syntaxes}\\
\midrule
negation & $\NOT f$ & $\NOTALT f$ \\
disjunction & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ \\
conjunction & $f\AND g$ & $f\ANDALT g$ & $f\ANDALTT g$ \\
implication & $f\IMPLIES g$ & $f\IMPLIESALT g$ & $f\IMPLIESALTT g$\\
exclusion & $f\XOR g$ & $f\XORALT g$ \\
equivalence & $f\EQUIV g$ & $f\EQUIVALT g$ & $f\EQUIVALTT g$ \\
\begin{tabular}{cccccrl}
& preferred & \multicolumn{2}{c}{other supported}&& \multicolumn{2}{l}{UTF8 characters supported}\\
operation & syntax & \multicolumn{2}{c}{syntaxes} && preferred & others\\
\cmidrule(r){1-5} \cmidrule(l){6-7}
negation & $\NOT f$ & $\NOTALT f$ & & & $\lnot$ \uni{00AC} \\
disjunction & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ & $\lor$ \uni{2228} & $\cup$ \uni{222A}\\
conjunction & $f\AND g$ & $f\ANDALT g$ & $f\ANDALTT g$ & & $\land$ \uni{2227} & $\cap$ \uni{2229}\\
implication & $f\IMPLIES g$ & $f\IMPLIESALT g$ & $f\IMPLIESALTT g$ & & $\limplies$ \uni{2192} & $\rightarrow$ \uni{27F6}, $\Rightarrow$ \uni{21D2} \uni{27F9} \\
exclusion & $f\XOR g$ & $f\XORALT g$ & & & $\oplus$ \uni{2295} \\
equivalence & $f\EQUIV g$ & $f\EQUIVALT g$ & $f\EQUIVALTT g$ & & $\liff$ \uni{2194} & $\Leftrightarrow$ \uni{21D4}\\
\end{tabular}
\end{center}
@ -455,19 +459,19 @@ following rewritings:
f\XOR g &\equiv (f\AND\NOT g)\OR (g\AND\NOT f)\\
\end{align*}
\section{Temporal Operators}
\section{Temporal Operators}\label{sec:ltlops}
Given two temporal formul\ae{} $f$, and $g$, the following
temporal operators can be used to construct another temporal formula.
\begin{center}
\begin{tabular}{ccc}
& preferred & \multicolumn{1}{c}{other supported} \\
operator & syntax & \multicolumn{1}{c}{syntaxes}\\
\midrule
Next & $\X f$ & $\XALT f$ \\
Eventually & $\F f$ & $\FALT f$ \\
Always & $\G f$ & $\GALT f$ \\
\begin{tabular}{cccrl}
& preferred & \multicolumn{1}{c}{other supported} & \multicolumn{2}{l}{UTF8 characters supported} \\
operator & syntax & \multicolumn{1}{c}{syntaxes} & preferred & others \\
\cmidrule(r){1-3} \cmidrule(l){4-5}
Next & $\X f$ & $\XALT f$ & $\Circle$ \uni{25CB} & $\Circle$ \uni{25EF}\\
Eventually & $\F f$ & $\FALT f$ & $\Diamond$ \uni{25C7} & $\Diamond$ \uni{22C4} \uni{2662}\\
Always & $\G f$ & $\GALT f$ & $\Square$ \uni{25A1} & $\Square$ \uni{2B1C} \uni{25FB}\\
(Strong) Until & $f \U g$ \\
Weak Until & $f \W g$ \\
(Weak) Release & $f \R g$ & $f \RALT g$ \\
@ -562,27 +566,29 @@ be further combined with the following operators, where $f$ and $g$
denote arbitrary SERE and $b$ denotes a Boolean formula.
\begin{center}
\begin{tabular}{ccccc}
& preferred & \multicolumn{2}{c}{other supported} \\
operation & syntax & \multicolumn{2}{c}{syntaxes}\\
\midrule
\begin{tabular}{cccccrl}
& preferred & \multicolumn{2}{c}{other supported} && \multicolumn{2}{l}{UTF8 characters supported}\\
operation & syntax & \multicolumn{2}{c}{syntaxes} && preferred & others \\
\cmidrule(r){1-5}\cmidrule(l){6-7}
empty word & $\eword$ \\
union & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ \\
(synchronized) intersection & $f\ANDALT g$ & $f\ANDALTT g$ \\
unsynchronized intersection & $f\AND g$ \\
union & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ && $\lor$ \uni{2228} $\cup$ \uni{222A}\\
intersection & $f\ANDALT g$ & $f\ANDALTT g$ &&& $\cap$ \uni{2229} & $\land$ \uni{2227}\\
NLM intersection\footnotemark & $f\AND g$ \\
concatenation & $f\CONCAT g$ \\
fusion & $f\FUSION g$ \\
bounded repetition & $f\STAR{\mvar{i}..\mvar{j}}$
& $f\STAR{\mvar{i}:\mvar{j}}$
& $f\STAR{\mvar{i} to \mvar{j}}$
& $f\STAR{\mvar{i},\mvar{j}}$\\
unbounded repetition & $f\STAR{\mvar{i}..}$
\llap{un}bounded repetition & $f\STAR{\mvar{i}..}$
& $f\STAR{\mvar{i}:}$
& $f\STAR{\mvar{i} to}$
& $f\STAR{\mvar{i},}$\\
\end{tabular}
\end{center}
\footnotetext{\emph{Non-Length-Matching} interesction.}
The character \samp{\$} or the string \samp{inf} can also be used as
value for $\mvar{j}$ in the above operators to denote an unbounded
range.\footnote{SVA uses \samp{\$} while PSL uses \samp{inf}.} For
@ -772,6 +778,14 @@ For technical reasons, the negated weak closure is actually implemented as
an operator, even if it is syntactically and semantically equal to the
combination of $\NOT$ and $\sere{r}$.
UTF-8 input may combine one box or diamond character from
section~\ref{sec:ltlops} with one arrow character from
section~\ref{sec:boolops} to replace the operators $\Asuffix$,
$\Esuffix$, as well as the operators $\AsuffixEQ$ and $\EsuffixEQ$
that will be defined in \ref{sec:pslsugar}. Additionally,
$\AsuffixALT$ may be replaced by $\mapsto$ \uni{21A6}, and
$\AsuffixALTEQ$ by $\Mapsto$ \uni{2907}.
\subsection{Semantics}
The following semantics assume that $r$ is a SERE,
@ -792,7 +806,7 @@ is a model of $r$. An infinite sequence $\texttt{a;a;a;a;a;}\ldots$
is therefore a model of the formula \samp{$\sere{a\PLUS{};\NOT
a}$} even though it never sees \samp{$\NOT a$}.
\subsection{Syntactic Sugar}
\subsection{Syntactic Sugar}\label{sec:pslsugar}
The syntax on the left is equivalent to the syntax on the right.
These rewritings are performed from left to right when parsing a