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tl: formul\ae -> formulas

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* doc/tl/tl.tex: Use "formulas" everywhere in this file.
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Alexandre Duret-Lutz 2015-01-15 10:32:18 +01:00
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doc/tl/tl.tex
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@ -86,6 +86,7 @@
\newcommand{\0}{\texttt{0}} \newcommand{\0}{\texttt{0}}
\newcommand{\1}{\texttt{1}} \newcommand{\1}{\texttt{1}}
\newcommand{\STAR}[1]{\texttt{[*#1]}} \newcommand{\STAR}[1]{\texttt{[*#1]}}
\newcommand{\FSTAR}[1]{\texttt{[:*#1]}}
\newcommand{\STARALT}{\texttt{*}} \newcommand{\STARALT}{\texttt{*}}
\newcommand{\STARBIN}{\mathbin{\texttt{*}}} \newcommand{\STARBIN}{\mathbin{\texttt{*}}}
\newcommand{\EQUAL}[1]{\texttt{[=#1]}} \newcommand{\EQUAL}[1]{\texttt{[=#1]}}
@ -169,7 +170,7 @@
\newcommand\msamp[1]{#1} \newcommand\msamp[1]{#1}
\def\manualtitle{Spot's Temporal Logic Formul\ae} \def\manualtitle{Spot's Temporal Logic Formulas}
%% ---------- %% %% ---------- %%
%% Document. %% %% Document. %%
@ -226,7 +227,7 @@ starting at letter $\sigma(i)$.
\section{Usage in Model Checking} \section{Usage in Model Checking}
The temporal formul\ae{} described in this document, should be The temporal formulas described in this document, should be
interpreted on behaviors (or executions, or scenarios) of the system interpreted on behaviors (or executions, or scenarios) of the system
to verify. In model checking we want to ensure that a formula (the to verify. In model checking we want to ensure that a formula (the
property to verify) holds on all possible behaviors of the system. property to verify) holds on all possible behaviors of the system.
@ -304,7 +305,7 @@ reserved keyword, or one that starts with a digit, is to use double quotes.
The reason we deal with leading \samp{F}, \samp{G}, and \samp{X} The reason we deal with leading \samp{F}, \samp{G}, and \samp{X}
specifically in rule~\ref{rule:ap2} is that these are unary LTL specifically in rule~\ref{rule:ap2} is that these are unary LTL
operators and we want to be able to write compact formul\ae{} like operators and we want to be able to write compact formulas like
\samp{GFa} instead of the equivalent \samp{G(F(a))} or \samp{GFa} instead of the equivalent \samp{G(F(a))} or
\samp{G~F~a}. If you want to name an atomic proposition \samp{GFa}, \samp{G~F~a}. If you want to name an atomic proposition \samp{GFa},
you will have to quote it as \samp{"GFa"}. you will have to quote it as \samp{"GFa"}.
@ -320,7 +321,7 @@ applied to \samp{1}). On the other hand, having numbered versions of
a variable is pretty common, so it makes sense to favor this a variable is pretty common, so it makes sense to favor this
interpretation. interpretation.
If you are typing in formul\ae{} by hand, we suggest you name all your If you are typing in formulas by hand, we suggest you name all your
atomic propositions in lower case, to avoid clashes with the uppercase atomic propositions in lower case, to avoid clashes with the uppercase
operators. operators.
@ -357,9 +358,9 @@ For any atomic proposition $a$, we have
In other words $a$ holds if and only if it is true in the first In other words $a$ holds if and only if it is true in the first
configuration of $\sigma$. configuration of $\sigma$.
\section{Boolean Operators (for Temporal Formul\ae{})}\label{sec:boolops} \section{Boolean Operators (for Temporal Formulas)}\label{sec:boolops}
Two temporal formul\ae{} $f$ and $g$ can be combined using the Two temporal formulas $f$ and $g$ can be combined using the
following Boolean operators: following Boolean operators:
\begin{center} \begin{center}
@ -377,14 +378,14 @@ following Boolean operators:
\end{center} \end{center}
\footnotetext{The $\STARALT$-form of the conjunction operator \footnotetext{The $\STARALT$-form of the conjunction operator
(allowing better compatibility with Wring and VIS) may only used in (allowing better compatibility with Wring and VIS) may only used in
temporal formul\ae. Boolean expressions that occur inside SERE (see temporal formulas. Boolean expressions that occur inside SERE (see
Section~\ref{sec:sere}) may not use this form because the $\STARALT$ Section~\ref{sec:sere}) may not use this form because the $\STARALT$
symbol is used as the Kleen star.} symbol is used as the Kleen star.}
Additionally, an atomic proposition $a$ can be negated using the Additionally, an atomic proposition $a$ can be negated using the
syntax \samp{$a$=0}, which is equivalent to \samp{$\NOT a$}. Also syntax \samp{$a$=0}, which is equivalent to \samp{$\NOT a$}. Also
\samp{$a$=1} is equivalent to just \samp{a}. These two syntaxes help \samp{$a$=1} is equivalent to just \samp{a}. These two syntaxes help
us read formul\ae{} written using Wring's syntax. us read formulas written using Wring's syntax.
When using UTF-8 input, a \samp{=0} that follow a single-letter atomic When using UTF-8 input, a \samp{=0} that follow a single-letter atomic
proposition may be replaced by a combining overline \uni{0305} or a proposition may be replaced by a combining overline \uni{0305} or a
@ -477,7 +478,7 @@ using the following rewritings:
\section{Temporal Operators}\label{sec:ltlops} \section{Temporal Operators}\label{sec:ltlops}
Given two temporal formul\ae{} $f$, and $g$, the following Given two temporal formulas $f$, and $g$, the following
temporal operators can be used to construct another temporal formula. temporal operators can be used to construct another temporal formula.
\begin{center} \begin{center}
@ -662,7 +663,7 @@ $a$ is an atomic proposition.
Notes: Notes:
\begin{itemize} \begin{itemize}
\item The semantics of $\ANDALT$ and $\AND$ coincide if both \item The semantics of $\ANDALT$ and $\AND$ coincide if both
operands are Boolean formul\ae. operands are Boolean formulas.
\item The SERE $f\FUSION g$ will never hold on $\eword$, \item The SERE $f\FUSION g$ will never hold on $\eword$,
regardless of the value of $f$ and $g$. For instance regardless of the value of $f$ and $g$. For instance
$a\STAR{}\FUSION b\STAR{}$ is actually equivalent to $a\STAR{}\FUSION b\STAR{}$ is actually equivalent to
@ -710,7 +711,7 @@ it for output. $b$ must be a Boolean formula.
The following identities also hold if $j$ or $l$ are missing (assuming The following identities also hold if $j$ or $l$ are missing (assuming
they are then equal to $\infty$). $f$ can be any SERE, while $b$, they are then equal to $\infty$). $f$ can be any SERE, while $b$,
$b_1$, $b_2$ are assumed to be Boolean formul\ae. $b_1$, $b_2$ are assumed to be Boolean formulas.
\begin{align*} \begin{align*}
\0\STAR{0..\mvar{j}} &\equiv \eword & \0\STAR{0..\mvar{j}} &\equiv \eword &
@ -1022,22 +1023,22 @@ instance using the following methods:
\newfootnotetext{1}{\url{http://www.tcs.hut.fi/Software/maria/tools/lbt/}} \newfootnotetext{1}{\url{http://www.tcs.hut.fi/Software/maria/tools/lbt/}}
\section{Pure Eventualities and Purely Universal Formul\ae{}} \section{Pure Eventualities and Purely Universal Formulas}
\label{sec:eventuniv} \label{sec:eventuniv}
These two syntactic classes of formul\ae{} were introduced These two syntactic classes of formulas were introduced
by~\citet{etessami.00.concur} to simplify LTL formul\ae{}. We shall by~\citet{etessami.00.concur} to simplify LTL formulas. We shall
present the associated simplification rules in present the associated simplification rules in
Section~\ref{sec:eventunivrew}, for now we only define these two Section~\ref{sec:eventunivrew}, for now we only define these two
classes. classes.
Pure eventual formul\ae{} describe properties that are left-append Pure eventual formulas describe properties that are left-append
closed, i.e., any accepted (infinite) sequence can be prefixed by a closed, i.e., any accepted (infinite) sequence can be prefixed by a
finite sequence and remain accepted. From an LTL standpoint, if finite sequence and remain accepted. From an LTL standpoint, if
$\varphi$ is a left-append closed formula, then $\F\varphi \equiv $\varphi$ is a left-append closed formula, then $\F\varphi \equiv
\varphi$. \varphi$.
Purely universal formul\ae{} describe properties that are Purely universal formulas describe properties that are
suffix-closed, i.e., if you remove any finite prefix of an accepted suffix-closed, i.e., if you remove any finite prefix of an accepted
(infinite) sequence, it remains accepted. From an LTL standpoint if (infinite) sequence, it remains accepted. From an LTL standpoint if
$\varphi$ is a suffix-closed formula, then $\G\varphi \equiv \varphi$. $\varphi$ is a suffix-closed formula, then $\G\varphi \equiv \varphi$.
@ -1086,10 +1087,10 @@ rules:
\begin{captionbeside}{ \begin{captionbeside}{
The temporal hierarchy of \citet{manna.87.podc} with their The temporal hierarchy of \citet{manna.87.podc} with their
associated classes of automata~\citep{cerna.03.mfcs}. The associated classes of automata~\citep{cerna.03.mfcs}. The
formul\ae{} associated to each class are more than canonical formulas associated to each class are more than canonical
examples: they show the normal forms under which any LTL formula 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 of the class can be rewritten, assuming that $p,p_i,q,q_i$ denote
subformul\ae{} involving only Boolean operators, $\X$, and past subformulas involving only Boolean operators, $\X$, and past
temporal operators (Spot does not support the temporal operators (Spot does not support the
latter). \label{fig:hierarchy}}[o] latter). \label{fig:hierarchy}}[o]
\centering \centering
@ -1149,7 +1150,7 @@ The symbols $\varphi_G$, $\varphi_S$, $\varphi_O$, $\varphi_P$,
$\varphi_R$ denote any formula belonging respectively to the $\varphi_R$ denote any formula belonging respectively to the
Guarantee, Safety, Obligation, Persistence, or Recurrence classes. Guarantee, Safety, Obligation, Persistence, or Recurrence classes.
$\varphi_F$ denotes a finite LTL formula (the unnamed class at the $\varphi_F$ denotes a finite LTL formula (the unnamed class at the
intersection of Safety and Guarantee formul\ae{} on intersection of Safety and Guarantee formulas on
Fig.~\ref{fig:hierarchy}). $v$ denotes any variable, $r$ any SERE, Fig.~\ref{fig:hierarchy}). $v$ denotes any variable, $r$ any SERE,
$r_F$ any bounded SERE (no loops), and $r_I$ any unbounded SERE. $r_F$ any bounded SERE (no loops), and $r_I$ any unbounded SERE.
@ -1217,10 +1218,10 @@ equivalent to $\G q\OR \G r$. Such a formula is usually said
The LTL rewritings described in this section are all implemented in The LTL rewritings described in this section are all implemented in
the `\verb|ltl_simplifier|' class defined in the `\verb|ltl_simplifier|' class defined in
\texttt{spot/ltlvisit/simplify.hh}. This class implements several \texttt{spot/ltlvisit/simplify.hh}. This class implements several
caches in order to quickly rewrite formul\ae{} that have already been caches in order to quickly rewrite formulas that have already been
rewritten previously. For this reason, it is suggested that you reuse rewritten previously. For this reason, it is suggested that you reuse
your instance of `\verb|ltl_simplifier|' as much as possible. If you your instance of `\verb|ltl_simplifier|' as much as possible. If you
write an algorithm that will simplify LTL formul\ae{}, we suggest you write an algorithm that will simplify LTL formulas, we suggest you
accept an optional `\verb|ltl_simplifier|' argument, so that you can accept an optional `\verb|ltl_simplifier|' argument, so that you can
benefit from an existing instance. benefit from an existing instance.
@ -1278,7 +1279,7 @@ Section~\ref{sec:unabbbool}. Therefore it is never necessary to apply
`\verb|ltl_simplifier::negative_normal_form|`. `\verb|ltl_simplifier::negative_normal_form|`.
If the option `\verb|nenoform_stop_on_boolean|' is set, the above If the option `\verb|nenoform_stop_on_boolean|' is set, the above
recursive rewritings are not applied to Boolean subformul\ae{}. For recursive rewritings are not applied to Boolean subformulas. For
instance calling `\verb|ltl_simplifier::negative_normal_form|` on instance calling `\verb|ltl_simplifier::negative_normal_form|` on
$\NOT\F\G(a \XOR b)$ will produce $\G\F(((\NOT a)\AND(\NOT $\NOT\F\G(a \XOR b)$ will produce $\G\F(((\NOT a)\AND(\NOT
b))\OR(a\AND b))$ if `\verb|nenoform_stop_on_boolean|' is unset, while b))\OR(a\AND b))$ if `\verb|nenoform_stop_on_boolean|' is unset, while
@ -1296,10 +1297,10 @@ The goals in most of these simplification are to:
\item remove useless terms and operator. \item remove useless terms and operator.
\item move the $\X$ operators to the front of the formula (e.g., $\X\G \item move the $\X$ operators to the front of the formula (e.g., $\X\G
f$ is better than the equivalent $\G\X f$). This is because LTL f$ is better than the equivalent $\G\X f$). This is because LTL
translators will usually want to rewrite LTL formul\ae{} in translators will usually want to rewrite LTL formulas in
a kind of disjunctive form: $\displaystyle\bigvee_i a kind of disjunctive form: $\displaystyle\bigvee_i
\left(\beta_i\land\X\psi_i\right)$ where $\beta_i$s are Boolean \left(\beta_i\land\X\psi_i\right)$ where $\beta_i$s are Boolean
formul\ae{} and $\psi_i$s are LTL formul\ae{}. Moving $\X$ to the formulas and $\psi_i$s are LTL formulas. Moving $\X$ to the
front therefore simplifies the translation. front therefore simplifies the translation.
\item move the $\F$ operators to the front of the formula (e.g., $\F(f \item move the $\F$ operators to the front of the formula (e.g., $\F(f
\OR g)$ is better than the equivalent $(\F f)\OR (\F g)$), but not \OR g)$ is better than the equivalent $(\F f)\OR (\F g)$), but not
@ -1417,7 +1418,7 @@ The following more complicated rules are generalizations of $f\AND
f\OR \X(\F(f)\OR h\ldots) &\equiv \F(f) \OR \X(h\ldots) f\OR \X(\F(f)\OR h\ldots) &\equiv \F(f) \OR \X(h\ldots)
\end{align*} \end{align*}
The latter rule for $f\OR \X(\F(f)\OR h\ldots)$ is only applied if all The latter rule for $f\OR \X(\F(f)\OR h\ldots)$ is only applied if all
$\F$-formul\ae{} can be removed from the argument of $\X$ with the $\F$-formulas can be removed from the argument of $\X$ with the
rewriting. For instance $a \OR b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$ rewriting. For instance $a \OR b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$
will be rewritten to $\F(a \OR b \OR c) \OR \X\G d$ but will be rewritten to $\F(a \OR b \OR c) \OR \X\G d$ but
$b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$ will only become $b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$ will only become
@ -1467,7 +1468,7 @@ SERE.
\sere{b_1\FUSION r_1}\AND \sere{b_2\FUSION r_2} &\equiv \sere{b_1\ANDALT b_2}\FUSION\sere{r_1\AND r_2} \mathrlap{\quad\text{if~}\varepsilon\nVDash r_1\land\varepsilon\nVDash r_2}\\ \sere{b_1\FUSION r_1}\AND \sere{b_2\FUSION r_2} &\equiv \sere{b_1\ANDALT b_2}\FUSION\sere{r_1\AND r_2} \mathrlap{\quad\text{if~}\varepsilon\nVDash r_1\land\varepsilon\nVDash r_2}\\
\end{align*} \end{align*}
Starred subformul\ae{} are rewritten in Star Normal Starred subformulas are rewritten in Star Normal
Form~\cite{bruggeman.96.tcs} with: Form~\cite{bruggeman.96.tcs} with:
\[r\STAR{} \equiv r^\circ\STAR{} \] \[r\STAR{} \equiv r^\circ\STAR{} \]
where $r^\circ$ is recursively defined as follows: where $r^\circ$ is recursively defined as follows:
@ -1581,14 +1582,14 @@ Here are basic the rewritings for the weak closure and its negation:
\nsere{r_1\OR r_2}&\equiV\nsere{r_1}\AND\nsere{r_2} \nsere{r_1\OR r_2}&\equiV\nsere{r_1}\AND\nsere{r_2}
\end{align*} \end{align*}
\subsection{Simplifications for Eventual and Universal Formul\ae} \subsection{Simplifications for Eventual and Universal Formulas}
\label{sec:eventunivrew} \label{sec:eventunivrew}
The class of \textit{pure eventuality} and \textit{purely universal} The class of \textit{pure eventuality} and \textit{purely universal}
formul\ae{} are described in section~\ref{sec:eventuniv}. formulas are described in section~\ref{sec:eventuniv}.
In the following rewritings, we use the following In the following rewritings, we use the following
notation to distinguish the class of subformul\ae: notation to distinguish the class of subformulas:
\begin{center} \begin{center}
\begin{tabular}{rl} \begin{tabular}{rl}
@ -1747,7 +1748,7 @@ only familiar with some of the operators.
\end{align*}} \end{align*}}
These equivalences make it possible to build artificially complex These equivalences make it possible to build artificially complex
formul\ae{}. For instance by applying the above rules to successively formulas. For instance by applying the above rules to successively
rewrite $\U\to\M\to\R\to\U$ we get rewrite $\U\to\M\to\R\to\U$ we get
\begin{align*} \begin{align*}
f\U g &\equiv ((\X g)\M f)\OR g \\ f\U g &\equiv ((\X g)\M f)\OR g \\
@ -1772,7 +1773,7 @@ Syntactic implications are used for the rewriting of
Section~\ref{sec:syntimpl}. The rules presented bellow extend those Section~\ref{sec:syntimpl}. The rules presented bellow extend those
first presented by~\citet{somenzi.00.cav}. first presented by~\citet{somenzi.00.cav}.
A few words about implication first. For two PSL formul\ae{} $f$ and A few words about implication first. For two PSL formulas $f$ and
$g$, we say that $f\implies g$ if $\forall\sigma,\,(\sigma\vDash $g$, we say that $f\implies g$ if $\forall\sigma,\,(\sigma\vDash
f)\implies(\sigma\vDash g)$. For two SERE $f$ and f)\implies(\sigma\vDash g)$. For two SERE $f$ and
$g$, we say that $f\implies g$ if $\forall\pi,\,(\pi\VDash $g$, we say that $f\implies g$ if $\forall\pi,\,(\pi\VDash
@ -1800,7 +1801,7 @@ conclusion by chaining two rules: ``$\F f \simp \underbrace{\F
``$f \simp \F g$ \textit{if} $f\simp g$''. ``$f \simp \F g$ \textit{if} $f\simp g$''.
The rules from table~\ref{tab:syntimpl} should be completed by the The rules from table~\ref{tab:syntimpl} should be completed by the
following cases, where $f_b$ and $g_b$ denote Boolean formul\ae{}: following cases, where $f_b$ and $g_b$ denote Boolean formulas:
\begin{center} \begin{center}
\begin{tabular}{rl} \begin{tabular}{rl}
we have & if \\ we have & if \\