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formula: track Δ₁, Σ₂, Π₂, and Δ₂ membership

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* spot/tl/formula.hh, spot/tl/formula.cc: Update the properties
and track them.
* tests/core/kind.test: Augment the test case.
* doc/tl/tl.tex, doc/spot.bib, NEWS: Document these new classes.
This commit is contained in:
Alexandre Duret-Lutz 2024-07-19 17:04:21 +02:00
parent 0c52c49079
commit 7901a37747
6 changed files with 473 additions and 175 deletions
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5
NEWS
View file

@ -18,6 +18,11 @@ New in spot 2.12.0.dev (not yet released)
in its highest setting. This can be fine-tuned with the "rde" in its highest setting. This can be fine-tuned with the "rde"
extra option, see the spot-x (7) man page for detail. extra option, see the spot-x (7) man page for detail.
- The formula class now keeps track of membership to the Δ₁, Σ₂,
Π₂, and Δ₂ syntactic class. This can be tested with
formula::is_delta1(), formula::is_sigma2(), formula::is_pi2(),
formula::is_delta2(). See doc/tl/tl.pdf from more discussion.
New in spot 2.12 (2024-05-16) New in spot 2.12 (2024-05-16)
Build: Build:

15
doc/spot.bib
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@ -1,3 +1,4 @@
@InProceedings{ babiak.12.tacas, @InProceedings{ babiak.12.tacas,
author = {Tom{\'a}{\v{s}} Babiak and Mojm{\'i}r author = {Tom{\'a}{\v{s}} Babiak and Mojm{\'i}r
K{\v{r}}et{\'i}nsk{\'y} and Vojt{\v{e}}ch {\v{R}}eh{\'a}k K{\v{r}}et{\'i}nsk{\'y} and Vojt{\v{e}}ch {\v{R}}eh{\'a}k
@ -470,6 +471,20 @@
doi = {10.1145/3209108.3209161} doi = {10.1145/3209108.3209161}
} }
@Article{ esparza.24.acm,
author = {Javier Esparza and Rub\'{e}n Rubio and Salomon Sickert},
title = {Efficient Normalization of Linear Temporal Logic},
year = 2024,
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {71},
number = {2},
issn = {0004-5411},
doi = {10.1145/3651152},
journal = {Journal of the ACM},
month = apr
}
@InProceedings{ etessami.00.concur, @InProceedings{ etessami.00.concur,
author = {Kousha Etessami and Gerard J. Holzmann}, author = {Kousha Etessami and Gerard J. Holzmann},
title = {Optimizing {B\"u}chi Automata}, title = {Optimizing {B\"u}chi Automata},

205
doc/tl/tl.tex
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@ -1121,6 +1121,14 @@ instance using the following methods:
recurrence property. recurrence property.
\\\texttt{is\_syntactic\_persistence()}& Whether the formula is a syntactic \\\texttt{is\_syntactic\_persistence()}& Whether the formula is a syntactic
persistence property. persistence property.
\\\texttt{is\_syntactic\_delta1()}& Whether the formula belongs to
the $\Delta_1$ class.
\\\texttt{is\_syntactic\_pi2()}& Whether the formula belongs to
the $\Pi_2$ class.
\\\texttt{is\_syntactic\_sigma2()}& Whether the formula belongs to
the $\Sigma_2$ class.
\\\texttt{is\_syntactic\_delta2()}& Whether the formula belongs to
the $\Delta_2$ class.
\\\texttt{is\_marked()}& Whether the formula contains a special \\\texttt{is\_marked()}& Whether the formula contains a special
``marked'' version of the $\Esuffix$ or $\nsere{r}$ operators.\newfootnotemark{1} ``marked'' version of the $\Esuffix$ or $\nsere{r}$ operators.\newfootnotemark{1}
\\\texttt{accepts\_eword()}& Whether the formula accepts \\\texttt{accepts\_eword()}& Whether the formula accepts
@ -1198,6 +1206,9 @@ rules:
\mid \varphi_U\M \varphi_U \mid \varphi_U\M \varphi_U
\end{align*} \end{align*}
Given a formula \texttt{f}, its membership to these two classes can be
tested with \texttt{f.is\_eventual()} and \texttt{f.is\_universal()}.
\section{Syntactic Hierarchy Classes} \section{Syntactic Hierarchy Classes}
\begin{figure}[tbp] \begin{figure}[tbp]
@ -1221,12 +1232,13 @@ rules:
\path[fill=green!40,fill opacity=.5] (6,0) -- (1.5,0) -- (6,3); \path[fill=green!40,fill opacity=.5] (6,0) -- (1.5,0) -- (6,3);
\draw (0,0) rectangle (6,7); \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] (rea) at (3,5.9) {Reactivity\\ $\bigwedge\G\F p_i\lor \F\G q_i$ \\ $\Delta_2$};
\node[align=center] (rec) at (1,4.5) {Recurrence\\ $\G\F p$}; \node[align=center] (rec) at (1,4.4) {Recurrence\\ $\G\F p$ \\ $\Pi_2$};
\node[align=center] (per) at (5,4.5) {Persistence\\ $\F\G p$}; \node[align=center] (per) at (5,4.4) {Persistence\\ $\F\G p$ \\ $\Sigma_2$};
\node[align=center] (obl) at (3,2.85) {Obligation\\ $\bigwedge\G p_i\lor \F q_i$}; \node[align=center] (obl) at (3,2.85) {Obligation\\ $\bigwedge\G p_i\lor \F q_i$ \\ $\Delta_1$};
\node[align=center] (saf) at (1,1) {Safety\\ $\G p$}; \node[align=center] (saf) at (1,1) {Safety\\ $\G p$ \\ $\Pi_1$};
\node[align=center] (gua) at (5,1) {Guarantee\\ $\F p$}; \node[align=center] (gua) at (5,1) {Guarantee\\ $\F p$ \\ $\Sigma_1$};
\node[align=center] (bas) at (3,0.4) {$\Delta_0$};
\node[align=right,below left] (det) at (-.2,6.7) {Deterministic\\Büchi\\Automata}; \node[align=right,below left] (det) at (-.2,6.7) {Deterministic\\Büchi\\Automata};
\node[align=left,below right](weak) at (6.2,6.7) {Weak Büchi\\Automata}; \node[align=left,below right](weak) at (6.2,6.7) {Weak Büchi\\Automata};
@ -1254,24 +1266,107 @@ presented by~\citet{chang.92.icalp}, but other presentations have been
done including negation~\citep{cerna.03.mfcs} and weak done including negation~\citep{cerna.03.mfcs} and weak
until~\citep{schneider.01.lpar}. until~\citep{schneider.01.lpar}.
The following grammar rules extend the aforementioned work slightly by Spot implements two versions of a syntactic hierarchy, and extend them
dealing with PSL operators. These are the rules used by Spot to to deal with PSL operators.
decide upon construction to which class a formula belongs (see the
methods \texttt{is\_syntactic\_safety()},
\texttt{is\_syntactic\_guarantee()},
\texttt{is\_syntactic\_obligation()},
\texttt{is\_syntactic\_recurrence()}, and
\texttt{is\_syntactic\_persistence()} listed on
page~\pageref{property-methods}).
The symbols $\varphi_G$, $\varphi_S$, $\varphi_O$, $\varphi_P$,
$\varphi_R$ denote any formula belonging respectively to the The first hierarchy, usually denoted with names such as $\Sigma_i$ and
Guarantee, Safety, Obligation, Persistence, or Recurrence classes. $\Pi_i$, as shown in Fig.~\ref{fig:hierarchy}. Following Esparza et
Additionally $\varphi_B$ denotes a finite LTL formula (the unnamed al.~\cite{esparza.24.acm}, we also introduce the $\Delta_0$,
class at the intersection of Safety and Guarantee formulas, at the $\Delta_1$, and $\Delta_2$ classes.
\textbf{b}ottom of Fig.~\ref{fig:hierarchy}). $v$ denotes any
variable, $r$ any SERE, $r_F$ any bounded SERE (no loops), and $r_I$ Intuitively, those classes are related to how the weak operators
any unbounded SERE. ($\G$, $\W$, $\R$) alternate with the strong operators ($\U$, $\F$,
$\M$) in formula:
\begin{itemize}
\item the class $\Delta_0$ contains all formulas that may only
use $\X$ as temporal operator,
\item formulas in $\Pi_1$ contains no strong operators,
\item formulas in $\Sigma_1$ contains no weak operators,
\item the class $\Delta_1$ contains all boolean combinations of
$\Pi_1$ and $\Sigma_1$,
\item in each branch of a formula of $\Pi_2$ that contains both types
of operator, weak operators are all above strong operators,
\item in each branch of a formula of $\Sigma_2$ that contains both types
of operator, strong operators are all above weak operators,
\item the class $\Delta_2$ contains all boolean combinations of
$\Pi_2$ and $\Sigma_2$.
\end{itemize}
Those classes can be captured by the following grammar rules, where
$v$ denotes any variable, $r$ any SERE, $r_F$ any bounded SERE (no
loops), and $r_I$ any unbounded SERE.
\begin{align*}
\varphi_{\Delta_0} ::={}& \0\mid\1\mid v\mid\NOT\varphi_{\Delta_0}\mid\varphi_{\Delta_0}\AND\varphi_{\Delta_0}
\mid(\varphi_{\Delta_0}\OR\varphi_{\Delta_0})\mid\varphi_{\Delta_0}\EQUIV\varphi_{\Delta_0}
\mid\varphi_{\Delta_0}\XOR\varphi_{\Delta_0}\mid\varphi_{\Delta_0}\IMPLIES\varphi_{\Delta_0}
\mid\X\varphi_{\Delta_0}\\
\mid{}& \sere{r_F}\mid \nsere{r_F}\\
\varphi_{\Pi_1} ::={}& \varphi_{\Delta_0}\mid \NOT\varphi_S\mid
\varphi_{\Pi_1}\AND \varphi_{\Pi_1}\mid (\varphi_{\Pi_1}\OR \varphi_{\Pi_1})
\mid\varphi_S\IMPLIES\varphi_{\Pi_1}\mid
\X\varphi_{\Pi_1} \mid \F\varphi_{\Pi_1}\mid
\varphi_{\Pi_1}\U\varphi_{\Pi_1}\mid \varphi_{\Pi_1}\M\varphi_{\Pi_1}\\
\mid{}& \nsere{r}\mid
\sere{r}\Esuffix \varphi_{\Pi_1}\mid
\sere{r_F}\Asuffix \varphi_{\Pi_1} \\
\varphi_{\Sigma_1} ::={}& \varphi_{\Delta_0}\mid \NOT\varphi_{\Pi_1}\mid
\varphi_{\Sigma_1}\AND \varphi_{\Sigma_1}\mid (\varphi_{\Sigma_1}\OR \varphi_{\Sigma_1})
\mid\varphi_{\Pi_1}\IMPLIES\varphi_{\Sigma_1}\mid
\X\varphi_{\Sigma_1} \mid \G\varphi_{\Sigma_1}\mid
\varphi_{\Sigma_1}\R\varphi_{\Sigma_1}\mid \varphi_{\Sigma_1}\W\varphi_{\Sigma_1}\\
\mid{}& \sere{r}\mid
\sere{r_F}\Esuffix \varphi_{\Sigma_1}\mid
\sere{r}\Asuffix \varphi_{\Sigma_1}\\
\varphi_{\Delta_1} ::={}& \varphi_{\Pi_1} \mid \varphi_{\Sigma_1}\mid \NOT\varphi_{\Delta_1}\mid
\varphi_{\Delta_1}\AND \varphi_{\Delta_1}\mid (\varphi_{\Delta_1}\OR \varphi_{\Delta_1})\mid
\varphi_{\Delta_1}\EQUIV \varphi_{\Delta_1}\mid \varphi_{\Delta_1}\XOR \varphi_{\Delta_1}\mid
\varphi_{\Delta_1}\IMPLIES \varphi_{\Delta_1}\\
\mid{}& \X\varphi_{\Delta_1} \mid{}
\sere{r_F}\Esuffix \varphi_{\Delta_1} \mid
\sere{r_F}\Asuffix \varphi_{\Delta_1}\\
\varphi_{\Sigma_2} ::={}& \varphi_{\Delta_1} \mid \NOT\varphi_{\Pi_2}\mid
\varphi_{\Sigma_2}\AND \varphi_{\Sigma_2}\mid (\varphi_{\Sigma_2}\OR \varphi_{\Sigma_2})\mid
\varphi_{\Pi_2}\IMPLIES \varphi_{\Sigma_2}\\
\mid{}& \X\varphi_{\Sigma_2} \mid \F\varphi_{\Sigma_2} \mid
\varphi_{\Sigma_2}\U\varphi_{\Sigma_2}\mid\varphi_{\Sigma_2}\M\varphi_{\Sigma_2}
\mid{} \sere{r}\Esuffix \varphi_{\Sigma_2}\mid \sere{r_F}\Asuffix \varphi_{\Sigma_2}\\
\varphi_{\Pi_2} ::={}& \varphi_{\Delta_1} \mid \NOT\varphi_{\Sigma_2}\mid
\varphi_{\Pi_2}\AND \varphi_{\Pi_2}\mid (\varphi_{\Pi_2}\OR \varphi_{\Pi_2})\mid
\varphi_{\Sigma_2}\IMPLIES \varphi_{\Pi_2}\\
\mid{}& \X\varphi_{\Pi_2} \mid \G\varphi_{\Pi_2} \mid
\varphi_{\Pi_2}\R\varphi_{\Pi_2}\mid
\varphi_{\Pi_2}\W\varphi_{\Pi_2} \mid{} \sere{r}\Asuffix \varphi_{\Pi_2}\mid \sere{r_F}\Esuffix \varphi_{\Pi_2}\\
\varphi_{\Delta_2} ::={}& \varphi_{\Pi_2} \mid \varphi_{\Sigma_2}\mid \NOT\varphi_{\Delta_2}\mid
\varphi_{\Delta_2}\AND \varphi_{\Delta_2}\mid (\varphi_{\Delta_2}\OR \varphi_{\Delta_2})\mid
\varphi_{\Delta_2}\EQUIV \varphi_{\Delta_2}\mid \varphi_{\Delta_2}\XOR \varphi_{\Delta_2}\mid
\varphi_{\Delta_2}\IMPLIES \varphi_{\Delta_2}\\
\mid{}& \X\varphi_{\Delta_2} \mid{} \sere{r_F}\Esuffix \varphi_{\Delta_2} \mid
\sere{r_F}\Asuffix \varphi_{\Delta_2}\\
\end{align*}
A nice property of these classes, is that they are as expressive as
their corresponding automata classes. For instance any LTL/PSL
property that is representable by a deterministic Büchi automaton (the
recurrence class) can be represented by an LTL/PSL formula in the
$\Pi_2$ fragment, even if the original formula isn't in the $\Pi_2$
fragment originally.
If the objective is to classify properties syntactically, it is useful
to use some slightly more complete grammar rules. In the following
list, the rules the initial $G$, $S$, $O$, $P$, $R$ of their
corresponding property clases, as listed in Fig.~\ref{fig:hierarchy}
(i.e., Guarantee, Safety, Obligation, Persistence, Recurrence).
Additionally, $B$ denotes the ``bottom'' class (a.k.a. $\Delta_0$).
Note that $\varphi_B$, $\varphi_G$, and $\varphi_S$ are rigorously
equivalent to $\varphi_{\Delta_0}$, $\varphi_{\Pi_1}$, and
$\varphi_{\Sigma_1}$. The difference in the higher classes are
\colorbox{yellow}{highlighted}. There is no generalization of
$\varphi_{\Delta_2}$ since any LTL/PSL formula is a reactivity
property.
\begin{align*} \begin{align*}
\varphi_B ::={}& \0\mid\1\mid v\mid\NOT\varphi_B\mid\varphi_B\AND\varphi_B \varphi_B ::={}& \0\mid\1\mid v\mid\NOT\varphi_B\mid\varphi_B\AND\varphi_B
@ -1300,36 +1395,70 @@ any unbounded SERE.
\varphi_O\EQUIV \varphi_O\mid \varphi_O\XOR \varphi_O\mid \varphi_O\EQUIV \varphi_O\mid \varphi_O\XOR \varphi_O\mid
\varphi_O\IMPLIES \varphi_O\\ \varphi_O\IMPLIES \varphi_O\\
\mid{}& \X\varphi_O \mid \mid{}& \X\varphi_O \mid
\varphi_O\U\varphi_G\mid\varphi_O\R\varphi_S \mid \colorbox{yellow}{$\varphi_O\U\varphi_G$}\mid
\varphi_S\W\varphi_O\mid \varphi_G\M\varphi_O\\ \colorbox{yellow}{$\varphi_O\R\varphi_S$}\mid
\mid{}& \sere{r} \mid \nsere{r}\mid \colorbox{yellow}{$\varphi_S\W\varphi_O$}\mid
\sere{r_F}\Esuffix \varphi_O \mid \sere{r_I}\Esuffix \varphi_G\mid \colorbox{yellow}{$\varphi_G\M\varphi_O$}\\
\mid{}& \sere{r_F}\Esuffix \varphi_O \mid \colorbox{yellow}{$\sere{r_I}\Esuffix \varphi_G$}\mid
\sere{r_F}\Asuffix \varphi_O\mid \sere{r_F}\Asuffix \varphi_O\mid
\sere{r_I}\Asuffix \varphi_S\\ \colorbox{yellow}{$\sere{r_I}\Asuffix \varphi_S$}\\
\varphi_P ::={}& \varphi_O \mid \NOT\varphi_R\mid \varphi_P ::={}& \varphi_O \mid \NOT\varphi_R\mid
\varphi_P\AND \varphi_P\mid (\varphi_P\OR \varphi_P)\mid \varphi_P\AND \varphi_P\mid (\varphi_P\OR \varphi_P)\mid
\varphi_R\IMPLIES \varphi_P\\ \varphi_R\IMPLIES \varphi_P\\
\mid{}& \X\varphi_P \mid \F\varphi_P \mid \mid{}& \X\varphi_P \mid \F\varphi_P \mid
\varphi_P\U\varphi_P\mid\varphi_P\R\varphi_S\mid \varphi_P\U\varphi_P\mid\colorbox{yellow}{$\varphi_P\R\varphi_S$}\mid
\varphi_S\W\varphi_P\mid\varphi_P\M\varphi_P\\ \colorbox{yellow}{$\varphi_S\W\varphi_P$}\mid\varphi_P\M\varphi_P\\
\mid{}& \sere{r}\Esuffix \varphi_P\mid \mid{}& \sere{r}\Esuffix \varphi_P\mid
\sere{r_F}\Asuffix \varphi_P\mid \sere{r_F}\Asuffix \varphi_P\mid
\sere{r_I}\Asuffix \varphi_S\\ \colorbox{yellow}{$\sere{r_I}\Asuffix \varphi_S$}\\
\varphi_R ::={}& \varphi_O \mid \NOT\varphi_P\mid \varphi_R ::={}& \varphi_O \mid \NOT\varphi_P\mid
\varphi_R\AND \varphi_R\mid (\varphi_R\OR \varphi_R)\mid \varphi_R\AND \varphi_R\mid (\varphi_R\OR \varphi_R)\mid
\varphi_P\IMPLIES \varphi_R\\ \varphi_P\IMPLIES \varphi_R\\
\mid{}& \X\varphi_R \mid \G\varphi_R \mid \mid{}& \X\varphi_R \mid \G\varphi_R \mid
\varphi_R\U\varphi_G\mid\varphi_R\R\varphi_R\mid \colorbox{yellow}{$\varphi_R\U\varphi_G$}\mid\varphi_R\R\varphi_R\mid
\varphi_R\W\varphi_R\mid\varphi_G\M\varphi_R\\ \varphi_R\W\varphi_R\mid\colorbox{yellow}{$\varphi_G\M\varphi_R$}\\
\mid{}& \sere{r}\Asuffix \varphi_R\mid \sere{r_F}\Esuffix \varphi_R \mid \sere{r_I}\Esuffix \varphi_G\\ \mid{}& \sere{r}\Asuffix \varphi_R\mid \sere{r_F}\Esuffix \varphi_R \mid \colorbox{yellow}{$\sere{r_I}\Esuffix \varphi_G$}\\
\end{align*} \end{align*}
It should be noted that a formula can belong to a class of the 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 temporal hierarchy even if it does not syntactically appears so. For
instance the formula $(\G(q\OR \F\G p)\AND \G(r\OR \F\G\NOT p))\OR\G instance the formula
q\OR \G r$ is not syntactically safe, yet it is a safety formula $(\G(q\OR \F\G p)\AND \G(r\OR \F\G\NOT p))\OR\G q\OR \G r$ is not
equivalent to $\G q\OR \G r$. Such a formula is usually said syntactically safe (and isn't even in $\Delta_2$), yet it is a safety
\emph{pathologically safe}. formula equivalent to $\G q\OR \G r$ (which is in $\Pi_1$, the
syntactical class of safety formulas). Such a formula is usually said
to be a \emph{pathological safety} formula.
To illustrate the difference in the grammar for the higher classes,
consider the formula $\G((\G a) \U b)$. This formula can be converted
to a deterministic Büchi automaton, so it specifies a recurrence
property. It is captured by the grammar rule for $\varphi_R$ above,
yet it does not belong to the $\Pi_2$ class because of the alternation
between weak ($\G$), strong ($\U$), and weak ($\G$) operators.
However the equivalent formula $\G((\G a) \W b))\land \G\F b$ belongs
to $\Pi_2$.
Spot computes the membership to each of those class whenever a formula
$f$ is constructed. Here is how the membership to each of those class
can be tested:
\begin{center}
\begin{tabular}{cl}
\toprule
$f\in \Delta_0$ & \texttt{f.is\_syntactic\_safety() \&\& f.is\_syntactic\_guarantee()} \\
$f\in \Pi_1$, $f\in S$ & \texttt{f.is\_syntactic\_safety()} \\
$f\in \Sigma_1$, $f\in G$ & \texttt{f.is\_syntactic\_guarantee()} \\
$f\in \Delta_1$ & \texttt{f.is\_delta1()} \\
$f\in O$ & \texttt{f.is\_syntactic\_obligation()} \\
$f\in \Pi_2$ & \texttt{f.is\_pi2()} \\
$f\in R$ & \texttt{f.is\_syntactic\_recurrence()} \\
$f\in \Sigma_2$ & \texttt{f.is\_sigma2()} \\
$f\in P$ & \texttt{f.is\_syntactic\_persistence()} \\
$f\in \Delta_2$ & \texttt{f.is\_delta2()} \\
\bottomrule
\end{tabular}
\end{center}
\chapter{Rewritings} \chapter{Rewritings}

154
spot/tl/formula.cc
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@ -1290,6 +1290,10 @@ namespace spot
is_.accepting_eword = false; is_.accepting_eword = false;
is_.lbt_atomic_props = true; is_.lbt_atomic_props = true;
is_.spin_atomic_props = true; is_.spin_atomic_props = true;
is_.delta1 = true;
is_.sigma2 = true;
is_.pi2 = true;
is_.delta2 = true;
break; break;
case op::eword: case op::eword:
is_.boolean = false; is_.boolean = false;
@ -1312,6 +1316,10 @@ namespace spot
is_.accepting_eword = true; is_.accepting_eword = true;
is_.lbt_atomic_props = true; is_.lbt_atomic_props = true;
is_.spin_atomic_props = true; is_.spin_atomic_props = true;
is_.delta1 = true;
is_.sigma2 = true;
is_.pi2 = true;
is_.delta2 = true;
break; break;
case op::ap: case op::ap:
is_.boolean = true; is_.boolean = true;
@ -1348,6 +1356,10 @@ namespace spot
is_.lbt_atomic_props = lbtap; is_.lbt_atomic_props = lbtap;
is_.spin_atomic_props = lbtap || is_spin_ap(n.c_str()); is_.spin_atomic_props = lbtap || is_spin_ap(n.c_str());
} }
is_.delta1 = true;
is_.sigma2 = true;
is_.pi2 = true;
is_.delta2 = true;
break; break;
case op::Not: case op::Not:
props = children[0]->props; props = children[0]->props;
@ -1364,6 +1376,10 @@ namespace spot
is_.syntactic_persistence = children[0]->is_syntactic_recurrence(); is_.syntactic_persistence = children[0]->is_syntactic_recurrence();
is_.accepting_eword = false; is_.accepting_eword = false;
// is_.delta1 inherited
is_.sigma2 = children[0]->is_pi2();
is_.pi2 = children[0]->is_sigma2();
// is_.delta2 inherited
break; break;
case op::X: case op::X:
case op::strong_X: case op::strong_X:
@ -1382,6 +1398,10 @@ namespace spot
// we could make sense of it if we start supporting LTL over // we could make sense of it if we start supporting LTL over
// finite traces. // finite traces.
is_.accepting_eword = false; is_.accepting_eword = false;
// is_.delta1 inherited
// is_.sigma2 inherited
// is_.pi2 inherited
// is_.delta2 inherited
break; break;
case op::F: case op::F:
props = children[0]->props; props = children[0]->props;
@ -1397,6 +1417,10 @@ namespace spot
is_.syntactic_recurrence = is_.syntactic_guarantee; is_.syntactic_recurrence = is_.syntactic_guarantee;
// is_.syntactic_persistence inherited // is_.syntactic_persistence inherited
is_.accepting_eword = false; is_.accepting_eword = false;
is_.delta1 = is_.syntactic_guarantee;
is_.pi2 = is_.syntactic_guarantee;
// is_.sigma2 inherited
is_.delta2 = is_.pi2 | is_.sigma2;
break; break;
case op::G: case op::G:
props = children[0]->props; props = children[0]->props;
@ -1412,6 +1436,10 @@ namespace spot
// is_.syntactic_recurrence inherited // is_.syntactic_recurrence inherited
is_.syntactic_persistence = is_.syntactic_safety; is_.syntactic_persistence = is_.syntactic_safety;
is_.accepting_eword = false; is_.accepting_eword = false;
is_.delta1 = is_.syntactic_safety;
is_.sigma2 = is_.syntactic_safety;
// is_.pi2 inherited
is_.delta2 = is_.pi2 | is_.sigma2;
break; break;
case op::NegClosure: case op::NegClosure:
case op::NegClosureMarked: case op::NegClosureMarked:
@ -1427,6 +1455,10 @@ namespace spot
is_.syntactic_recurrence = true; is_.syntactic_recurrence = true;
is_.syntactic_persistence = true; is_.syntactic_persistence = true;
is_.accepting_eword = false; is_.accepting_eword = false;
is_.delta1 = true;
is_.sigma2 = true;
is_.pi2 = true;
is_.delta2 = true;
assert(children[0]->is_sere_formula()); assert(children[0]->is_sere_formula());
assert(!children[0]->is_boolean()); assert(!children[0]->is_boolean());
break; break;
@ -1443,6 +1475,10 @@ namespace spot
is_.syntactic_recurrence = true; is_.syntactic_recurrence = true;
is_.syntactic_persistence = true; is_.syntactic_persistence = true;
is_.accepting_eword = false; is_.accepting_eword = false;
is_.delta1 = true;
is_.sigma2 = true;
is_.pi2 = true;
is_.delta2 = true;
assert(children[0]->is_sere_formula()); assert(children[0]->is_sere_formula());
assert(!children[0]->is_boolean()); assert(!children[0]->is_boolean());
break; break;
@ -1473,6 +1509,17 @@ namespace spot
is_.syntactic_recurrence = false; is_.syntactic_recurrence = false;
is_.syntactic_persistence = false; is_.syntactic_persistence = false;
} }
if (is_.delta1)
{
assert(is_.pi2 == true);
assert(is_.sigma2 == true);
assert(is_.delta2 == true);
}
else
{
is_.pi2 = false;
is_.sigma2 = false;
}
break; break;
case op::Implies: case op::Implies:
props = children[0]->props & children[1]->props; props = children[0]->props & children[1]->props;
@ -1494,6 +1541,10 @@ namespace spot
is_.syntactic_recurrence = children[0]->is_syntactic_persistence() is_.syntactic_recurrence = children[0]->is_syntactic_persistence()
&& children[1]->is_syntactic_recurrence(); && children[1]->is_syntactic_recurrence();
is_.accepting_eword = false; is_.accepting_eword = false;
// is_.delta1 inherited
is_.sigma2 = children[0]->is_pi2() && children[1]->is_sigma2();
is_.pi2 = children[0]->is_sigma2() && children[1]->is_pi2();
// is_.delta2 inherited
break; break;
case op::EConcatMarked: case op::EConcatMarked:
case op::EConcat: case op::EConcat:
@ -1507,18 +1558,25 @@ namespace spot
is_.syntactic_guarantee = children[1]->is_syntactic_guarantee(); is_.syntactic_guarantee = children[1]->is_syntactic_guarantee();
is_.syntactic_persistence = children[1]->is_syntactic_persistence(); is_.syntactic_persistence = children[1]->is_syntactic_persistence();
if (children[0]->is_finite()) is_.sigma2 = children[1]->is_sigma2();
if (children[0]->is_finite()) // behaves like X
{ {
is_.syntactic_safety = children[1]->is_syntactic_safety(); is_.syntactic_safety = children[1]->is_syntactic_safety();
is_.syntactic_obligation = children[1]->is_syntactic_obligation(); is_.syntactic_obligation = children[1]->is_syntactic_obligation();
is_.syntactic_recurrence = children[1]->is_syntactic_recurrence(); is_.syntactic_recurrence = children[1]->is_syntactic_recurrence();
is_.delta1 = children[1]->is_delta1();
is_.pi2 = children[1]->is_pi2();
is_.delta2 = children[1]->is_delta2();
} }
else else // behaves like F
{ {
is_.syntactic_safety = false; is_.syntactic_safety = false;
bool g = children[1]->is_syntactic_guarantee(); bool g = children[1]->is_syntactic_guarantee();
is_.syntactic_obligation = g; is_.syntactic_obligation = g;
is_.syntactic_recurrence = g; is_.syntactic_recurrence = g;
is_.delta1 = g;
is_.pi2 = g;
is_.delta2 = g | is_.sigma2;
} }
assert(children[0]->is_sere_formula()); assert(children[0]->is_sere_formula());
assert(children[1]->is_psl_formula()); assert(children[1]->is_psl_formula());
@ -1536,19 +1594,25 @@ namespace spot
is_.syntactic_safety = children[1]->is_syntactic_safety(); is_.syntactic_safety = children[1]->is_syntactic_safety();
is_.syntactic_recurrence = children[1]->is_syntactic_recurrence(); is_.syntactic_recurrence = children[1]->is_syntactic_recurrence();
if (children[0]->is_finite()) is_.pi2 = children[1]->is_pi2();
if (children[0]->is_finite()) // behaves like X
{ {
is_.syntactic_guarantee = children[1]->is_syntactic_guarantee(); is_.syntactic_guarantee = children[1]->is_syntactic_guarantee();
is_.syntactic_obligation = children[1]->is_syntactic_obligation(); is_.syntactic_obligation = children[1]->is_syntactic_obligation();
is_.syntactic_persistence = is_.syntactic_persistence = children[1]->is_syntactic_persistence();
children[1]->is_syntactic_persistence(); is_.delta1 = children[1]->is_delta1();
is_.sigma2 = children[1]->is_sigma2();
is_.delta2 = children[1]->is_delta2();
} }
else else // behaves like G
{ {
is_.syntactic_guarantee = false; is_.syntactic_guarantee = false;
bool s = children[1]->is_syntactic_safety(); bool s = children[1]->is_syntactic_safety();
is_.syntactic_obligation = s; is_.syntactic_obligation = s;
is_.syntactic_persistence = s; is_.syntactic_persistence = s;
is_.delta1 = s;
is_.sigma2 = s;
is_.delta2 = is_.pi2 | s;
} }
assert(children[0]->is_sere_formula()); assert(children[0]->is_sere_formula());
assert(children[1]->is_psl_formula()); assert(children[1]->is_psl_formula());
@ -1596,6 +1660,10 @@ namespace spot
children[0]->is_syntactic_recurrence() children[0]->is_syntactic_recurrence()
&& children[1]->is_syntactic_guarantee(); && children[1]->is_syntactic_guarantee();
// is_.syntactic_persistence = Persistence U Persistance // is_.syntactic_persistence = Persistence U Persistance
is_.delta1 = is_.syntactic_guarantee;
// is_.sigma2 = Σ₂ U Σ₂
is_.pi2 = is_.syntactic_guarantee;
is_.delta2 = is_.sigma2 | is_.pi2;
break; break;
case op::W: case op::W:
// See comment for op::U. // See comment for op::U.
@ -1617,7 +1685,10 @@ namespace spot
is_.syntactic_persistence = // Safety W Persistance is_.syntactic_persistence = // Safety W Persistance
children[0]->is_syntactic_safety() children[0]->is_syntactic_safety()
&& children[1]->is_syntactic_persistence(); && children[1]->is_syntactic_persistence();
is_.delta1 = is_.syntactic_safety;
is_.sigma2 = is_.syntactic_safety;
// is_.pi2 = Π₂ U Π₂
is_.delta2 = is_.sigma2 | is_.pi2;
break; break;
case op::R: case op::R:
// See comment for op::U. // See comment for op::U.
@ -1640,7 +1711,10 @@ namespace spot
is_.syntactic_persistence = // Persistence R Safety is_.syntactic_persistence = // Persistence R Safety
children[0]->is_syntactic_persistence() children[0]->is_syntactic_persistence()
&& children[1]->is_syntactic_safety(); && children[1]->is_syntactic_safety();
is_.delta1 = is_.syntactic_safety;
is_.sigma2 = is_.syntactic_safety;
// is_.pi2 = Π₂ U Π₂
is_.delta2 = is_.sigma2 | is_.pi2;
break; break;
case op::M: case op::M:
// See comment for op::U. // See comment for op::U.
@ -1662,7 +1736,10 @@ namespace spot
children[0]->is_syntactic_guarantee() children[0]->is_syntactic_guarantee()
&& children[1]->is_syntactic_recurrence(); && children[1]->is_syntactic_recurrence();
// is_.syntactic_persistence = Persistence M Persistance // is_.syntactic_persistence = Persistence M Persistance
is_.delta1 = is_.syntactic_guarantee;
// is_.sigma2 = Σ₂ M Σ₂
is_.pi2 = is_.syntactic_guarantee;
is_.delta2 = is_.sigma2 | is_.pi2;
break; break;
case op::Or: case op::Or:
{ {
@ -1787,6 +1864,10 @@ namespace spot
is_.syntactic_obligation = false; is_.syntactic_obligation = false;
is_.syntactic_recurrence = false; is_.syntactic_recurrence = false;
is_.syntactic_persistence = false; is_.syntactic_persistence = false;
is_.delta1 = false;
is_.pi2 = false;
is_.sigma2 = false;
is_.delta2 = false;
switch (op_) switch (op_)
{ {
@ -1832,6 +1913,10 @@ namespace spot
is_.syntactic_obligation = false; is_.syntactic_obligation = false;
is_.syntactic_recurrence = false; is_.syntactic_recurrence = false;
is_.syntactic_persistence = false; is_.syntactic_persistence = false;
is_.delta1 = false;
is_.pi2 = false;
is_.sigma2 = false;
is_.delta2 = false;
break; break;
} }
} }
@ -2011,31 +2096,38 @@ namespace spot
return strverscmp(f->ap_name().c_str(), g->ap_name().c_str()); return strverscmp(f->ap_name().c_str(), g->ap_name().c_str());
} }
#define printprops \ #define printprops \
proprint(is_boolean, "B", "Boolean formula"); \ proprint(is_boolean, "B", "Boolean formula"); \
proprint(is_sugar_free_boolean, "&", "without Boolean sugar"); \ proprint(is_sugar_free_boolean, "&", "without Boolean sugar"); \
proprint(is_in_nenoform, "!", "in negative normal form"); \ proprint(is_in_nenoform, "!", "in negative normal form"); \
proprint(is_syntactic_stutter_invariant, "x", \ proprint(is_syntactic_stutter_invariant, "x", \
"syntactic stutter invariant"); \ "syntactic stutter invariant"); \
proprint(is_sugar_free_ltl, "f", "without LTL sugar"); \ proprint(is_sugar_free_ltl, "f", "without LTL sugar"); \
proprint(is_ltl_formula, "L", "LTL formula"); \ proprint(is_ltl_formula, "L", "LTL formula"); \
proprint(is_psl_formula, "P", "PSL formula"); \ proprint(is_psl_formula, "P", "PSL formula"); \
proprint(is_sere_formula, "S", "SERE formula"); \ proprint(is_sere_formula, "S", "SERE formula"); \
proprint(is_finite, "F", "finite"); \ proprint(is_finite, "F", "finite"); \
proprint(is_eventual, "e", "pure eventuality"); \ proprint(is_eventual, "e", "pure eventuality"); \
proprint(is_universal, "u", "purely universal"); \ proprint(is_universal, "u", "purely universal"); \
proprint(is_syntactic_safety, "s", "syntactic safety"); \ proprint(is_syntactic_safety, "s", "syntactic safety"); \
proprint(is_syntactic_guarantee, "g", "syntactic guarantee"); \ proprint(is_syntactic_guarantee, "g", "syntactic guarantee"); \
proprint(is_syntactic_obligation, "o", "syntactic obligation"); \ proprint(is_syntactic_obligation, "o", "syntactic obligation"); \
proprint(is_syntactic_persistence, "p", "syntactic persistence"); \ proprint(is_syntactic_persistence, "p", "syntactic persistence"); \
proprint(is_syntactic_recurrence, "r", "syntactic recurrence"); \ proprint(is_syntactic_recurrence, "r", "syntactic recurrence"); \
proprint(is_marked, "+", "marked"); \ proprint(is_marked, "+", "marked"); \
proprint(accepts_eword, "0", "accepts the empty word"); \ proprint(accepts_eword, "0", "accepts the empty word"); \
proprint(has_lbt_atomic_props, "l", \ proprint(has_lbt_atomic_props, "l", \
"has LBT-style atomic props"); \ "has LBT-style atomic props"); \
proprint(has_spin_atomic_props, "a", \ proprint(has_spin_atomic_props, "a", \
"has Spin-style atomic props"); "has Spin-style atomic props"); \
proprint(is_delta1, "O", "delta1"); \
proprint(is_sigma2, "P", "sigma2"); \
proprint(is_pi2, "R", "pi2"); \
proprint(is_delta2, "D", "delta2");
// O (for Δ₁), P (for Σ₂), R (for Π₂) are the uppercase versions of
// o (obligation), p (persistence), r (recurrence) because they are
// stricter subsets of those.
std::list<std::string> std::list<std::string>
list_formula_props(const formula& f) list_formula_props(const formula& f)

75
spot/tl/formula.hh
View file

@ -522,6 +522,30 @@ namespace spot
return is_.spin_atomic_props; return is_.spin_atomic_props;
} }
/// \see formula::is_sigma2
bool is_sigma2() const
{
return is_.sigma2;
}
/// \see formula::is_pi2
bool is_pi2() const
{
return is_.pi2;
}
/// \see formula::is_delta1
bool is_delta1() const
{
return is_.delta1;
}
/// \see formula::is_delta2
bool is_delta2() const
{
return is_.delta2;
}
private: private:
static size_t bump_next_id(); static size_t bump_next_id();
void setup_props(op o); void setup_props(op o);
@ -627,15 +651,19 @@ namespace spot
bool finite:1; // Finite SERE formulae, or Bool+X forms. bool finite:1; // Finite SERE formulae, or Bool+X forms.
bool eventual:1; // Purely eventual formula. bool eventual:1; // Purely eventual formula.
bool universal:1; // Purely universal formula. bool universal:1; // Purely universal formula.
bool syntactic_safety:1; // Syntactic Safety Property. bool syntactic_safety:1; // Syntactic Safety Property (S).
bool syntactic_guarantee:1; // Syntactic Guarantee Property. bool syntactic_guarantee:1; // Syntactic Guarantee Property (G).
bool syntactic_obligation:1; // Syntactic Obligation Property. bool syntactic_obligation:1; // Syntactic Obligation Property (O).
bool syntactic_recurrence:1; // Syntactic Recurrence Property. bool syntactic_recurrence:1; // Syntactic Recurrence Property (R).
bool syntactic_persistence:1; // Syntactic Persistence Property. bool syntactic_persistence:1; // Syntactic Persistence Property (P).
bool not_marked:1; // No occurrence of EConcatMarked. bool not_marked:1; // No occurrence of EConcatMarked.
bool accepting_eword:1; // Accepts the empty word. bool accepting_eword:1; // Accepts the empty word.
bool lbt_atomic_props:1; // Use only atomic propositions like p42. bool lbt_atomic_props:1; // Use only atomic propositions like p42.
bool spin_atomic_props:1; // Use only spin-compatible atomic props. bool spin_atomic_props:1; // Use only spin-compatible atomic props.
bool delta1:1; // Boolean combination of (S) and (G).
bool sigma2:1; // Boolean comb. of (S) with X/F/U/M possibly applied.
bool pi2:1; // Boolean comb. of (G) with X/G/R/W possibly applied.
bool delta2:1; // Boolean combination of (Σ₂) and (Π₂).
}; };
union union
{ {
@ -1698,16 +1726,43 @@ namespace spot
/// universal formula also satisfies the formula. /// universal formula also satisfies the formula.
/// \cite etessami.00.concur /// \cite etessami.00.concur
SPOT_DEF_PROP(is_universal); SPOT_DEF_PROP(is_universal);
/// Whether a PSL/LTL formula is syntactic safety property. /// \brief Whether a PSL/LTL formula is syntactic safety property.
///
/// Is class is also called Π₁.
SPOT_DEF_PROP(is_syntactic_safety); SPOT_DEF_PROP(is_syntactic_safety);
/// Whether a PSL/LTL formula is syntactic guarantee property. /// \brief Whether a PSL/LTL formula is syntactic guarantee property.
///
/// Is class is also called Σ₁.
SPOT_DEF_PROP(is_syntactic_guarantee); SPOT_DEF_PROP(is_syntactic_guarantee);
/// Whether a PSL/LTL formula is syntactic obligation property. /// \brief Whether a PSL/LTL formula is in the Δ₁ syntactic frament
///
/// A formula is in Δ₁ if it is a boolean combination of syntactic
/// safety and syntactic guarantee properties.
SPOT_DEF_PROP(is_delta1);
/// \brief Whether a PSL/LTL formula is syntactic obligation property.
///
/// This class is a proper syntactic superset of Δ₁, but has the
/// same expressive power.
SPOT_DEF_PROP(is_syntactic_obligation); SPOT_DEF_PROP(is_syntactic_obligation);
/// Whether a PSL/LTL formula is syntactic recurrence property. /// Whether a PSL/LTL formula is in Σ₂
SPOT_DEF_PROP(is_sigma2);
/// Whether a PSL/LTL formula is in Π₂
SPOT_DEF_PROP(is_pi2);
/// \brief Whether a PSL/LTL formula is syntactic recurrence property.
///
/// This class is a proper syntactic superset of Σ₂ syntactically,
/// expressive power.
SPOT_DEF_PROP(is_syntactic_recurrence); SPOT_DEF_PROP(is_syntactic_recurrence);
/// Whether a PSL/LTL formula is syntactic persistence property. /// \brief Whether a PSL/LTL formula is syntactic persistence property.
///
/// This class is a proper syntactic superset of Π₂, but has the
/// same expressive power.
SPOT_DEF_PROP(is_syntactic_persistence); SPOT_DEF_PROP(is_syntactic_persistence);
/// \brief Whether a PSL/LTL formula is in the Δ₂ syntactic frament
///
/// A formula is in Δ₂ if it is a boolean combination of Σ₂ and Π₂
/// properties.
SPOT_DEF_PROP(is_delta2);
/// \brief Whether the formula has an occurrence of EConcatMarked /// \brief Whether the formula has an occurrence of EConcatMarked
/// or NegClosureMarked /// or NegClosureMarked
SPOT_DEF_PROP(is_marked); SPOT_DEF_PROP(is_marked);

194
tests/core/kind.test
View file

@ -25,118 +25,120 @@
set -e set -e
cat >input<<EOF cat >input<<EOF
a,B&!xfLPSFsgopra a,B&!xfLPSFsgopraOPRD
a<->b,BxfLPSFsgopra a<->b,BxfLPSFsgopraOPRD
!a,B&!xfLPSFsgopra !a,B&!xfLPSFsgopraOPRD
!(a|b),B&xfLPSFsgopra !(a|b),B&xfLPSFsgopraOPRD
F(a),&!xLPegopra F(a),&!xLPegopraOPRD
G(a),&!xLPusopra G(a),&!xLPusopraOPRD
a U b,&!xfLPgopra a U b,&!xfLPgopraOPRD
a U Fb,&!xLPegopra a U Fb,&!xLPegopraOPRD
Ga U b,&!xLPopra Ga U b,&!xLPopraPD
1 U a,&!xfLPegopra 1 U a,&!xfLPegopraOPRD
a W b,&!xfLPsopra a W b,&!xfLPsopraOPRD
a W 0,&!xfLPusopra a W 0,&!xfLPusopraOPRD
a M b,&!xfLPgopra a M b,&!xfLPgopraOPRD
a M 1,&!xfLPegopra a M 1,&!xfLPegopraOPRD
a R b,&!xfLPsopra a R b,&!xfLPsopraOPRD
0 R b,&!xfLPusopra 0 R b,&!xfLPusopraOPRD
a R (b R (c R d)),&!xfLPsopra a R (b R (c R d)),&!xfLPsopraOPRD
a U (b U (c U d)),&!xfLPgopra a U (b U (c U d)),&!xfLPgopraOPRD
a W (b W (c W d)),&!xfLPsopra a W (b W (c W d)),&!xfLPsopraOPRD
a M (b M (c M d)),&!xfLPgopra a M (b M (c M d)),&!xfLPgopraOPRD
Fa -> Fb,xLPopra Fa -> Fb,xLPopraOPRD
Ga -> Fb,xLPegopra Ga -> Fb,xLPegopraOPRD
Fa -> Gb,xLPusopra Fa -> Gb,xLPusopraOPRD
(Ga|Fc) -> Fb,xLPopra (Ga|Fc) -> Fb,xLPopraOPRD
(Ga|Fa) -> Gb,xLPopra (Ga|Fa) -> Gb,xLPopraOPRD
{a;c*;b}|->!Xb,&fPsopra {a;c*;b}|->!Xb,&fPsopraOPRD
{a;c*;b}|->X!b,&!fPsopra {a;c*;b}|->X!b,&!fPsopraOPRD
{a;c*;b}|->!Fb,&Psopra {a;c*;b}|->!Fb,&PsopraOPRD
{a;c*;b}|->G!b,&!Psopra {a;c*;b}|->G!b,&!PsopraOPRD
{a;c*;b}|->!Gb,&Pra {a;c*;b}|->!Gb,&PraRD
{a;c*;b}|->F!b,&!Pra {a;c*;b}|->F!b,&!PraRD
{a;c*;b}|->GFa,&!Pra {a;c*;b}|->GFa,&!PraRD
{a;c*;b}|->FGa,&!Pa {a;c*;b}|->FGa,&!Pa
{a[+];c[+];b*}|->!Fb,&Psopra {a[+];c[+];b*}|->!Fb,&PsopraOPRD
{a*;c[+];b*}|->!Fb,&xPsopra {a*;c[+];b*}|->!Fb,&xPsopraOPRD
{a[+];c*;b[+]}|->G!b,&!Psopra {a[+];c*;b[+]}|->G!b,&!PsopraOPRD
{a*;c[+];b[+]}|->!Gb,&Pra {a*;c[+];b[+]}|->!Gb,&PraRD
{a[+];c*;b[+]}|->F!b,&!Pra {a[+];c*;b[+]}|->F!b,&!PraRD
{a[+];c[+];b*}|->GFa,&!Pra {a[+];c[+];b*}|->GFa,&!PraRD
{a*;c[+];b[+]}|->FGa,&!Pa {a*;c[+];b[+]}|->FGa,&!Pa
{a;c;b|(d;e)}|->!Xb,&fPFsgopra {a;c;b|(d;e)}|->!Xb,&fPFsgopraOPRD
{a;c;b|(d;e)}|->X!b,&!fPFsgopra {a;c;b|(d;e)}|->X!b,&!fPFsgopraOPRD
{a;c;b|(d;e)}|->!Fb,&Psopra {a;c;b|(d;e)}|->!Fb,&PsopraOPRD
{a;c;b|(d;e)}|->G!b,&!Psopra {a;c;b|(d;e)}|->G!b,&!PsopraOPRD
{a;c;b|(d;e)}|->!Gb,&Pgopra {a;c;b|(d;e)}|->!Gb,&PgopraOPRD
{a;c;b|(d;e)}|->F!b,&!Pgopra {a;c;b|(d;e)}|->F!b,&!PgopraOPRD
{a;c;b|(d;e)}|->GFa,&!Pra {a;c;b|(d;e)}|->GFa,&!PraRD
{a;c;b|(d;e)}|->FGa,&!Ppa {a;c;b|(d;e)}|->FGa,&!PpaPD
{a[+] && c[+]}|->!Xb,&fPsopra {a[+] && c[+]}|->!Xb,&fPsopraOPRD
{a[+] && c[+]}|->X!b,&!fPsopra {a[+] && c[+]}|->X!b,&!fPsopraOPRD
{a[+] && c[+]}|->!Fb,&xPsopra {a[+] && c[+]}|->!Fb,&xPsopraOPRD
{a[+] && c[+]}|->G!b,&!xPsopra {a[+] && c[+]}|->G!b,&!xPsopraOPRD
{a[+] && c[+]}|->!Gb,&xPra {a[+] && c[+]}|->!Gb,&xPraRD
{a[+] && c[+]}|->F!b,&!xPra {a[+] && c[+]}|->F!b,&!xPraRD
{a[+] && c[+]}|->GFa,&!xPra {a[+] && c[+]}|->GFa,&!xPraRD
{a[+] && c[+]}|->FGa,&!xPa {a[+] && c[+]}|->FGa,&!xPa
{a;c*;b}<>->!Gb,&Pgopra {a;c*;b}<>->!Gb,&PgopraOPRD
{a;c*;b}<>->F!b,&!Pgopra {a;c*;b}<>->F!b,&!PgopraOPRD
{a;c*;b}<>->FGb,&!Ppa {a;c*;b}<>->FGb,&!PpaPD
{a;c*;b}<>->!GFb,&Ppa {a;c*;b}<>->!GFb,&PpaPD
{a;c*;b}<>->GFb,&!Pa {a;c*;b}<>->GFb,&!Pa
{a;c*;b}<>->!FGb,&Pa {a;c*;b}<>->!FGb,&Pa
{a*;c[+];b[+]}<>->!FGb,&Pa {a*;c[+];b[+]}<>->!FGb,&Pa
{a;c|d;b}<>->!Gb,&Pgopra {a;c|d;b}<>->!Gb,&PgopraOPRD
{a;c|d;b}<>->G!b,&!Psopra {a;c|d;b}<>->G!b,&!PsopraOPRD
{a;c|d;b}<>->FGb,&!Ppa {a;c|d;b}<>->FGb,&!PpaPD
{a;c|d;b}<>->!GFb,&Ppa {a;c|d;b}<>->!GFb,&PpaPD
{a;c|d;b}<>->GFb,&!Pra {a;c|d;b}<>->GFb,&!PraRD
{a;c|d;_b}<>->!FGb,&Pr {a;c|d;_b}<>->!FGb,&PrRD
# Equivalent to a&b&c&d # Equivalent to a&b&c&d
{a:b:c:d}!,B&!xfLPSFsgopra {a:b:c:d}!,B&!xfLPSFsgopraOPRD
a&b&c&d,B&!xfLPSFsgopra a&b&c&d,B&!xfLPSFsgopraOPRD
(Xa <-> XXXc) U (b & Fe),LPgopra (Xa <-> XXXc) U (b & Fe),LPgopraOPRD
(!X(a|X(!b))&(FX(g xor h)))U(!G(a|b)),LPegopra (!X(a|X(!b))&(FX(g xor h)))U(!G(a|b)),LPegopraOPRD
(!X(a|X(!b))&(GX(g xor h)))R(!F(a|b)),LPusopra (!X(a|X(!b))&(GX(g xor h)))R(!F(a|b)),LPusopraOPRD
(!X(a|X(!b))&(GX(g xor h)))U(!G(a|b)),LPeopra (!X(a|X(!b))&(GX(g xor h)))U(!G(a|b)),LPeopraPD
(!X(a|X(!b))&(FX(g xor h)))R(!F(a|b)),LPuopra (!X(a|X(!b))&(FX(g xor h)))R(!F(a|b)),LPuopraRD
(!X(a|X(!b))&(GX(g xor h)))U(!F(a|b)),LPpa (!X(a|X(!b))&(GX(g xor h)))U(!F(a|b)),LPpaPD
(!X(a|X(!b))&(FX(g xor h)))R(!G(a|b)),LPra (!X(a|X(!b))&(FX(g xor h)))R(!G(a|b)),LPraRD
(!X(a|GXF(!b))&(FGX(g xor h)))U(!F(a|b)),LPpa (!X(a|GXF(!b))&(FGX(g xor h)))U(!F(a|b)),LPpaPD
(!X(a|GXF(!b))&(FGX(g xor h)))R(!F(a|b)),LPupa (!X(a|GXF(!b))&(FGX(g xor h)))R(!F(a|b)),LPupa
(!X(a|FXG(!b))&(GFX(g xor h)))R(!G(a|b)),LPra (!X(a|FXG(!b))&(GFX(g xor h)))R(!G(a|b)),LPraRD
(!X(a|FXG(!b))&(GFX(g xor h)))U(!G(a|b)),LPera (!X(a|FXG(!b))&(GFX(g xor h)))U(!G(a|b)),LPera
(!X(a|GXF(!b))&(FGX(g xor h)))U(!G(a|Fb)),LPepa (!X(a|GXF(!b))&(FGX(g xor h)))U(!G(a|Fb)),LPepaPD
(!X(a|GXF(!b))&(FGX(g xor h)))U(!F(a|Gb)),LPa (!X(a|GXF(!b))&(FGX(g xor h)))U(!F(a|Gb)),LPa
(!X(a|FXG(!b))&(GFX(g xor h)))R(!F(a|Gb)),LPura (!X(a|FXG(!b))&(GFX(g xor h)))R(!F(a|Gb)),LPuraRD
(!X(a|FXG(!b))&(GFX(g xor h)))R(!G(a|Fb)),LPa (!X(a|FXG(!b))&(GFX(g xor h)))R(!G(a|Fb)),LPa
GFa M GFb,&!xLPeua GFa M GFb,&!xLPeua
FGa M FGb,&!xLPeupa FGa M FGb,&!xLPeupaPD
Fa M GFb,&!xLPera Fa M GFb,&!xLPera
GFa W GFb,&!xLPeura GFa W GFb,&!xLPeuraRD
FGa W FGb,&!xLPeua FGa W FGb,&!xLPeua
Ga W FGb,&!xLPupa Ga W FGb,&!xLPupa
Ga W b,&!xLPsopra Ga W b,&!xLPsopraOPRD
Fa M b,&!xLPgopra Fa M b,&!xLPgopraOPRD
{a;b*;c},&!fPsopra {a;b*;c},&!fPsopraOPRD
{a;b*;c}!,&!fPgopra {a;b*;c}!,&!fPgopraOPRD
# The negative normal form is {a;b*;c}[]->1 # The negative normal form is {a;b*;c}[]->1
!{a;b*;c}!,&fPsopra !{a;b*;c}!,&fPsopraOPRD
{a;b*;p112}[]->0,&!fPsopra {a;b*;p112}[]->0,&!fPsopraOPRD
!{a;b*;c.2},&!fPgopr !{a;b*;c.2},&!fPgoprOPRD
!{a[+];b*;c[+]},&!fPgopra !{a[+];b*;c[+]},&!fPgopraOPRD
!{a[+];b*;c*},&!xfPgopra !{a[+];b*;c*},&!xfPgopraOPRD
{a[+];b*;c[+]},&!fPsopra {a[+];b*;c[+]},&!fPsopraOPRD
{a[+] && b || c[+]},&!fPsopra {a[+] && b || c[+]},&!fPsopraOPRD
{a[+] && b[+] || c[+]},&!xfPsopra {a[+] && b[+] || c[+]},&!xfPsopraOPRD
{p[+]:p[+]},&!xfPsoprla {p[+]:p[+]},&!xfPsoprlaOPRD
(!p W Gp) | ({(!p[*];(p[+]:(p[*];!p[+])))[:*4][:+]}<>-> (!p W Gp)),&!xPpla (!p W Gp) | ({(!p[*];(p[+]:(p[*];!p[+])))[:*4][:+]}<>-> (!p W Gp)),&!xPplaPD
{b[+][:*0..3]},&!fPsopra {b[+][:*0..3]},&!fPsopraOPRD
{a->c[*]},xfPsopra {a->c[*]},xfPsopraOPRD
{(a[+];b*);c*}<>->d,&!xfPgopra {(a[+];b*);c*}<>->d,&!xfPgopraOPRD
{first_match(a[+];b*);c*}<>->d,&!fPgopra {first_match(a[+];b*);c*}<>->d,&!fPgopraOPRD
G(Ga U b),&!xLPura
GFb & G(Ga R b),&!xLPuraRD
EOF EOF
run 0 ../kind input run 0 ../kind input