diff --git a/doc/tl/tl.tex b/doc/tl/tl.tex index 141daa7b8..80c47e973 100644 --- a/doc/tl/tl.tex +++ b/doc/tl/tl.tex @@ -1473,10 +1473,10 @@ The goals in most of these simplification are to: \end{itemize} Rewritings defined with $\equivEU$ are applied only when -\verb|tl_simplifier_options::favor_event_univ|' is \texttt{true}: +`\verb|tl_simplifier_options::favor_event_univ|' is \texttt{true}: they try to lift subformulas that are both eventual and universal \emph{higher} in the syntax tree. Conversely, rules defined with $\equivNeu$ -are applied only when \verb|favor_event_univ|' is \texttt{false}: they +are applied only when `\verb|favor_event_univ|' is \texttt{false}: they try to \textit{lower} subformulas that are both eventual and universal. Currently all these simplifications assume LTL semantics, so they make @@ -1486,10 +1486,10 @@ only listed with $\X$. \subsection{Basic Simplifications}\label{sec:basic-simp} These simplifications are enabled with -\verb|tl_simplifier_options::reduce_basics|'. A couple of them may +`\verb|tl_simplifier_options::reduce_basics|'. A couple of them may enlarge the size of the formula: they are denoted using $\equiV$ instead of $\equiv$, and they can be disabled by setting the -\verb|tl_simplifier_options::reduce_size_strictly|' option to +`\verb|tl_simplifier_options::reduce_size_strictly|' option to \texttt{true}. \subsubsection{Basic Simplifications for Temporal Operators} @@ -1715,7 +1715,7 @@ $\Esuffix$. They assume that $b$, denote a Boolean formula. As noted at the beginning for section~\ref{sec:basic-simp}, rewritings denoted with $\equiV$ can be disabled by setting the -\verb|tl_simplifier_options::reduce_size_strictly|' option to +`\verb|tl_simplifier_options::reduce_size_strictly|' option to \texttt{true}. \begin{align*} @@ -1818,7 +1818,7 @@ $q,\,q_i$ & a pure eventuality that is also purely universal \\ \G(f_1\AND\ldots\AND f_n \AND q_1 \AND \ldots \AND q_p)&\equivEU \G(f_1\AND\ldots\AND f_n)\AND q_1 \AND \ldots \AND q_p \\ \G\F(f_1\AND\ldots\AND f_n \AND q_1 \AND \ldots \AND q_p)&\equiv \G(\F(f_1\AND\ldots\AND f_n)\AND q_1 \AND \ldots \AND q_p) \\ \G(f_1\AND\ldots\AND f_n \AND e_1 \AND \ldots \AND e_m \AND \G(e_{m+1}) \AND \ldots\AND \G(e_p))&\equivEU \G(f_1\AND\ldots\AND f_n)\AND \G(e_1 \AND \ldots \AND e_p) \\ - \G(f_1\AND\ldots\AND f_n \AND \G(g_1) \AND \ldots \AND \G(g_m) &\equiv \G(f_1\AND\ldots\AND f_n\AND g_1 \AND \ldots \AND g_m) \\ + \G(f_1\AND\ldots\AND f_n \AND \G(g_1) \AND \ldots \AND \G(g_m)) &\equiv \G(f_1\AND\ldots\AND f_n\AND g_1 \AND \ldots \AND g_m) \\ \F(f_1 \OR \ldots \OR f_n \OR u_1 \OR \ldots \OR u_m \OR \F(u_{m+1})\OR\ldots\OR \F(u_p)) &\equivEU \F(f_1\OR \ldots\OR f_n) \OR \F(u_1 \OR \ldots \OR u_p)\\ \F(f_1 \OR \ldots \OR f_n \OR \F(g_1) \OR \ldots \OR \G(g_m)) &\equiv \F(f_1\OR \ldots\OR f_n \OR g_1 \OR \ldots \OR g_m)\\ \G(f_1)\AND\ldots\AND \G(f_n) \AND \G(e_1) \AND \ldots\AND \G(e_p)&\equivEU \G(f_1\AND\ldots\AND f_n)\AND \G(e_1 \AND \ldots \AND e_p) \\ @@ -1837,19 +1837,19 @@ implication can be done in two ways: \begin{description} \item[Syntactic Implication Checks] were initially proposed by~\citet{somenzi.00.cav}. This detection is enabled by the - ``\verb|tl_simplifier_options::synt_impl|'' option. This is a + `\verb|tl_simplifier_options::synt_impl|' option. This is a cheap way to detect implications, but it may miss some. The rules we implement are described in Appendix~\ref{ann:syntimpl}. \item[Language Containment Checks] were initially proposed by~\citet{tauriainen.03.tr}. This detection is enabled by the - ``\verb|tl_simplifier_options::containment_checks|'' option. + `\verb|tl_simplifier_options::containment_checks|' option. \end{description} In the following rewritings rules, $f\simp g$ means that $g$ was proved to be implied by $f$ using either of the above two methods. Additionally, implications denoted by $f\Simp g$ are only checked if -the ``\verb|tl_simplifier_options::containment_checks_stronger|'' +the `\verb|tl_simplifier_options::containment_checks_stronger|' option is set (otherwise the rewriting rule is not applied). We write $f\simpe g$ iff $f\simp g$ and $g\simp f$. @@ -1936,7 +1936,7 @@ The first six rules, about n-ary operators $\AND$ and $\OR$, are implemented for $n$ operands by testing each operand against all other. To prevent the complexity to escalate, this is only performed with up to 16 operands. That value can be changed in -``\verb|tl_simplifier_options::containment_max_ops|''. +`\verb|tl_simplifier_options::containment_max_ops|'. The following rules mix implication-based checks with formulas that are pure eventualities ($e$) or that are purely universal ($u$).