* doc/tl/tl.tex: Various minor improvements.
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Alexandre Duret-Lutz
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1 changed files with 83 additions and 69 deletions
152
doc/tl/tl.tex
152
doc/tl/tl.tex
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@ -16,7 +16,9 @@
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\usepackage{tabulary}
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\usepackage[numbers]{natbib}
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\usepackage{rotating}
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\usepackage{booktabs}
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\usepackage{tikz}
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\usepackage{calc}
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\usetikzlibrary{backgrounds}
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\usetikzlibrary{shadows}
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\usetikzlibrary{arrows}
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@ -24,6 +26,9 @@
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\pgfdeclarelayer{foreground}
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\pgfsetlayers{background,main,foreground}
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\addtokomafont{caption}{\small}
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\setkomafont{captionlabel}{\sffamily\bfseries}
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% TODO
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\usepackage{todonotes}
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\newcommand{\stodo}[2][]
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@ -291,7 +296,7 @@ reserved keyword, or one that starts with a digit, is to use double quotes.
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The reason we deal with leading \samp{F}, \samp{G}, and \samp{X}
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specifically in rule~\ref{rule:ap2} is that these are unary LTL
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operators and we want to be able to write compact formul\ae{} like
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$\samp{GFa}$ instead of the equivalent \samp{G(F(a))} or
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\samp{GFa} instead of the equivalent \samp{G(F(a))} or
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\samp{G~F~a}. If you want to name an atomic proposition \samp{GFa},
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you will have to quote it as \samp{"GFa"}.
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@ -352,9 +357,9 @@ following Boolean operators:
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\begin{tabular}{ccccc}
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& preferred & \multicolumn{2}{c}{other supported} \\
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operation & syntax & \multicolumn{2}{c}{syntaxes}\\
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\hline
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\midrule
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negation & $\NOT f$ & $\NOTALT f$ \\
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disjunction & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT$ \\
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disjunction & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ \\
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conjunction & $f\AND g$ & $f\ANDALT g$ & $f\ANDALTT g$ \\
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implication & $f\IMPLIES g$ & $f\IMPLIESALT g$ & $f\IMPLIESALTT g$\\
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exclusion & $f\XOR g$ & $f\XORALT g$ \\
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@ -454,7 +459,7 @@ temporal operators can be used to construct another temporal formula.
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\begin{tabular}{ccc}
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& preferred & \multicolumn{1}{c}{other supported} \\
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operator & syntax & \multicolumn{1}{c}{syntaxes}\\
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\hline
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\midrule
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Next & $\X f$ & $\XALT f$ \\
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Eventually & $\F f$ & $\FALT f$ \\
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Always & $\G f$ & $\GALT f$ \\
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@ -527,6 +532,7 @@ The operator \samp{F}, \samp{G}, \samp{U}, \samp{R}, \samp{M}, and
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and one of \samp{U}, \samp{W}, \samp{R}, or \samp{M}. This property
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is usually used to simplify proofs.
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{\allowdisplaybreaks
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\begin{align*}
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\intertext{Equivalences using $\U$:}
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\F f &\equiv \1\U f\\
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@ -555,7 +561,7 @@ is usually used to simplify proofs.
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f\U g&\equiv ((\X g)\M f)\OR g \\
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f \W g&\equiv (f\U g)\OR \G f \equiv ((\X g)\M f)\OR g\OR \NOT((\NOT f)\M\1)\\
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f \R g&\equiv (f \M g)\OR\G f \equiv (f \M g)\OR \NOT((\NOT f)\M\1)
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\end{align*}
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\end{align*}}
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\ltltodo[noline]{Why do we have two ways to rewrite $f\W g$ with $\U$,
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and two ways to rewrite $f\M g$ with $\R$, but no similar rules for other operators? What are we missing?}
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@ -606,7 +612,7 @@ denote arbitrary SERE and $b$ denotes a Boolean formula.
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\begin{tabular}{ccccc}
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& preferred & \multicolumn{2}{c}{other supported} \\
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operation & syntax & \multicolumn{2}{c}{syntaxes}\\
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\hline
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\midrule
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empty word & $\eword$ \\
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union & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ \\
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(synchronized) intersection & $f\ANDALT g$ & $f\ANDALTT g$ \\
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@ -649,6 +655,7 @@ the above operators to denote an unbounded range. For instance
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The following semantics assume that $f$ and $g$ are two SEREs, while
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$b$ is a Boolean formula.
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{\allowdisplaybreaks
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\begin{align*}
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\sigma\VDash \eword & \iff |\sigma| = 0 \\
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\sigma\VDash a & \iff \sigma(0)(a) = 1 \\
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@ -689,7 +696,7 @@ $b$ is a Boolean formula.
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\sigma\VDash\mathtt{\{\{}\NOT b\,\mathtt\}\STAR{0..}\CONCAT\ b\,\mathtt\}\STAR{\mvar{i}..\mvar{j}}\\
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\sigma\VDash b\GOTO{\mvar{i}..} & \iff
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\sigma\VDash\mathtt{\{\{}\NOT b\,\mathtt\}\STAR{0..}\CONCAT\ b\,\mathtt\}\STAR{\mvar{i}..}\\
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\end{align*}
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\end{align*}}
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Notes:
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\begin{itemize}
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@ -799,7 +806,7 @@ formula $f$ to form another PSL formula.
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\begin{tabular}{ccccc}
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& preferred & \multicolumn{2}{c}{other supported} \\
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operation & syntax & \multicolumn{2}{c}{syntaxes}\\
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\hline
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\midrule
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(universal) suffix implication
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& $\sere{r}\Asuffix{} f$
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& $\sere{r}\AsuffixALT{} f$
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@ -892,30 +899,30 @@ operator, even if the operator has multiple synonyms (like \samp{|},
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\samp{||}, and {`\verb=\/='}).
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\begin{align*}
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\mathit{constant} ::=&\, \0 \mid \1 & \mathit{tformula} ::=&\,\mathit{bformula}\\
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\mathit{atomic\_prop} ::=&\, \text{see section~\ref{sec:ap}} & \mid&\,\tsamp{(}\,\mathit{tformula}\,\tsamp{)} \\
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\mathit{bformula} ::=&\, \mathit{constant} & \mid&\,\msamp{\NOT}\,\mathit{tformula}\,\\
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\mid&\,\mathit{atomic\_prop} & \mid&\,\mathit{tformula}\,\msamp{\AND}\,\mathit{tformula} \\
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\mid&\,\mathit{atomic\_prop}\code{=0} & \mid&\,\mathit{tformula}\,\msamp{\OR}\,\mathit{tformula} \\
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\mid&\,\mathit{atomic\_prop}\code{=1} & \mid&\,\mathit{tformula}\,\msamp{\IMPLIES}\,\mathit{tformula} \\
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\mid&\,\tsamp{(}\,\mathit{bformula}\,\tsamp{)} & \mid&\,\mathit{tformula}\,\msamp{\XOR}\,\mathit{tformula} \\
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\mid&\,\msamp{\NOT}\,\mathit{bformula} & \mid&\,\mathit{tformula}\,\msamp{\EQUIV}\,\mathit{tformula} \\
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\mid&\,\mathit{bformula}\,\msamp{\AND}\,\mathit{bformula} & \mid&\,\msamp{\X}\,\mathit{tformula}\\
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\mid&\,\mathit{bformula}\,\msamp{\OR}\,\mathit{bformula} & \mid&\,\msamp{\F}\,\mathit{tformula}\\
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\mid&\,\mathit{bformula}\,\msamp{\IMPLIES}\,\mathit{bformula} & \mid&\,\msamp{\G}\,\mathit{tformula}\\
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\mid&\,\mathit{bformula}\,\msamp{\XOR}\,\mathit{bformula} & \mid&\,\mathit{tformula}\,\msamp{\U}\,\mathit{tformula} \\
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\mid&\,\mathit{bformula}\,\msamp{\EQUIV}\,\mathit{bformula} & \mid&\,\mathit{tformula}\,\msamp{\W}\,\mathit{tformula} \\
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\mathit{rformula} ::=&\, \mathit{bformula} & \mid&\,\mathit{tformula}\,\msamp{\R}\,\mathit{tformula} \\
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\mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}} & \mid&\,\mathit{tformula}\,\msamp{\M}\,\mathit{tformula} \\
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\mid&\,\mathit{rformula}\msamp{\OR}\mathit{rformula} & \mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}}\,\msamp{\Asuffix}\,\mathit{tformula}\\
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\mid&\,\mathit{rformula}\msamp{\AND}\mathit{rformula} & \mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}}\,\msamp{\AsuffixEQ}\,\mathit{tformula}\\
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\mid&\,\mathit{rformula}\msamp{\ANDALT}\mathit{rformula} & \mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}}\,\msamp{\Esuffix}\,\mathit{tformula}\\
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\mid&\,\mathit{rformula}\msamp{\CONCAT}\mathit{rformula} & \mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}}\,\msamp{\EsuffixEQ}\,\mathit{tformula} \\
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\mid&\,\mathit{rformula}\msamp{\FUSION}\mathit{rformula} & \mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}} \\
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\mid&\,\mathit{rformula}\msamp{\STAR{\mvar{i}..\mvar{j}}} & \mid&\,\tsamp{\{}\mathit{rformula}\tsamp{\}}\msamp{\NOT} \\
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\mid&\,\mathit{rformula}\msamp{\PLUS} \\
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\mid&\,\mathit{rformula}\msamp{\EQUAL{\mvar{i}..\mvar{j}}} \\
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\mid&\,\mathit{rformula}\msamp{\GOTO{\mvar{i}..\mvar{j}}} \\
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\mathit{constant} ::={}& \0 \mid \1 & \mathit{tformula} ::={}&\mathit{bformula}\\
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\mathit{atomic\_prop} ::={}& \text{see section~\ref{sec:ap}} & \mid{}&\tsamp{(}\,\mathit{tformula}\,\tsamp{)} \\
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\mathit{bformula} ::={}& \mathit{constant} & \mid{}&\msamp{\NOT}\,\mathit{tformula}\,\\
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\mid{}&\mathit{atomic\_prop} & \mid{}&\mathit{tformula}\,\msamp{\AND}\,\mathit{tformula} \\
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\mid{}&\mathit{atomic\_prop}\code{=0} & \mid{}&\mathit{tformula}\,\msamp{\OR}\,\mathit{tformula} \\
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\mid{}&\mathit{atomic\_prop}\code{=1} & \mid{}&\mathit{tformula}\,\msamp{\IMPLIES}\,\mathit{tformula} \\
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\mid{}&\tsamp{(}\,\mathit{bformula}\,\tsamp{)} & \mid{}&\mathit{tformula}\,\msamp{\XOR}\,\mathit{tformula} \\
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\mid{}&\msamp{\NOT}\,\mathit{bformula} & \mid{}&\mathit{tformula}\,\msamp{\EQUIV}\,\mathit{tformula} \\
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\mid{}&\mathit{bformula}\,\msamp{\AND}\,\mathit{bformula} & \mid{}&\msamp{\X}\,\mathit{tformula}\\
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\mid{}&\mathit{bformula}\,\msamp{\OR}\,\mathit{bformula} & \mid{}&\msamp{\F}\,\mathit{tformula}\\
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\mid{}&\mathit{bformula}\,\msamp{\IMPLIES}\,\mathit{bformula} & \mid{}&\msamp{\G}\,\mathit{tformula}\\
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\mid{}&\mathit{bformula}\,\msamp{\XOR}\,\mathit{bformula} & \mid{}&\mathit{tformula}\,\msamp{\U}\,\mathit{tformula} \\
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\mid{}&\mathit{bformula}\,\msamp{\EQUIV}\,\mathit{bformula} & \mid{}&\mathit{tformula}\,\msamp{\W}\,\mathit{tformula} \\
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\mathit{sere} ::={}&\mathit{bformula} & \mid{}&\mathit{tformula}\,\msamp{\R}\,\mathit{tformula} \\
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\mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}} & \mid{}&\mathit{tformula}\,\msamp{\M}\,\mathit{tformula} \\
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\mid{}&\mathit{sere}\msamp{\OR}\mathit{sere} & \mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}}\,\msamp{\Asuffix}\,\mathit{tformula}\\
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\mid{}&\mathit{sere}\msamp{\AND}\mathit{sere} & \mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}}\,\msamp{\AsuffixEQ}\,\mathit{tformula}\\
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\mid{}&\mathit{sere}\msamp{\ANDALT}\mathit{sere} & \mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}}\,\msamp{\Esuffix}\,\mathit{tformula}\\
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\mid{}&\mathit{sere}\msamp{\CONCAT}\mathit{sere} & \mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}}\,\msamp{\EsuffixEQ}\,\mathit{tformula} \\
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\mid{}&\mathit{sere}\msamp{\FUSION}\mathit{sere} & \mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}} \\
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\mid{}&\mathit{sere}\msamp{\STAR{\mvar{i}..\mvar{j}}} & \mid{}&\tsamp{\{}\mathit{sere}\tsamp{\}}\msamp{\NOT} \\
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\mid{}&\mathit{sere}\msamp{\PLUS} \\
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\mid{}&\mathit{sere}\msamp{\EQUAL{\mvar{i}..\mvar{j}}} \\
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\mid{}&\mathit{sere}\msamp{\GOTO{\mvar{i}..\mvar{j}}} \\
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\end{align*}
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\section{Operator precedence}
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@ -944,10 +951,10 @@ operator, even if the operator has multiple synonyms (like \samp{|},
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We want to get closer to the PSL standard eventually.}
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\begin{center}
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\begin{tabular}{cll}
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\begin{tabular}{clc}
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assoc. & operators & priority \\
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\hline
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left & $\Asuffix,\,\AsuffixEQ,\,\Esuffix,\,\EsuffixEQ$ & lowest \\
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\midrule
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left & $\Asuffix,\,\AsuffixEQ,\,\Esuffix,\,\EsuffixEQ$ & \tikz[remember picture,baseline]\node (lowest){lowest}; \\
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left & $\CONCAT,\,\FUSION$ & \\
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left & $\IMPLIES,\,\EQUIV$ & \\
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left & $\XOR$ & \\
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@ -958,9 +965,12 @@ left & $\U,\,\W,\,\M,\,\R$
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& $\X$ & \\
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& $\STAR{\mvar{i}..\mvar{j}},\,\PLUS,\,\EQUAL{\mvar{i}..\mvar{j}},\,\GOTO{\mvar{i}..\mvar{j}}$ & \\
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& $\NOT$ & \\
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& $\code{=0},\,\code{=1}$ & highest \\
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& $\code{=0},\,\code{=1}$ & \tikz[remember picture,baseline]\node (highest){highest}; \\
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\end{tabular}
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\end{center}
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\begin{tikzpicture}[remember picture,overlay,>=stealth',semithick]
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\draw[->] (lowest) -- (highest);
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\end{tikzpicture}
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\chapter{Properties}
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@ -971,7 +981,7 @@ instance using the following methods:
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\noindent
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\label{property-methods}
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\begin{tabulary}{\textwidth}{lL}
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\begin{tabulary}{.995\textwidth}{@{}lJ@{}}
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\texttt{is\_boolean()}& Whether the formula uses only Boolean
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operators.
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\\\texttt{is\_sugar\_free\_boolean()}& Whether the formula uses
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@ -1010,6 +1020,7 @@ instance using the following methods:
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\\\texttt{accepts\_eword()}& Whether the formula accepts
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$\eword$. (This can only be true for a SERE formula.)
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\end{tabulary}
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\footnotetext{This special operator is used when translating recurring
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$\Esuffix$, it is rendered as $\EsuffixMarked$ and it obeys the same
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simplification rules and properties as $\Esuffix$ (except for the
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@ -1042,7 +1053,7 @@ Let $\varphi$ designate any arbitrary formula and $\varphi_E$
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rules:
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\begin{align*}
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\varphi_E ::=&\, \0
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\varphi_E ::={}& \0
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\mid \1
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\mid \X \varphi_E
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\mid \F \varphi
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@ -1050,13 +1061,13 @@ rules:
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\mid \varphi_E\AND \varphi_E
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\mid (\varphi_E\OR \varphi_E)
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\mid \NOT\varphi_U\\
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\mid&\,\varphi \U \varphi_E
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\mid{}&\varphi \U \varphi_E
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\mid 1 \U \varphi
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\mid \varphi_E\R \varphi_E
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\mid \varphi_E\W \varphi_E
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\mid \varphi_E\M \varphi_E
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\mid \varphi \M \1\\
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\varphi_U ::=&\, \0
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\varphi_U ::={}& \0
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\mid \1
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\mid \X \varphi_U
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\mid \F \varphi_U
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@ -1064,7 +1075,7 @@ rules:
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\mid \varphi_U\AND \varphi_U
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\mid (\varphi_U\OR \varphi_U)
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\mid \NOT\varphi_E\\
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\mid&\,\varphi_U\U \varphi_U
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\mid{}&\varphi_U\U \varphi_U
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\mid \varphi \R \varphi_U
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\mid \0 \R \varphi
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\mid \varphi_U\W \varphi_U
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@ -1075,6 +1086,16 @@ rules:
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\section{Syntactic Hierarchy Classes}
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\begin{figure}[tbp]
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\setcapindent{1em}
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\begin{captionbeside}{
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The temporal hierarchy of \citet{manna.87.podc} with their
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associated classes of automata~\citep{cerna.03.mfcs}. The
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formul\ae{} associated to each class are more than canonical
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examples: they show the normal forms under which any LTL formula
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of the class can be rewritten, assuming that $p,p_i,q,q_i$ denote
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subformul\ae{} involving only Boolean operators, $\X$, and past
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temporal operators (Spot does not support the
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latter). \label{fig:hierarchy}}[o]
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\centering
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\begin{tikzpicture}
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\draw[drop shadow,fill=white] (0,0) rectangle (6,7);
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@ -1092,29 +1113,22 @@ rules:
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\node[align=center] (saf) at (1,1) {Safety\\ $\G p$};
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\node[align=center] (gua) at (5,1) {Guarantee\\ $\F p$};
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\node[align=center,below left] (det) at (-.2,6.7) {Deterministic\\Büchi Automata};
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\node[align=center,below right](weak) at (6.2,6.7) {Weak Büchi\\Automata};
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\node[align=center,right](wdba) at (6.2,4.3) {Weak Deteterministic\\Büchi Automata};
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\node[align=center,above right](ter) at (6.2,2) {Terminal\\Büchi Automata};
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\node[align=center,above left](occ) at (-.2,2) {Terminal\\co-Büchi Automata};
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\node[align=right,below left] (det) at (-.2,6.7) {Deterministic\\Büchi\\Automata};
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\node[align=left,below right](weak) at (6.2,6.7) {Weak Büchi\\Automata};
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\node[align=left,right](wdba) at (6.2,4.3) {Weak\\Deterministic\\Büchi\\Automata};
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\node[align=left,above right](ter) at (6.2,1.7) {Terminal\\Büchi\\Automata};
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\node[align=right,above left](occ) at (-.2,2) {Terminal\\co-Büchi\\Automata};
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\node[above right] (buc) at (3.35,7) {General Büchi Automata};
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\draw[annote] (rec) -- (det);
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\draw[annote] (per) -- (weak);
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\draw[annote] (obl.east) -- (wdba);
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\draw[annote] (gua) -- (ter);
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\draw[annote] (obl.east) -- (wdba.west);
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\draw[annote] (gua.north) -- (ter.west);
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\draw[annote] (saf) -- (occ);
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\draw[annote] (rea.north) .. controls +(north:5mm) and +(west:5mm) .. (buc.west);
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\end{tikzpicture}
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\caption{The temporal hierarchy of \citet{manna.87.podc} with their
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associated classes of automata~\citep{cerna.03.mfcs}. The
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formul\ae{} associated to each class are more than canonical
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examples: they show the normal forms under which any LTL formula of
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the class can be rewritten, assuming that $p,p_i,q,q_i$ denote
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subformul\ae{} involving only Boolean operators, $\X$, and past
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temporal operators (Spot does not support the
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latter). \label{fig:hierarchy}}
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\end{captionbeside}
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\end{figure}
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The hierarchy of linear temporal properties was introduced
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@ -1142,43 +1156,43 @@ $v$ denotes any variable, $r$ any SERE, $r_F$ any bounded SERE (no
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loops), and $r_I$ any unbounded SERE.
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\begin{align*}
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\varphi_G ::=&\, \0\mid\1\mid v\mid \NOT\varphi_S\mid
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\varphi_G ::={}& \0\mid\1\mid v\mid \NOT\varphi_S\mid
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\varphi_G\AND \varphi_G\mid (\varphi_G\OR \varphi_G)\mid
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\X\varphi_G \mid \F\varphi_G\mid
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\varphi_G\U\varphi_G\mid \varphi_G\M\varphi_G\\
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\mid&\, \sere{r_F}\mid
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\mid{}& \sere{r_F}\mid
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\sere{r}\Esuffix \varphi_G\mid
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\sere{r_F}\Asuffix \varphi_G \\
|
||||
\varphi_S ::=&\, \0\mid\1\mid v\mid \NOT\varphi_G\mid
|
||||
\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&\, \nsere{r_F}\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 ::={}& \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} \mid \nsere{r}\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
|
||||
\varphi_P ::={}& \varphi_O \mid \NOT\varphi_R\mid
|
||||
\varphi_P\AND \varphi_P\mid (\varphi_P\OR \varphi_P)\mid
|
||||
\X\varphi_P \mid \F\varphi_P \mid
|
||||
\varphi_P\U\varphi_P\mid\varphi_P\R\varphi_S\mid
|
||||
\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_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
|
||||
\X\varphi_R \mid \G\varphi_R \mid
|
||||
\varphi_R\U\varphi_G\mid\varphi_R\R\varphi_R\mid
|
||||
\varphi_R\W\varphi_R\mid\varphi_G\M\varphi_R\\
|
||||
\mid&\, \sere{r_F}\Esuffix \varphi_R \mid \sere{r_I}\Esuffix \varphi_G\mid \sere{r}\Asuffix \varphi_P\\
|
||||
\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
|
||||
|
|
@ -1413,12 +1427,12 @@ notation to distinguish the class of subformul\ae:
|
|||
|
||||
\begin{center}
|
||||
\begin{tabular}{rl}
|
||||
\hline
|
||||
\toprule
|
||||
$f,\,f_i,\,g,\,g_i$ & any PSL formula \\
|
||||
$e,\,e_i$ & a pure eventuality \\
|
||||
$u,\,u_i$ & a purely universal formula \\
|
||||
$q,\,q_i$ & a pure eventuality that is also purely universal \\
|
||||
\hline
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
|
|
@ -1528,7 +1542,7 @@ following cases, where $f_b$ and $g_b$ designate Boolean formul\ae{}:
|
|||
\begin{center}
|
||||
\begin{tabular}{rl}
|
||||
we have & if \\
|
||||
\hline
|
||||
\midrule
|
||||
$f\simp \1$ & always \\
|
||||
$\0\simp g$ & always \\
|
||||
$f_b \simp g_b$ & $\mathrm{BDD}(f_b)\land \mathrm{BDD}(g_b) =
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue