\newcommand{\TRUE}{\mathit{true}}
\newcommand{\FALSE}{\mathit{false}}
\newcommand{\FUN}[2]{\lambda#1.#2}
+\newcommand{\LET}[3]{\mathbf{let}~#1=#2~\mathbf{in}~#3}
+\newcommand{\APPLY}[2]{(#1\;#2)}
+\newcommand{\AND}{\wedge}
+\newcommand{\OR}{\vee}
+\newcommand{\AAND}{\,\vec{\AND}\,}
+\newcommand{\AOR}{\,\vec{\OR}\,}
\[
\begin{array}{rcl}
\PSEM{q} &=& \FUN{x}{\QSEM{q}{x}} \\
\PSEM{..} &=& \FUN{x}{\PARENT{x}\ne\emptyset}\\
\PSEM{/} &=& \FUN{x}{\CHILDREN{x}\ne\emptyset}\\
- \PSEM{c_1\;c_2} &=& \IFV{\PREDICATE{c_1}}{\FUN{x}{(\PSEM{c_1}\;x)\wedge(\PSEM{c_2}\;x)}}{\FSEM{c_1}{\PSEM{c_2}}}\\
+ \PSEM{c_1\;c_2} &=& \IFV{\PREDICATE{c_1}}{\FUN{x}{(\PSEM{c_1}\;x)\wedge(\PSEM{c_2}\;x)}}{\FSEM{c_1}{\PSEM{c_2},\FUN{\_}{\FALSE}}}\\
\PSEM{(c)} &=& \PSEM{c}\\
- \PSEM{c_1\&c_2} &=& \IFV{\PREDICATE{c_1}\wedge\PREDICATE{c_2}}{\FUN{x}{(\PSEM{c_1}\;x)\wedge(\PSEM{c_2}\;x)}}{\FSEM{c_1\&c_2}{\FUN{\_}{\TRUE}}}\\
+ \PSEM{c_1\&c_2} &=& \IFV{\PREDICATE{c_1}\wedge\PREDICATE{c_2}}{\FUN{x}{(\PSEM{c_1}\;x)\wedge(\PSEM{c_2}\;x)}}{\FSEM{c_1\&c_2}{\FUN{\_}{\TRUE},\FUN{\_}{\FALSE}}}\\
\PSEM{c_1\mid c_2} &=& \FUN{x}{(\PSEM{c_1}\;x)\vee(\PSEM{c_2}\;x)}\\
\PSEM{c+} &=& \PSEM{c}\\
\PSEM{c?} &=& \FUN{\_}{\TRUE}\\
\PSEM{c*} &=& \FUN{\_}{\TRUE}\\[3ex]
- \FSEM{q}{l} &=& \FUN{x}{\IFH{(\PSEM{q}\;x)}{(l\;x)}{\FALSE}}\\
- \FSEM{..}{l} &=& \FUN{x}{\IFH{\PARENT{x}=\{y\}}{(l\;y)}{\FALSE}}\\
- \FSEM{/}{l} &=& \FUN{x}{\vee_{p\in\CHILDREN{x}} (l\;p)}\\
- \FSEM{(c)}{l} &=& \FSEM{c}{l}\\
- \FSEM{c_1\;c_2}{l} &=& \FSEM{c_1}{\FSEM{c_2}{l}}\\
- \FSEM{c_1\&c_2}{l} &=& \FUN{x}{(\FSEM{c_1}{\FUN{y}{(l\;y)\wedge(\FSEM{c_2}{\FUN{z}{z=y}}\;x)}}\;x)}\\
- \FSEM{c_1\mid c_2}{l} &=& \FUN{x}{(\FSEM{c_1}{l}\;x)\vee(\FSEM{c_2}{l}\;x)}\\
- \FSEM{c+}{l} &=& \FSEM{c}{\FUN{y}{(l\;y)\vee(\FSEM{c+}{l}\;y)}}\\
- \FSEM{c?}{l} &=& \FUN{x}{(l\;x)\vee(\FSEM{c}{l}\;x)}\\
- \FSEM{c*}{l} &=& \FSEM{{c+}?}{l}\\
+ \FSEM{q}{t,f} &=& \FUN{x}{(\APPLY{\PSEM{q}}{x}\AAND\APPLY{t}{x})\AOR\APPLY{f}{x}}\\
+ \FSEM{..}{t,f} &=& \FUN{x}{\LET{\{y\}}{\PARENT{x}}{\APPLY{t}{y}\AOR\APPLY{f}{x}}}\\
+ \FSEM{/}{t,f} &=& \FUN{x}{(\vee_{y\in\CHILDREN{x}} \APPLY{t}{y})\AOR\APPLY{f}{x}}\\
+ \FSEM{(c)}{t,f} &=& \FSEM{c}{t,f}\\
+ \FSEM{c_1\;c_2}{t,f} &=& \FSEM{c_1}{\FSEM{c_2}{t,\FUN{\_}{\FALSE}},f}\\
+ \FSEM{c_1\&c_2}{t,f} &=& \FUN{x}{\APPLY{\FSEM{c_1}{\FUN{y}{\APPLY{t}{y}\AAND\APPLY{\FSEM{c_2}{\FUN{z}{z=y},\FUN{\_}{\FALSE}}}{x}},f}}{x}}\\
+ \FSEM{c_1\mid c_2}{t,f} &=& \FSEM{c_1}{t,\FSEM{c_2}{t,f}}\\
+ \FSEM{c+}{t,f} &=& \FSEM{c}{\FSEM{c+}{t,t},f}\\
+ \FSEM{c?}{t,f} &=& \FSEM{c}{t,t}\\
+ \FSEM{c*}{t,f} &=& \FSEM{{c+}?}{t,f}\\
\end{array}
\]