- \QSEM{\langle!*\rangle}{x} &=& \neg\QSEM{\langle*\rangle}{x}\\
- \QSEM{\langle n_1\mid\cdots\mid n_k\rangle}{x} &=& \exists i\in\{1,\dots,k\}:\NAME{x}=n_i\\
- \QSEM{\langle !n_1\mid\cdots\mid n_k\rangle}{x} &=& \neg\QSEM{\langle n_1\mid\cdots\mid n_k\rangle}{x}\\
- \QSEM{q[@n]}{x} &=& \QSEM{q}{x} \wedge \HASATTRIBUTE{x}{n}\\
- \QSEM{q[!@n]}{x} &=& \QSEM{q}{x} \wedge \HASNOATTRIBUTE{x}{n}\\
- \QSEM{q[@n=v]}{x} &=& \QSEM{q}{x} \wedge \ATTRIBUTE{x}{n}= v\\
- \QSEM{q[!@n=v]}{x} &=& \QSEM{q}{x} \wedge \ATTRIBUTE{x}{n}\ne v\\
- \QSEM{q[p]}{x} &=& \QSEM{q}{x} \wedge \PSEM{p}{x}\\
- \QSEM{q[!p]}{x} &=& \QSEM{q}{x} \wedge \neg\PSEM{p}{x}\\[3ex]
- \PSEM{p_1\#p_2}{x} &=& \PPSEM{p_1}{*,\PREV{x}}\wedge\PPSEM{p_2}{\NEXT{x},*}\\
- \PSEM{\cent p_1\#p_2}{x} &=& \PPSEM{p_1}{\cent,\PREV{x}}\wedge\PPSEM{p_2}{\NEXT{x},*}\\
- \PSEM{p_1\#p_2\$}{x} &=& \PPSEM{p_1}{*,\PREV{x}}\wedge\PPSEM{p_2}{\NEXT{x},\$}\\
- \PSEM{\cent p_1\#p_2\$}{x} &=& \PPSEM{p_1}{\cent,\PREV{x}}\wedge\PPSEM{p_2}{\NEXT{x},\$}\\[3ex]
- \PPSEM{}{*,\alpha} &=& \mathit{true}\\
- \PPSEM{}{\cent,\alpha} &=& \alpha=\emptyset\\
- \PPSEM{p\;c}{\alpha,\emptyset} &=& \mathit{false}\\
- \PPSEM{p\;c}{\alpha,\{x\}} &=& \CSEM{c}{x}\ne\emptyset\wedge\PPSEM{p}{\alpha,\PREV{x}}\\
- \PPSEM{}{\alpha,*} &=& \mathit{true}\\
- \PPSEM{}{\alpha,\$} &=& \alpha=\emptyset\\
- \PPSEM{c\;p}{\emptyset,\alpha} &=& \mathit{false}\\
- \PPSEM{c\;p}{\{x\},\alpha} &=& \CSEM{c}{x}\ne\emptyset\wedge\PPSEM{p}{\NEXT{x},\alpha}\\
+ \QSEM{\langle n\rangle}{x} &=& \ISELEMENT{x}\wedge\NAME{x}=n\\
+ \QSEM{@n}{x} &=& \ISELEMENT{x}\wedge\HASATTRIBUTE{x}{n}\\
+ \QSEM{@n=v}{x} &=& \ISELEMENT{x}\wedge\ATTRIBUTE{x}{n}=v\\
+ \QSEM{[p_1\#p_2]}{x} &=& \ISELEMENT{x}\wedge\LSEM{p_1}{\PREV{x}}\wedge\RSEM{p_2}{\NEXT{x}}\\[3ex]
+ \LSEM{}{\alpha} &=& \TRUE\\
+ \LSEM{\cent}{\alpha} &=& \alpha=\emptyset\\
+ \LSEM{p\;q}{\emptyset} &=& \mathit{false}\\
+ \LSEM{p\;q}{\{x\}} &=& \QSEM{q}{x}\wedge\LSEM{p}{\PREV{x}}\\[3ex]
+ \RSEM{}{\alpha} &=& \TRUE\\
+ \RSEM{\$}{\alpha} &=& \alpha=\emptyset\\
+ \RSEM{q\;p}{\emptyset} &=& \mathit{false}\\
+ \RSEM{q\;p}{\{x\}} &=& \QSEM{q}{x}\wedge\RSEM{p}{\NEXT{x}}\\
+\end{array}
+\]
+
+\[
+\begin{array}{rcl}
+ \PREDICATE{q} &=& \TRUE\\
+ \PREDICATE{..} &=& \FALSE\\
+ \PREDICATE{/} &=& \FALSE\\
+ \PREDICATE{c_1\;c_2} &=& \PREDICATE{c_1}\wedge\PREDICATE{c_2}\\
+ \PREDICATE{(c)} &=& \PREDICATE{c}\\
+ \PREDICATE{c_1\&c_2} &=& \PREDICATE{c_1}\wedge\PREDICATE{c_2}\\
+ \PREDICATE{c_1\mid c_2} &=& \PREDICATE{c_1}\wedge\PREDICATE{c_2}\\
+ \PREDICATE{c+} &=& \PREDICATE{c}\\
+ \PREDICATE{c?} &=& \PREDICATE{c}\\
+ \PREDICATE{c*} &=& \PREDICATE{c}
+\end{array}
+\]
+
+\[
+\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},\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},\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}{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}\\