\newcommand{\TSEMUP}[2]{\mathcal{T}^\uparrow\llbracket#1\rrbracket#2}
\newcommand{\TSEMDOWN}[2]{\mathcal{T}_\downarrow\llbracket#1\rrbracket#2}
\newcommand{\NSEM}[2]{\mathcal{N}\llbracket#1\rrbracket#2}
-\newcommand{\PSEM}[2]{\mathcal{P}\llbracket#1\rrbracket#2}
-\newcommand{\PPSEM}[2]{\mathcal{P'}\llbracket#1\rrbracket(#2)}
+\newcommand{\PSEM}[1]{\mathcal{P}\llbracket#1\rrbracket}
+\newcommand{\LSEM}[2]{\mathcal{L}\llbracket#1\rrbracket#2}
+\newcommand{\RSEM}[2]{\mathcal{R}\llbracket#1\rrbracket#2}
+\newcommand{\FSEM}[2]{\mathcal{F}\llbracket#1\rrbracket(#2)}
\newcommand{\PARENT}[1]{\mathit{parent}(#1)}
\newcommand{\CHILDREN}[1]{\mathit{children}(#1)}
+\newcommand{\CHILD}[1]{\mathit{child}(#1)}
\newcommand{\ANCESTORS}[1]{\mathit{ancestors}(#1)}
\newcommand{\DESCENDANTS}[1]{\mathit{descendants}(#1)}
\newcommand{\HASATTRIBUTE}[2]{\mathit{hasAttribute}(#1,#2)}
\newcommand{\NAME}[1]{\mathit{name}(#1)}
\newcommand{\PREV}[1]{\mathit{prev}(#1)}
\newcommand{\NEXT}[1]{\mathit{next}(#1)}
+\newcommand{\PREDICATE}[1]{\mathit{predicate}(#1)}
+\newcommand{\IFV}[3]{\begin{array}[t]{@{}l}\mathbf{if}~#1~\mathbf{then}\\\quad#2\\\mathbf{else}\\\quad#3\end{array}}
+\newcommand{\IFH}[3]{\mathbf{if}~#1~\mathbf{then}~#2~\mathbf{else}~#3}
+\newcommand{\TRUE}{\mathbf{true}}
+\newcommand{\FALSE}{\mathbf{false}}
+\newcommand{\FUN}[2]{\lambda#1.#2}
+\newcommand{\LET}[3]{\mathbf{let}~#1=#2~\mathbf{in}~#3}
+\newcommand{\REC}[2]{\mathbf{rec}~#1=#2}
+\newcommand{\APPLY}[2]{(#1\;#2)}
+\newcommand{\APPLYX}[3]{(#1\;#2\;#3)}
+\newcommand{\AND}{\wedge}
+\newcommand{\OR}{\vee}
+\newcommand{\AAND}{\,\vec{\AND}\,}
+\newcommand{\AOR}{\,\vec{\OR}\,}
+\newcommand{\MATCH}[4]{\begin{array}[t]{@{}c@{~\to~}l@{}l@{}}\multicolumn{2}{@{}l@{}}{\mathbf{match}~#1~\mathbf{with}}\\\phantom{|}\quad\{#2\}\\|\quad\emptyset\end{array}}
\[
\begin{array}{rcl}
- \CSEM{.}{x} &=& \{x\}\\
+ \CSEM{q}{x} &=& \{x_1\mid x_1\in\{x\} \wedge \QSEM{q}{x_1}\}\\
\CSEM{..}{x} &=& \PARENT{x}\\
\CSEM{/}{x} &=& \CHILDREN{x}\\
- \CSEM{q}{x} &=& \{x_1\mid x_1\in\{x\} \wedge \QSEM{q}{x_1}\}\\
+ \CSEM{c_1\;c_2}{x} &=& \CSEM{c_2}{\CSEM{c_1}{x}}\\
\CSEM{(c)}{x} &=& \CSEM{c}{x}\\
\CSEM{\{c:\alpha\}}{x} &=& \alpha(x,\CSEM{c}{x})\\
\CSEM{c_1\&c_2}{x} &=& \CSEM{c_1}{x} \cap \CSEM{c_2}{x}\\
\CSEM{c_1\mid c_2}{x} &=& \CSEM{c_1}{x} \cup \CSEM{c_2}{x}\\
\CSEM{c+}{x} &=& \CSEM{c}{x} \cup \CSEM{c+}{\CSEM{c}{x}}\\
- \CSEM{c?}{x} &=& \CSEM{.\mid c}{x}\\
- \CSEM{c*}{x} &=& \CSEM{{c+}?}{x}\\
- \CSEM{c_1\;c_2}{x} &=& \CSEM{c_2}{\CSEM{c_1}{x}}\\
- \CSEM{!c}{x} &=& \{x_1\mid x_1\in\{x\} \wedge \CSEM{c}{x}=\emptyset\}\\[3ex]
- \QSEM{\langle*\rangle}{x} &=& \ISELEMENT{x}\\
- \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}\\
+ \CSEM{c?}{x} &=& \{x\}\cup\CSEM{c}{x}\\
+ \CSEM{c*}{x} &=& \CSEM{{c+}?}{x}\\[3ex]
+ \QSEM{c}{x} &=& \CSEM{c}{x}\ne\emptyset\\
+ \QSEM{!c}{x} &=& \CSEM{c}{x}=\emptyset\\
+ \QSEM{\langle*\rangle}{x} &=& \TRUE\\
+ \QSEM{\langle n\rangle}{x} &=& \NAME{x}=n\\
+ \QSEM{@n}{x} &=& \HASATTRIBUTE{x}{n}\\
+ \QSEM{@n=v}{x} &=& \ATTRIBUTE{x}{n}=v\\
+ \QSEM{[p_1\#p_2]}{x} &=& \LSEM{p_1}{\PREV{x}}\wedge\RSEM{p_2}{\NEXT{x}}\\[3ex]
+ \LSEM{}{\alpha} &=& \TRUE\\
+ \LSEM{\cent}{\alpha} &=& \alpha=\emptyset\\
+ \LSEM{p\;q}{\emptyset} &=& \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} &=& \FALSE\\
+ \RSEM{q\;p}{\{x\}} &=& \QSEM{q}{x}\wedge\RSEM{p}{\NEXT{x}}\\[3ex]
+ \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}{\APPLY{\QSEM{q}{}}{x}} \\
+ \PSEM{..} &=& \FUN{x}{\neg\APPLY{\mathit{null}}{\PARENT{x}}}\\
+ \PSEM{/} &=& \FUN{x}{\neg\APPLY{\mathit{null}}{\CHILD{x}}}\\
+ \PSEM{(c)} &=& \PSEM{c}\\
+ \PSEM{\{c:\alpha\}} &=& \FUN{x}{\APPLY{\PSEM{c}}{x}\AAND\APPLY{\alpha}{x}}\\
+ \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_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}{\MATCH{\PARENT{x}}{y}{\APPLY{t}{y}}{\APPLY{f}{x}}}\\
+% \FSEM{/}{t,f} &=& \FUN{x}{(\vee_{y\in\CHILDREN{x}} \APPLY{t}{y})\AOR\APPLY{f}{x}}\\
+ \FSEM{/}{t,f} &=& \FUN{x}{\APPLYX{\mathit{exists}}{t}{\CHILD{x}}\AOR\APPLY{f}{x}}\\
+ \FSEM{(c)}{t,f} &=& \FSEM{c}{t,f}\\
+ \FSEM{\{c:\alpha\}}{t,f} &=& \FSEM{c}{\FUN{x}{\PSEM{c}\AAND\APPLY{\alpha}{x}\AAND\APPLY{t}{x},f}}\\
+ \FSEM{c_1\;c_2}{t,f} &=& \FUN{x}{\APPLY{\FSEM{c_1}{\FSEM{c_2}{t,\FUN{\_}{\APPLY{f}{x}}},f}}{x}}\\
+ \FSEM{c_1\&c_2}{t,f} &=& \FUN{x}{\APPLY{\FSEM{c_1}{\FUN{y}{\APPLY{\FSEM{c_2}{\FUN{z}{(y=z)\AAND\APPLY{t}{z}},\FUN{\_}{\APPLY{f}{x}}}}{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}\\[3ex]
+ \QSEM{c}{} &=& \PSEM{c}\\
+ \QSEM{!c}{} &=& \FUN{x}{\neg\APPLY{\PSEM{c}}{x}}\\
+ \QSEM{\langle*\rangle}{} &=& \FUN{\_}{\TRUE}\\
+ \QSEM{\langle n\rangle}{} &=& \FUN{x}{\NAME{x}=n}\\
+ \QSEM{@n}{} &=& \FUN{x}{\HASATTRIBUTE{x}{n}}\\
+ \QSEM{@n=v}{} &=& \FUN{x}{\ATTRIBUTE{x}{n}=v}\\
+ \QSEM{[p_1\#p_2]}{} &=& \FUN{x}{\APPLY{\LSEM{p_1}{}}{\PREV{x}}\wedge\APPLY{\RSEM{p_2}{}}{\NEXT{x}}}\\[3ex]
+ \LSEM{}{} &=& \FUN{\_}{\TRUE}\\
+ \LSEM{\cent}{} &=& \mathit{null}\\
+ \LSEM{p\;q}{} &=& \FUN{x}{\MATCH{x}{y}{\QSEM{q}{y}\AAND\APPLY{\LSEM{p}}{\PREV{y}}}{\FALSE}}\\
+ \RSEM{}{} &=& \FUN{\_}{\TRUE}\\
+ \RSEM{\$}{} &=& \mathit{null}\\
+ \RSEM{p\;q}{} &=& \FUN{x}{\MATCH{x}{y}{\QSEM{q}{y}\AAND\APPLY{\RSEM{p}}{\NEXT{y}}}{\FALSE}}\\
+ \mathit{null} &=& \FUN{x}{\MATCH{x}{\_}{\FALSE}{\TRUE}}\\
+ \mathit{exists} &=& \FUN{t}{\REC{a}{\FUN{x}{\MATCH{x}{y}{\APPLY{t}{y}\AOR\APPLY{a}{\NEXT{x}}}{\FALSE}}}}
+\end{array}
+\]
+
+
+
\end{document}
projects / helm.git / blobdiff