-\documentclass[a4paper]{article}
+\documentclass[a4paper,draft]{article}
\usepackage{manfnt}
\usepackage{a4wide}
\newcommand{\NVAR}[1]{#1:\mathtt{number}}
\newcommand{\IVAR}[1]{#1:\mathtt{name}}
\newcommand{\FENCED}[1]{\texttt{\char'050}#1\texttt{\char'051}}
-\newcommand{\IOT}[2]{|[#1|]_{\mathcal#2}^1}
+\newcommand{\ITO}[2]{|[#1|]_{\mathcal#2}^1}
+\newcommand{\IOT}[2]{|[#1|]_{\mathcal#2}^2}
\newcommand{\ADDPARENS}[2]{\llparenthesis#1\rrparenthesis^{#2}}
\newcommand{\NAMES}{\mathit{names}}
\newcommand{\DOMAIN}{\mathit{domain}}
\end{table}
\[
-\IOT{\cdot}{{}} : P -> \mathit{Env} -> T
+\ITO{\cdot}{{}} : P -> \mathit{Env} -> T
\]
\begin{table}
\hrule
\[
\begin{array}{rcll}
- \IOT{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\IOT{(P_1)}{E},\dots,\IOT{(P_n)}{E} ] \\
- \IOT{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
- \IOT{\BREAK}{E} & = & \BREAK \\
- \IOT{(P)}{E} & = & \IOT{P}{E} \\
- \IOT{(P_1\cdots P_n)}{E} & = & B_H^{00}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
- \IOT{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
- \IOT{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
- \IOT{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
- \IOT{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
- \IOT{\mathtt{opt}~P}{E} & = & \IOT{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
+ \ITO{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\ITO{(P_1)}{E},\dots,\ITO{(P_n)}{E} ] \\
+ \ITO{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\ITO{P_1}{E}\cdots\ITO{P_n}{E}] \\
+ \ITO{\BREAK}{E} & = & \BREAK \\
+ \ITO{(P)}{E} & = & \ITO{P}{E} \\
+ \ITO{(P_1\cdots P_n)}{E} & = & B_H^{00}[\ITO{P_1}{E}\cdots\ITO{P_n}{E}] \\
+ \ITO{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
+ \ITO{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
+ \ITO{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
+ \ITO{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
+ \ITO{\mathtt{opt}~P}{E} & = & \ITO{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
& & & \mathcal{E}'(x)=\left\{
\begin{array}{@{}ll}
v, & \mathcal{E}(x) = \mathtt{Some}~v \\
\mathcal{E}(x), & \mbox{otherwise}
\end{array}
\right. \\
- \IOT{\mathtt{list}k~P~l?}{E} & = & \IOT{P}{{E}_1}~{l?}\cdots {l?}~\IOT{P}{{E}_n} &
+ \ITO{\mathtt{list}k~P~l?}{E} & = & \ITO{P}{{E}_1}~{l?}\cdots {l?}~\ITO{P}{{E}_n} &
\mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
& & & n\ge k \\
& & & \mathcal{E}_i(x) = \left\{
\mathcal{E}(x), & \mbox{otherwise}
\end{array}
\right. \\
- \IOT{l}{E} & = & l \\
+ \ITO{l}{E} & = & l \\
%% & | & (P) & \mbox{(fenced)} \\
%% & | & M & \mbox{(magic)} \\
\newcommand{\ANNPOS}[2]{\mathit{pos}(#1)_{#2}}
\begin{table}
-\caption{\label{tab:addparens} Where are parentheses added? Look here.\strut}
+\caption{\label{tab:addparens} Can't read the AST and need parentheses? Here you go!.\strut}
\hrule
\[
\begin{array}{rcll}
\newcommand{\NT}[1]{\langle\mathit{#1}\rangle}
\begin{table}
+\caption{\label{tab:synl2} Concrete syntax of level 2 patterns.\strut}
+\hrule
\[
\begin{array}{@{}rcll@{}}
\NT{term} & ::= & & \mbox{\bf terms} \\
& & x~\{\NT{ptname}\}~\verb+\Rightarrow+~\NT{term} \\[10ex]
\NT{meta} & ::= & & \mbox{\bf meta} \\
- & & \BLOB(\NT{term},\dots,\NT{term}) & (term blob) \\
+ & & \BLOB(\NT{term},\dots,\NT{term}) & \mbox{(term blob)} \\
& | & [\verb+term+]~x \\
& | & \verb+number+~x \\
& | & \verb+ident+~x \\
& | & \verb+fail+
\end{array}
\]
+\hrule
\end{table}
\begin{table}
\renewcommand{\arraystretch}{3.5}
\begin{array}{@{}c@{}}
\inference[\sc Constr]
- {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {P_i :: D_i}
{\BLOB[P_1,\dots,P_n] :: D_i \oplus \cdots \oplus D_j} \\
\inference[\sc TermVar]
{}
- {[\mathtt{term}]~x :: x : \mathtt{Term}}
+ {\mathtt{term}~x :: x : \mathtt{Term}}
\quad
\inference[\sc NumVar]
{}
\qquad
\inference[\sc Fail]
{}
- {\mathtt{fail} : \emptyset}
+ {\mathtt{fail} :: \emptyset}
%% & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
%% & | & \verb+fail+
\end{array}
\hrule
\end{table}
+\begin{table}
+\caption{...}
+\hrule
+\[
+\begin{array}{rcll}
+\IOT{C[t_1,\dots,t_n]}{E} & = & C[\IOT{t_1}{E},\dots,\IOT{t_n}{E}] \\
+\IOT{\mathtt{term}~x}{E} & = & t & \mathcal{E}(x) = \mathtt{Term}~t \\
+\IOT{\mathtt{number}~x}{E} & = & n & \mathcal{E}(x) = \mathtt{Number}~n \\
+\IOT{\mathtt{ident}~x}{E} & = & y & \mathcal{E}(x) = \mathtt{String}~y \\
+\IOT{\mathtt{fresh}~x}{E} & = & y & \mathcal{E}(x) = \mathtt{String}~y \\
+\IOT{\mathtt{default}~P_1~P_2}{E} & = & \IOT{P_2}{E'} & \mathcal{E}(\NAMES(P_1)\setminus\NAMES(P_2)))=\{\mathtt{None}\} \\
+& & & \mathcal{E}'(x) =
+ \left\{
+ \begin{array}{@{}ll}
+ \bot & x \in \NAMES(P_1)\setminus\NAMES(P_2) \\
+ \mathcal{E}(x) & \mbox{otherwise} \\
+ \end{array}
+ \right. \\
+\IOT{\mathtt{default}~P_1~P_2}{E} & = & \IOT{P_2}{E'} & \mathcal{E}(\NAMES(P_1)\setminus\NAMES(P_2)))=\{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
+& & & \mathcal{E}'(x) =
+ \left\{
+ \begin{array}{@{}ll}
+ v & x \in \NAMES(P_1)\setminus\NAMES(P_2) \wedge \mathcal{E}(x) = \mathtt{Some}~v\\
+ \mathcal{E}(x) & \mbox{otherwise}
+ \end{array}
+ \right. \\
+\IOT{\mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2}{E}
+& =
+& \IOT{P_2}{E'}
+& \mathcal{E}(\NAMES(P_2)\setminus\{x\}) = \{[]\} \\
+& & & \mathcal{E}'(y) =
+ \left\{
+ \begin{array}{@{}ll}
+ \bot & y \in \NAMES(P_2)\setminus\{x\} \\
+ \mathcal{E}(y) & \mbox{otherwise} \\
+ \end{array}
+ \right. \\
+\IOT{\mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2}{E}
+& =
+& \IOT{P_2}{E'}
+& \mathcal{E}(\NAMES(P_2)\setminus\{x\}) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
+& & & \mathcal{E}'(y) =
+ \left\{
+ \begin{array}{@{}ll}
+ \IOT{\mathtt{fold}~d~P_1~\mathtt{rec}~x~P_e}{E''} & y = x \\
+ v_1 & y\in\NAMES(P_2)\setminus\{x\} \wedge \mathcal{E}(y)=[v_1;\dots;v_n] \\
+ \mathcal{E}(y) & \mbox{otherwise} \\
+ \end{array}
+ \right. \\
+& & & \mathcal{E}''(y) =
+ \left\{
+ \begin{array}{@{}ll}
+ [v_2;\dots;v_n] & y\in\NAMES(P_2)\setminus\{x\} \wedge \mathcal{E}(y)=[v_1;\dots;v_n] \\
+ \mathcal{E}(y) & \mbox{otherwise} \\
+ \end{array}
+ \right. \\
+\end{array} \\
+\]
+\end{table}
+
\begin{table}
\caption{\label{tab:l2match} Pattern matching of level 2 terms.\strut}
\hrule
{t \not\in P_1 & t \in P_3 ~> \mathcal E}
{t \in \mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 ~> \mathcal E}
\\
- \inference[\sc FoldR]
- {t \in P_2 ~> \mathcal E & \mathcal{E}(x) \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
- {t \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E''}
- \quad
- \mathcal E''(y) = \left\{
+ \inference[\sc FoldRec]
+ {t \in P_2 ~> \mathcal E & \mathcal{E}(x) \in \mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
+ {t \in \mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E''}
+ \\
+ \mbox{where}~\mathcal{E}''(y) = \left\{
\renewcommand{\arraystretch}{1}
\begin{array}{ll}
- \mathcal{E}(y)::\mathcal{E}'(y) & y \in \NAMES(P_2) \setminus \{x\} \\
+ \mathcal{E}(y)::\mathcal{E}'(y) & y \in \NAMES(P_2) \setminus \{x\} \wedge d = \mathtt{right} \\
+ \mathcal{E}'(y)@[\mathcal{E}(y)] & y \in \NAMES(P_2) \setminus \{x\} \wedge d = \mathtt{left} \\
\mathcal{E}'(y) & \mbox{otherwise}
\end{array}
\right.
\\
- \inference[\sc FoldB]
+ \inference[\sc FoldBase]
{t \not\in P_2 & t \in P_1 ~> \mathcal E}
{t \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
\quad
\section{Type checking}
+\newcommand{\GUARDED}{\mathit{guarded}}
+\newcommand{\TRUE}{\mathit{true}}
+\newcommand{\FALSE}{\mathit{false}}
+
\newcommand{\TN}{\mathit{tn}}
-Assume that we have two corresponding patterns $P_1$ (level 1) and
-$P_2$ (level 2) and that we have to check whether they are
-``correct''. First we define the notion of \emph{top-level names} of
-$P_1$ and $P_2$, as follows:
+\begin{table}
+\caption{\label{tab:guarded} Guarded condition of level 2
+pattern. Note that the recursive case of the \texttt{fold} magic is
+not explicitly required to be guarded. The point is that it must
+contain at least two distinct names, and this guarantees that whatever
+is matched by the recursive pattern, the terms matched by those two
+names will be smaller than the whole matched term.\strut} \hrule
\[
-\begin{array}{rcl}
- \TN(C_1[P'_1,\dots,P'_2]) & = & \TN(P'_1) \cup \cdots \cup \TN(P'_2) \\
- \TN(\TVAR{x}) & = & \{x\} \\
- \TN(\NVAR{x}) & = & \{x\} \\
- \TN(\IVAR{x}) & = & \{x\} \\
- \TN(\mathtt{list0}~P'~l?) & = & \emptyset \\
- \TN(\mathtt{list1}~P'~l?) & = & \emptyset \\
- \TN(\mathtt{opt}~P') & = & \emptyset \\[3ex]
-
- \TN(\BLOB(P''_1,\dots,P''_2)) & = & \TN(P''_1) \cup \cdots \cup \TN(P''_2) \\
- \TN(\mathtt{term}~x) & = & \{x\} \\
- \TN(\mathtt{number}~x) & = & \{x\} \\
- \TN(\mathtt{ident}~x) & = & \{x\} \\
- \TN(\mathtt{fresh}~x) & = & \{x\} \\
- \TN(\mathtt{anonymous}) & = & \emptyset \\
- \TN(\mathtt{fold}~P''_1~\mathtt{rec}~x~P''_2) & = & \TN(P''_1) \\
- \TN(\mathtt{default}~P''_1~P''_2) & = & \TN(P''_1) \cap \TN(P''_2) \\
- \TN(\mathtt{if}~P''_1~\mathtt{then}~P''_2~\mathtt{else}~P''_3) & = & \TN(P''_2) \\
- \TN(\mathtt{fail}) & = & \emptyset
+\begin{array}{rcll}
+ \GUARDED(C(M(P))) & = & \GUARDED(P) \\
+ \GUARDED(C(t_1,\dots,t_n)) & = & \TRUE \\
+ \GUARDED(\mathtt{term}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{number}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{ident}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{fresh}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{anonymous}) & = & \TRUE \\
+ \GUARDED(\mathtt{default}~P_1~P_2) & = & \GUARDED(P_1) \wedge \GUARDED(P_2) \\
+ \GUARDED(\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3) & = & \GUARDED(P_2) \wedge \GUARDED(P_3) \\
+ \GUARDED(\mathtt{fail}) & = & \TRUE \\
+ \GUARDED(\mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2) & = & \GUARDED(P_2)
\end{array}
\]
+\hrule
+\end{table}
+
+%% Assume that we have two corresponding patterns $P_1$ (level 1) and
+%% $P_2$ (level 2) and that we have to check whether they are
+%% ``correct''. First we define the notion of \emph{top-level names} of
+%% $P_1$ and $P_2$, as follows:
+%% \[
+%% \begin{array}{rcl}
+%% \TN(C_1[P'_1,\dots,P'_2]) & = & \TN(P'_1) \cup \cdots \cup \TN(P'_2) \\
+%% \TN(\TVAR{x}) & = & \{x\} \\
+%% \TN(\NVAR{x}) & = & \{x\} \\
+%% \TN(\IVAR{x}) & = & \{x\} \\
+%% \TN(\mathtt{list0}~P'~l?) & = & \emptyset \\
+%% \TN(\mathtt{list1}~P'~l?) & = & \emptyset \\
+%% \TN(\mathtt{opt}~P') & = & \emptyset \\[3ex]
+%% \TN(\BLOB(P''_1,\dots,P''_2)) & = & \TN(P''_1) \cup \cdots \cup \TN(P''_2) \\
+%% \TN(\mathtt{term}~x) & = & \{x\} \\
+%% \TN(\mathtt{number}~x) & = & \{x\} \\
+%% \TN(\mathtt{ident}~x) & = & \{x\} \\
+%% \TN(\mathtt{fresh}~x) & = & \{x\} \\
+%% \TN(\mathtt{anonymous}) & = & \emptyset \\
+%% \TN(\mathtt{fold}~P''_1~\mathtt{rec}~x~P''_2) & = & \TN(P''_1) \\
+%% \TN(\mathtt{default}~P''_1~P''_2) & = & \TN(P''_1) \cap \TN(P''_2) \\
+%% \TN(\mathtt{if}~P''_1~\mathtt{then}~P''_2~\mathtt{else}~P''_3) & = & \TN(P''_2) \\
+%% \TN(\mathtt{fail}) & = & \emptyset
+%% \end{array}
+%% \]
We say that a \emph{bidirectional transformation}
\[
\]
is well-formed if:
\begin{itemize}
+ \item the pattern $P_2$ is guarded, that is $\GUARDED(P_2)=\TRUE$;
\item the two patterns are well-formed in the same context $D$, that
is $P_1 :: D$ and $P_2 :: D$;
\item for any direct sub-pattern $\mathtt{opt}~P'_1$ of $P_1$ such
Relationship with grammatical frameworks?
-\end{document}
\ No newline at end of file
+\end{document}
projects / helm.git / commitdiff