-\section {Operational semantics} \label {Operational}
+\section {Operational Semantics} \label {Operational}
This section describes {\MathQL} semantics, that we present in a natural
operational style \cite{Lan98,Win93}.
Here we use a simple type system that includes basic types such as strings and
Booleans, and some type constructors such as product and exponentiation.
$ y \oft Y $ will denote a typing judgement.
-Note that this semantics is not meant as a formal system \emph{per se}, but
-should serve as a reference for implementors.
+This semantics is not meant as a formal system \emph{per se}, but should be a
+reference for implementors.
-\subsection {Mathematical background}
-
-As a mathematical background for the semantics, we take the one presented in
-\cite{Gui03}.
-
-{\Str} denotes the type of strings and its elements are the finite sequences
-of Unicode \cite{Unicode} characters.
-Grammatical productions, represented as strings in angle brackets, denote the
-subtype of {\Str} containing the produced sequences of characters.
-
-{\Num} denotes the type of numbers and is the subtype of {\Str} defined by the
-regular expression: \TT{'0 - 9' [ '0 - 9' ]*}.
-In this type, numbers are represented by their decimal expansion.
-
-$ \Setof\ Y $ denotes the type of finite sets ({\ie} unordered finite
-sequences without repetitions) over $ Y $.
-$ \Listof\ Y $ denotes the type of lists ({\ie} ordered finite sequences)
-over $ Y $.
-We will use the notation $ [y_1, \cdots, y_m] $ for the list whose elements
-are $ y_1, \cdots, y_m $.
-
-{\Boole}, the type of Boolean values, is defined as
-$ \{\ES, \{("", \ES)\}\} \oft \Setof\ \Setof\ (\Str \times \Setof\ Y) $
-where the first element stands for \emph{false} (denoted by {\F}) and the
-second element stands for \emph{true} (denoted by {\T}).
-
-$ Y \times Z $ denotes the product of the types $ Y $ and $ Z $ whose elements
-are the ordered pairs $ (y, z) $ such that $ y \oft Y $ and $ z \oft Z $.
-The notation is also extended to a ternary product.
-
-$ Y \to Z $ denotes the type of functions from $ Y $ to $ Z $ and $ f\ y $
-denotes the application of $ f \oft Y \to Z $ to $ y \oft Y $.
-Relations over types, such as equality, are seen as functions to {\Boole}.
-
-With the above constructors we can give a formal meaning to most of the
-standard notation. For instance we will use the following:
-
-\begin{itemize}
-
-\item
-$ {\ES} \oft (\Setof\ Y) $
-
-\item
-$ {\lex} \oft ((\Setof\ Y) \to \Boole) \to \Boole $
-
-\item
-$ {\lall} \oft ((\Setof\ Y) \to \Boole) \to \Boole $
-
-\item
-$ {\in} \oft Y \to (\Setof\ Y) \to \Boole $ (infix)
-
-\item
-$ {\sub} \oft (\Setof\ Y) \to (\Setof\ Y) \to \Boole $ (infix)
-
-\item
-$ {\meet} \oft (\Setof\ Y) \to (\Setof\ Y) \to \Boole $ (infix)
-
-%\item
-$ {\sand} \oft (\Setof\ Y) \to (\Setof\ Y) \to (\Setof\ Y) $ (infix)
-
-\item
-$ {\sor} \oft (\Setof\ Y) \to (\Setof\ Y) \to (\Setof\ Y) $ (infix)
-
-\item
-$ {\sdor} \oft (\Setof\ Y) \to (\Setof\ Y) \to (\Setof\ Y) $
-(the disjoint union, infix)
-
-\item
-$ \le \oft \Num \to \Num \to \Boole $ (infix)
-
-\item
-$ < \oft \Num \to \Num \to \Boole $ (infix)
-
-\item
-$ \# \oft (\Setof\ Y) \to \Num $ (the size operator)
-
-\item
-$ \app \oft (\Listof\ Y) \to (\Listof\ Y) \to (\Listof\ Y) $
-(the concatenation, infix)
-
-\item
-$ \lnot \oft \Boole \to \Boole $
-
-\end{itemize}
-
-\noindent
-Note that $ \lall $ and $ \lex $ are always decidable because the sets are
-finite by definition.
-
-\noindent
-$ U \meet W $ means $ (\lex u \in U)\ u \in W $ and expresses the fact that
-$ U \sand W $ is inhabited as a primitive notion, {\ie} without mentioning
-intersection and equality as for $ U \sand W \neq \ES $, which is equivalent
-but may be implemented less efficiently in real cases%
-\footnote{As for the Boolean condition $ \a \lor \b $ which may have a more
-efficient implementation than $ \lnot(\lnot \a \land \lnot \b) $.}.
-
-$ U \meet W $ is a natural companion of $ U \sub W $ being its logical dual
-(recall that $ U \sub W $ means $ (\lall u \in U)\ u \in W $)
-and is already being used successfully in the context of a constructive
-({\ie} intuitionistic and predicative) approach to point-free topology
-\cite{Sam00}.
-
-Sets of couples play a central role in our formalization and in particular we
-will use:
-
-\begin{itemize}
-
-\item
-$ \Fst \oft (Y \times Z) \to Y $ such that $ \Fst\ (y, z) = y $.
-
-\item
-$ \Snd \oft (Y \times Z) \to Z $ such that $ \Snd\ (y, z) = z $.
-
-\item
-$ \Fsts \oft \Setof\ (Y \times Z) \to \Setof\ Z $ such that
-$ \Fsts\ U = \{\Fst\ u \st u \in U\} $.
-
-\item
-With the same notation, if $ W $ contains just one couple whose first
-component is $ y $, then $ \get{W}{y} $ is the second component of that couple.
-In the other cases $ \get{W}{y} $ is not defined.
-This operator has type $ (\Setof\ (Y \times Z)) \to Y \to Z $.
-
-\item
-Moreover $ \set{W}{y}{z} $ is the set obtained from $ W $ removing every
-couple whose first component is $ y $ and adding the couple $ (y, z) $.
-The type of this operator is \\
-$ (\Setof\ (Y \times Z)) \to Y \to Z \to (\Setof\ (Y \times Z)) $.
-
-\item
-Also $ U + W $ is the union of two sets of couples in the following sense:
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{rll}
-%
-$ U + \ES $ & rewrites to & $ U $ \\
-$ U + (W \sdor \{(y, z)\}) $ & rewrites to & $ \set{U}{y}{z} + W $
-%
-\end{tabular} \end{center}
-\end{footnotesize}
-
-\end{itemize}
-
-The last three operators are used to read, write and join association sets,
-which are sets of couples such that the first components of two different
-elements are always different.
-These sets will be exploited to formalize the memories appearing in evaluation
-contexts.
-
-\subsection {The core language}
-
-\subsubsection {Preliminaries}
-
-Wih the above background we are able to type the main objects needed in the
-formalization:
-
-\begin{itemize}
-
-\item
-A path $ s $ is a list of strings therefore its type is
-$ T_{0a} = \Listof\ \Str $.
-
-\item
-A multiple string value $ V $ is an object of type $ T_{0b} = \Setof\ \Str $.
-
-\item
-A attribute group $ G $ is an association set connecting the attribute names
-to their values, therefore its type is
-$ T_1 = \Setof\ (T_{0a} \times T_{0b}) $.
-
-\item
-A subject string $ r $ is an object of type $ \Str $.
-
-\item
-A set $ A $ of attribute groups is an object of type $ T_2 = \Setof\ T_1 $.
-
-\item
-An {\av} is a subject string with its attribute groups, so its type is
-$ T_3 = \Str \times T_2 $.
-
-\item
-A set $ S $ of {\av}'s is an association set of type $ T_4 = \Setof\ T_3 $.
-
-\item
-A triple of an attributed relation is of type
-$ T_5 = \Str \times \Str \times (T_{0a} \to \Str) $.
-
-\end{itemize}
-
-When a constant string appearing in a {\MathQL} expression is unquoted, the
-surrounding double quotes are deleted and each escaped sequence is translated
-according \figref{EscTS}.
-
-This operation is formally performed by the function
-$ \Unquote $ of type $ \Str \to \Str $.
-Moreover $ \Name \oft \GP{path} \to T_{0a} $ is a helper function that
-converts a linearized path in its structured representation.
-Formally $ \Name\ (\TT{/}q_1\TT{/} \cdots \TT{/}q_m) $
-rewrites to $ [\Unquote\ q_1, \cdots, \Unquote\ q_m] $.
-
-The semantics of {\MathQL} expressions denoting queries is given by the infix
-relation $ \daq $ that evaluates a query to an {\av} set.
-These expressions are evaluated in a context $ \G = \g $
-which is a triple of association sets that connect
-svar's to {\av} sets, avar's to {\av}'s and avar's to attribute groups.
-Therefore the type $ K $ of the context $ \G $ is:
-
-\begin{footnotesize} \begin{center}
-$
-\Setof\ (\GP{svar} \times T_4) \times
-\Setof\ (\GP{avar} \times T_3)\ \times % $ \\ $ \times\
-\Setof\ (\GP{avar} \times T_1)
-$
-\end{center} \end{footnotesize}
-
-\noindent
-and the evaluating relation is of the following type:
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{l}
-$ \mathord{\daq} \oft (K \times \GP{query}) \times (K \times T_4) \to \Boole $.
-\end{tabular} \end{center}
-\end{footnotesize}
-
-The context components $ \G_s $ and $ \G_a $ are used to store the contents of
-variables, while $ \G_g $ is used by the \TT{ex} operator to be presented
-below.
-
-In the following we will write $ (\G, x) \daq S $ to abbreviate
-$ (\G, x) \daq (\G, S) $.
-
-The semantics of {\MathQL} expressions denoting results is given by the infix
-relation $ \dar \oft \GP{avs} \times T_4 \to \Boole $ that evaluates a result
-to an {\av} set.
-
-\subsubsection{Queries}
-
-The first \GP{query} expressions include explicit {\av} sets and syntactic
-grouping:
-
-\begin{itemize}
-
-\item
-The syntactic grouping is obtained enclosing a \GP{query} between \TT{(}
-and \TT{)}.
-An explicit {\av} set can be represented by a single string, which is
-converted into a single {\av} with no attributes, or by a \GP{xavs}
-(extended {\av} set) expression enclosed between \TT{[} and \TT{]}.
-Such an expression describes all the components of an {\av} set and is
-evaluated using the rules given below.
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule{(\G, x) \daq S}{}{(\G, (x)) \daq S} \spc
-\irule{q \oft \GP{string}}{}{(\G, q) \daq \{(\Unquote\ q, \ES)\}}
-%
-\end{center}
-\begin{center}
-%
-\irule{x_1, \cdots, x_m \in \GP{xav} \spc
- (\G, TT{[} x_1 \TT{]}) \daq S_1 \spc \cdots \spc (\G, \TT{[} x_m \TT{]}) \daq S_m}{}
- {(\G, \TT{[} x_1 \TT{;} \cdots \TT{;} x_m \TT{]}) \daq S_1 \sum \cdots \sum S_m}
-%
-\end{center}
-\begin{center}
-%
-\irule{q \in \GP{string} \spc g_1, \cdots, g_m \in \GP{xgroup} \spc
- (\G, \TT{[} q\ \TT{attr}\ g_1 \TT{]}) \daq S_1 \spc \cdots \spc
- (\G, \TT{[} q\ \TT{attr}\ g_m \TT{]}) \daq S_m}{}
- {(\G, \TT{[} q\ \TT{attr}\ g_1 \TT{,} \cdots \TT{,} g_m \TT{]}) \daq S_1 \sum \cdots \sum S_m}
-%
-\end{center}
-\begin{center}
-%
-\irule{q \in \GP{string} \spc a_1, \cdots, a_m \in \GP{xatr} \spc
- (\G, \TT{[} q\ \TT{attr}\ \{ a_1 \} \TT{]}) \daq S_1 \spc \cdots \spc
- (\G, \TT{[} q\ \TT{attr}\ \{ a_m \} \TT{]}) \daq S_m}{}
- {(\G, \TT{[} q\ \TT{attr}\ \{ a_1 \TT{;} \cdots \TT{;} a_m \} \TT{]}) \daq S_1 \dsum \cdots \dsum S_m}
-%
-\end{center}
-\begin{center}
-%
-\irule{q \in \GP{string} \spc p \in \GP{path} \spc x \daq S}{}
- {(\G, \TT{[} q\ \TT{attr}\ \{ p = x \} \TT{]}) \daq
- \{(\Unquote\ q, \{ \{ (\Name\ p, \Fsts\ S) \} \})\}}
-%
-\end{center}
-\end{footnotesize}
-
-$ \dsum $ and $ \sum $ are helper functions describing the two union operations
-on {\av} sets: with and without attribute distribution respectively.
-$ \dsum $ and $ \sum $ have two rewrite rules each.
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{lrll}
-%
-1a &
-$ (S_1 \sdor \{(r, A_1)\}) \sum (S_2 \sdor \{(r, A_2)\}) $ & rewrites to &
-$ (S_1 \sum S_2) \sor \{(r, A_1 \sor A_2)\} $ \\
-1b & $ S_1 \sum S_2 $ & rewrites to & $ S_1 \sor S_2 $ \\
-2a &
-$ (S_1 \sdor \{(r, A_1)\}) \dsum (S_2 \sdor \{(r, A_2)\}) $ & rewrites to &
-$ (S_1 \dsum S_2) \sor \{(r, A_1 \distr A_2)\} $ \\
-2b & $ S_1 \dsum S_2 $ & rewrites to & $ S_1 \sor S_2 $
-%
-\end{tabular} \end{center}
-\end{footnotesize}
-
-Rules 1a, 2a override 1b, 2b respectively and
-$ A_1 \distr A_2 = \{G_1 \sum G_2 \st G_1 \in A_1, G_2 \in A_2\} $.
-
-\item
-The semantics of \TT{property} operator is described below.
-
-In the following rule,
-$s$ is ``$ \TT{property}\ h\ p_1\ \TT{main}\ p_2\ \RM{attr}\ e_1, \cdots,
-e_m\ \TT{in}\ k\ x $'', $P$ is $ \Property\ h $ and
-$A_2$ is $ \{ \Exp\ P\ p_1\ r_1\ \{e_1, \cdots, e_m\}\} $:
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule
-{h \oft \GP{refine} \spc p_1, p_2 \oft \GP{path} \spc
- e_1, \cdots, e_m \oft \GP{exp} \spc k \in \TT{["pattern"]?} \spc
- (\G, x) \daq S
-}{A}
-{(\G, s) \daq \bigsum \{ \{(r_2, A_2)\} \st
-(\lex r_1 \in \Src\ k\ P\ (\Fsts\ S))\
-(r_1, p_1 \app p_2, r_2) \in P
-\}}
-%
-\end{center}
-\end{footnotesize}
-
-When the \TT{main} clause is not present, we assume $ p_2 = \TT{/} $.
-
-Here $ \Property\ h $ gives the appropriate access relation according to
-the $ h $ flag (this is the primitive function that inspects the {\RDF} graph,
-see \subsecref{}).
-
-$ \Src\ k\ P\ V $ is a helper function giving the source set
-according to the $ k $ flag. $ \Src $ is based on $ \Match $, the helper
-function handling POSIX regular expressions. Formally:
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{rll}
-%
-$ \Src\ \TT{""}\ P\ V $ & rewrites to & $ V $ \\
-$ \Src\ \TT{"pattern"}\ P\ V $ & rewrites to &
-$ \Match\ \{r_1 \st (\lex p, r2)\ (r_1, p, r_2) \in P\} $\ V \\
-$ \Match\ W\ V $ & rewrites to & $ \bigsor \{\Pattern\ W\ s \st s \in V\} $
-%
-\end{tabular} \end{center}
-\end{footnotesize}
-
-Here $ \Pattern\ W\ s $ is the primitive function returning the subset of
-$ W \oft \Setof\ \Str $ whose element match the POSIX 1003.2-1992%
-\footnote{Included in POSIX 1003.1-2001:
-\URI{http://www.unix-systems.org/version3/ieee\_\,std.html}.}
-regular expression $ \verb+"^"+ \app s \app \TT{"\$"} $.
-
-$ \Exp\ P\ \p_1\ r_1\ E $ is the helper function that builds the group of
-attributes specified in the \TT{attr} clause.
-$ \Exp $ is based on $ \Exp\p $ which handles a single attribute. Formally:
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{rlll}
-%
-$ f\ P\ r_1\ p_1\ p $ & rewrites to &
-$ \{ r_2 \st (r_1, p_1 \app (\Name\ p), r_2) \in P \} $ &
-with $ p \oft \GP{path} $ \\
-$ \Exp\p\ P\ r_1\ p_1\ p $ & rewrites to &
-$ \{ (\Name\ p, f\ P\ r_1\ p_1\ p) \} $ &
-with $ p \oft \GP{path} $ \\
-$ \Exp\p\ P\ r_1\ p_1\ (p\ \TT{as}\ p\p) $ & rewrites to &
-$ \{ (\Name\ p\p, f\ P\ r_1\ p_1\ p) \} $ &
-with $ p, p\p \oft \GP{path} $ \\
-$ \Exp\ P\ r_1\ p_1\ E $ & rewrites to &
-$ \bigsum \{ \Exp\p\ P\ r_1\ p_1\ e \st e \in E \} $ &
-with $ E \oft \Setof\ \GP{exp} $
-\end{tabular} \end{center}
-\end{footnotesize}
-
-When $ c_1, \cdots, c_n \oft \GP{cons} $ and the clause
-``\TT{istrue} $ c_1, \cdots, c_n $'' is present, the set $ P $ must be replaced
-with $ \{ (r_1, p, r_2) \in P \st \Istrue\ P\ r_1\ p_1\ C \} $
-where $ C $ is $ \{c_1, \cdots, c_n\} $ and $ \Istrue $ is a helper function
-that checks the constraints in $ C $.
-$ \Istrue $ is based on $ \Istrue\p $ that handles a single constraint.
-Formally, if $ p \oft \GP{path} $ and $ (\G, x) \daq S $:
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{rll}
-%
-$ g\ P\ p_1\ p $ & rewrites to &
-$ \{ r_2 \st (\lex r_1)\ (r_1, p_1 \app (\Name\ p), r_2) \in P \} $ \\
-$ \Istrue\p\ P\ r_1\ p_1\ (p\ \TT{in}\ x) $ & rewrites to &
-$ (f\ P\ r_1\ p_1\ p) \meet \Fsts\ S $ \\
-$ \Istrue\p\ P\ r_1\ p_1\ (p\ \TT{match}\ x) $ & rewrites to &
-$ (f\ P\ r_1\ p_1\ p) \meet \Match\ (g\ P\ p_1\ p)\ (\Fsts\ S) $ \\
-$ \Istrue\ P\ r_1\ p_1\ C $ & rewrites to &
-$ (\lall c \in W)\ \Istrue\p\ P\ r_1\ p_1\ c $
-%
-\end{tabular} \end{center}
-\end{footnotesize}
-
-For each clause ``\TT{isfalse} $ c_1, \cdots, c_n $'' the set $ P $
-must be replaced with
-$ \{ (r_1, p, r_2) \in P \st \lnot (\Istrue\ P\ r_1\ p_1\ C) \} $
-(using the above notation).
-Note that these substitutions and the former must be composed if necessary.
-
-If the \TT{inverse} flag is present, also replace the instances of $ P $ in
-the rule $A$ and in the definition of $ \Src $ with
-$ \{ (r_2, p, r_1) \st (r_1, p, r_2) \in P \} $.
-
-\end{itemize}
-
-The second group of \GP{query} expressions includes the context manipulation
-facilities:
-
-\begin{itemize}
-
-\item
-The operators for reading variables:
-
-\begin{footnotesize} \begin{center}
-%
-\irule{i \oft \GP{svar}}{}{(\g, i) \daq \get{\G_s}{i}} \spc
-\irule{i \oft \GP{avar}}{}{(\g, i) \daq \{\get{\G_a}{i}\}}
-%
-\end{center} \end{footnotesize}
-
-$ \get{\G_s}{i} $ and $ \{\get{\G_a}{i}\} $ mean $ \ES $ if $ i $ is not defined.
-
-\item
-The \TT{let} operator assigns an {\av} set variable (svar):
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule{i \oft \GP{svar} \spc (\G_1, x_1) \daq (\g, S_1) \spc
- ((\set{\G_s}{i}{S_1}, \G_a, \G_g), x_2) \daq (\G_2, S_2)}
-{}{(\G_1, \TT{let}\ i\ \TT{=}\ x_1\ \TT{in}\ x_2) \daq (\G_2, S_2)}
-%
-\end{center}
-\end{footnotesize}
-
-The sequential composition operator \TT{;;} has the semantics of a \TT{let}
-introducing a fresh variable, so ``$ x_1\ \TT{;;}\ x_2 $'' revrites
-to ``$ \TT{let}\ i\ \TT{=}\ x_1\ \TT{in}\ x_2 $'' where $i$ does not occur in
-$x_2$.
-
-\item
-The \TT{ex} and ``dot'' operators provide a way to read the attributes stored
-in avar's.
-
-The \TT{ex} (exists) operator gives access to the groups of attributes
-associated to the {\av}'s in the $ \G_a $ part of the context and does
-this by loading its $ \G_g $ part, which is used by the ``dot'' operator
-described below.
-
-\TT{ex} is true if the query following it is successful for at least one
-pool of attribute groups, one for each {\av} in the $ \G_a $ part of the
-context. Formally we have the rules:
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule{(\lall \D_g \in \All\ \G_a)\ ((\G_s, \G_a, \G_g + \D_g), y) \daq \F}
- {1}{(\G, \TT{ex}\ y) \daq \F} \spc
-\irule{\Nop}{2}{(\G, \TT{ex}\ y) \daq \T} \spc
-%
-\end{center}
-\begin{center}
-%
-\irule {i \oft \GP{avar} \spc p \oft \GP{path} \spc \get{\get{\G_g}{i}}{\Name\ p} = \{s_1, \cdots, s_m\}}{}
- {(\G, i\TT{.}p) \daq \{(s_1, \ES), \cdots, (s_m, \ES)\}}
-%
-\end{center}
-\end{footnotesize}
-
-where%
-\footnote{$\D_g$ has the type of $ \G_g $.}
-$ \All\ \G_a = \{\D_g \st \get{\D_g}{i} = G\ \RM{iff}\ G \in \Snd\ \get{\G_a}{i} \} $,
-and $ \G = \g $.
-
-Moreover $ \get{\get{\G_g}{i}}{\Name\ p} $ means $ \ES $
-if $ i $ or $ \Name\ p $ are not defined where appropriate.
-
-Here the first rule has higher precedence than the second one does.
-
-\end{itemize}
-
-The third group of \GP{query} expressions includes the {\av} set manipulation
-facilities:
-
-\begin{itemize}
-
-\item
-The \TT{add} operator adds a given set of attribute groups to the {\av}'s
-of an {\av} set using a union with or without attribute distribution
-according to the \TT{distr} flag.
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule
-{h \in \TT{["distr"]?} \spc a \in \GP{xgroups} \spc
- (\G, \TT{[} ""\ \TT{attr}\ a \TT{]}) \daq \{("", A)\} \spc
- (\G, x) \daq \{(r_1, A_1), \cdots, (r_m, A_m)\}}{}
-{(\G, \TT{add}\ a\ \TT{in}\ x) \daq \{(r_1, A_1 \jolly A), \cdots, (r_m, A_m \jolly A)\}}
-%
-\end{center}
-\begin{center}
-%
-\irule
-{h \in \TT{["distr"]?} \spc i \in \GP{avar} \spc
- (\g, x) \daq \{(r_1, A_1), \cdots, (r_m, A_m)\}}{}
-{(\g, \TT{add}\ i\ \TT{in}\ x) \daq \{(r_1, A_1 \jolly \Snd\ \get{\G_a}{i}), \cdots, (r_m, A_m \jolly \Snd\ \get{\G_a}{i})\}}
-%
-\end{center}
-\end{footnotesize}
-
-Where $ \jolly_{\tt""} = \sor $ and $ \jolly_{\tt"distr"} = \distr $.
-Moreover $ \Snd\ \get{\G_a}{i} = \ES $ if $i$ is not defined.
-
-\item
-The semantics of the \TT{for} operator is given in terms of the {\For} helper
-function:
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule{i \oft \GP{avar} \spc (\G, x_1) \daq (\G_1, S_1) \spc h \in \TT{["sup"|"inf"]}}
-{}{(\G, \TT{for}\ i\ \TT{in}\ x_1\ h\ x_2) \daq \For\ h\ \G_1\ i\ x_2\ S_1} \spc
-\irule{i \oft \GP{avar} \spc x_2 \oft \GP{query}}{}
- {\For\ h\ \G\ i\ x_2\ \ES\ \RM{rewrites to}\ (\G, \ES)}
-%
-\end{center}
-\begin{center}
-%
-\irule{i \oft \GP{avar} \spc ((\G_s, \set{\G_a}{i}{R}, \G_g), x_2) \daq (\G_2, S_2)}
- {}{\For\ h\ \G\ i\ x_2\ (S_1 \sdor \{R\})\ \RM{rewrites to}\
- (\G_2 ,(\Snd\ (\For\ h\ \G_2\ i\ x_2\ S_1)) \jolly_h S_2)}
-%
-\end{center}
-\end{footnotesize}
-
-Here we have $ R \oft T_3 $, $ \G = \g $, $ \jolly_{\tt"sup"} = \sum $ and
-$ \jolly_{\tt"inf"} = \prod $.
-
-$ \dprod $ and $ \prod $ are helper functions describing the two intersection
-operations on {\av} sets: with and without attribute distribution respectively.
-They are dual to $ \dsum $ and $ \sum $. $ \dprod $ does not appear in this
-version of {\MathQL} but was used in the erlier versions
-\cite{Lor02, GS03, Gui03}.
-
-\begin{footnotesize}
-\begin{center} \begin{tabular}{lrll}
-%
-1a &
-$ (S_1 \sdor \{(r, A_1)\}) \prod (S_2 \sdor \{(r, A_2)\}) $ & rewrites to &
-$ (S_1 \prod S_2) \sor \{(r, A_1 \sor A_2)\} $ \\
-1b & $ S_1 \prod S_2 $ & rewrites to & $ \ES $ \\
-2a &
-$ (S_1 \sdor \{(r, A_1)\}) \dprod (S_2 \sdor \{(r, A_2)\}) $ & rewrites to &
-$ (S_1 \dprod S_2) \sor \{(r, A_1 \distr A_2)\} $ \\
-2b & $ S_1 \dprod S_2 $ & rewrites to & $ \ES $
-%
-\end{tabular} \end{center}
-\end{footnotesize}
-
-As for $ \sum $ and $ \dsum $, rules 1a, 2a override rules 1b, 2b respectively.
-
-\item
-The semantics of the \TT{while} operator is given by the rules below:
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule{h \in \TT{["sup"|"inf"]} \spc (\G, x_1) \daq (\G_1, \ES)}{1}
-{(\G, \TT{while}\ x_1\ h\ x_2) \daq (\G_1, \ES)}
-%
-\end{center}
-\begin{center}
-%
-\irule
-{h \in \TT{["sup"|"inf"]} \spc (\G, x_1) \daq (\G_1, S_1) \spc
- (\G_1, x_2) \daq (\G_2, S_2) \spc
- (\G_2, \TT{while}\ x_1\ h\ x_2) \daq (\G_3, S)}{2}
-{(\G, \TT{while}\ x_1\ h\ x_2) \daq (\G_3, S_2 \jolly_h S)}
-%
-\end{center}
-\end{footnotesize}
-
-Again $ \jolly_{\tt"sup"} = \sum $ and $ \jolly_{\tt"inf"} = \prod $.
-Moreover rule 1 takes precedence over rule 2.
-
-\end{itemize}
-
-The forth group of \GP{query} constructions make {\MathQL} an extensible
-language.
-
-\begin{itemize}
-
-\item
-The ``function'' construction invokes an external function returning an {\av}
-set. The function is identified by a \GP{path} and its arguments are a set of
-\GP{path}'s and a set of \GP{query}'s. It is a mistake to invoke a function
-with the wrong number of \GP{path}'s and \GP{query}'s as input (each
-particular function defines these numbers independently).
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule
-{p, p_1, \cdots, p_m \in \GP{path} \spc x_1, \cdots, x_n \in \GP{query}}{}
-{(\G, p\ \{p_1 \TT{,} \cdots \TT{,} p_m\}\ \{x_1 \TT{,} \cdots \TT{,} x_m\})
- \daq \Fun\ p\ [p_1, \cdots, p_m]\ [x_1, \cdots, x_n]\ \G}
-%
-\end{center}
-\end{footnotesize}
-
-where $ \Fun \oft \GP{path} \times (\Listof\ \GP{path}) \times (\Listof\
-\GP{query}) \times K \to T_4 $ is the primitive function performing the
-low level invocation.
-The core language does not include any external function and it is a mistake
-to invoke an undefined function.
-
-\item
-The \TT{gen} construction invokes an external function returning a \GP{query}
-The function is identified by a \GP{path} and its arguments are a set of
-\GP{query}'s. It is a mistake to invoke a function with the wrong number of
-\GP{query}'s as input (each particular function defines this number
-independently).
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule
-{p \in \GP{path} \spc x_1, \cdots, x_n \in \GP{query} \spc
- (\G, \Gen\ p\ [x_1, \cdots, x_n]\ \G) \daq (\G\p, S)}{}
-{(\G, \TT{gen}\ p\ \{x_1 \TT{,} \cdots \TT{,} x_m\}) \daq (\G\p, S)}
-%
-\end{center}
-\end{footnotesize}
-
-where $ \Gen \oft \GP{path} \times (\Listof\ \GP{query}) \times K \to
-\GP{query} $ is the primitive function performing the low level invocation.
-The core language does not include any external function of this kind and it
-is a mistake to invoke an undefined function.
-
-The construction ``\TT{gen} p \TT{in} x'' rewrites to ``\TT{gen} p \{x\}''
-for the user's convenience.
-
-\end{itemize}
-
-\subsubsection {Results}
-
-An \GP{avs} expression ({\ie} the explicit representation of an {\av} set that
-can denote a query result) is evaluated to an {\av} set according to the
-following rules.
-
-\begin{footnotesize}
-\begin{center}
-%
-\irule{x_1, \cdots, x_m \in \GP{av} \spc
- x_1 \dar S_1 \spc \cdots \spc x_m \dar S_m}{}
- {x_1 \TT{;} \cdots \TT{;} x_m \dar S_1 \sum \cdots \sum S_m}
-%
-\end{center}
-\begin{center}
-%
-\irule{q \in \GP{string} \spc g_1, \cdots, g_m \in \GP{group} \spc
- q\ \TT{attr}\ g_1 \dar S_1 \spc \cdots \spc
- q\ \TT{attr}\ g_m \dar S_m}{}
- {q\ \TT{attr}\ g_1 \TT{,} \cdots \TT{,} g_m \dar S_1 \sum \cdots \sum S_m}
-%
-\end{center}
-\begin{center}
-%
-\irule{q \in \GP{string} \spc a_1, \cdots, a_m \in \GP{atr} \spc
- q\ \TT{attr}\ \{ a_1 \} \dar S_1 \spc \cdots \spc
- q\ \TT{attr}\ \{ a_m \} \dar S_m}{}
- {(\G, q\ \TT{attr}\ \{ a_1 \TT{;} \cdots \TT{;} a_m \} \dar S_1 \dsum \cdots \dsum S_m}
-%
-\end{center}
-\begin{center}
-%
-\irule{q, q_1, \cdots, q_m \in \GP{string} \spc p \in \GP{path}}{}
- {q\ \TT{attr}\ \{ p = \{ q_1 \TT{,} \cdots \TT{,} q_m \} \} \dar
- \{(\Unquote\ q, \{ \{ (\Name\ p, \{ \Unquote\ q_1, \cdots, \Unquote\ q_m \}) \} \})\}}
-%
-\end{center}
-\end{footnotesize}
-
-
-\subsection {The basic library}
+\input{mathql_operational_background}
+\input{mathql_operational_core}
+\input{mathql_operational_basic}
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