+++ /dev/null
-\subsection {Sets of attributed values.} \label{AVSets}
-
-The data representation model used by {\MathQL} relies on the notion of
-\emph{set of attributed values} ({\av} set for short) that is, in practice,
-the only data type available in {\MathQL}.4. In this sense {\MathQL}.4 is a
-statically untyped language.%
-\footnote
-{A type system that fits {\MathQL} as an {\RDF}-oriented query language,
-should be driven from the {\RDFS} class system. This may be a future
-improvement.}
-Each {\av} in an {\av} set consists of a string%
-\footnote{When we say \emph{string}, we mean a finite sequence of characters.}
-(that we call the \emph{head string} or \emph{value}) and a (possibly empty)
-multiset of named attributes whose content is a set of strings.
-Attribute names are made of a (possibly empty) list of string components, so
-they can be hierarchically structured.
-Moreover the attributes of a value are partitioned into a set of \emph{groups}
-({\ie} subsets) to improve its structure.
-
-In the above description a \emph{set} is an \emph{unordered} finite
-sequence \emph{without} repetitions whereas a \emph{multiset} is an
-\emph{unordered} finite sequence \emph{with} repetitions.
-
-In the present context repetitions are defined as follows:
-two {\av}'s are repeated if they share the same head string without any
-condition on their attributes, two groups are repeated of they contain the
-same attributes (equal both in name and content), two attributes of a group
-are repeated if they share the same name without any condition on their
-content, and two strings are always compared in a case-sensitive manner.%
-\footnote
-{The Author's experience with {\MathQL} seems to show that the above
-definition of an {\av} set is just the right one among the many alternatives
-that were tried.}
-
-As we said, {\MathQL}.4 uses {\av} sets to represent many kinds of
-information:
-
-\begin{enumerate}
-
-\item
-A pool of {\RDF} triples having a common subject $r$, which in general is a
-{\URI} reference \cite{URI}%
-\footnote
-{A {\URI} \emph {reference} is a {\URI} with an optional fragment identifier.},
-is encoded in a single {\av} placing $r$ in the head string.
-The predicates of the triples are encoded as attribute names and their objects
-are placed in the attributes' contents.
-These contents are structured as multiple strings with the aim of holding the
-objects of repeated predicates.
-Moreover structured attribute names can encode various components of
-structured properties preserving their semantics.
-
-\begin{figure}
-\begin{footnotesize} \begin{verbatim}
-The RDF triples:
- ("protocol", "dc:creator", "Sandro Hawke")
- ("protocol", "dc:creator", "Eric Prud'hommeaux")
- ("protocol", "dc:date", "2002-01-08")
-
-The corresponding attributed value:
- "protocol" attr {/"dc:creator" = {"Sandro Hawke", "Eric Prud'hommeaux"};
- /"dc:date" = "2002-01-08"}
-\end{verbatim} \end{footnotesize}
-\vspace{-1pc}
-\caption{The representation of a pool of {\RDF} triples} \label{AVOne}
-\end{figure}
-
-\figref{AVOne} shows how a set of triples can be coded in an {\av}.
-Note that the word \TT{attr} separates the head string from its attributes,
-braces enclose an attribute group in which attributes are separated by
-semicolons, and an equal sign separates an attribute name from its contents.
-
-In this setting the grouping feature can be used to separate semantically
-different classes of properties associated to a resource (as for instance
-Dublin Core metadata, Euler metadata and user-defined metadata).
-
-\item
-A pool of arbitrarily chosen {\RDF} triples is encoded in an {\av} set
-placing in each {\av} the subset of triples sharing the same head string.
-
-Note that the use of {\av} sets to build query results allows {\MathQL} queries
-to return sets of {\RDF} triples instead of mere sets of resources, in the
-spirit of what is currently done by other {\RDF}-oriented query languages.
-
-If the {\av}'s of an {\av} set share the same attribute names and grouping
-structure, this set can be represented as a table in which each row encodes
-an {\av} and each column is associated to an attribute (except the first one
-which holds the head strings).
-\figref{Table} shows an {\av} set describing the properties of two resources
-``A'' and ``B'' giving its table representation, in which the columns
-corresponding to attributes in the same group are clustered between
-double-line delimiters.%
-\footnote{A table with grouped labelled columns like the one above resembles a
-set of relational database tables.}
-
-\begin{figure}
-\begin{footnotesize} \begin{verbatim}
-"A" attr {/"major" = "1"; /"minor" = "2"},
- {/"first" = "2002-01-01"; /"modified" = "2002-03-01"};
-"B" attr {/"major" = "1"; /"minor" = "7"},
- {/"first" = "2002-02-01"; /"modified" = "2002-04-01"}
-\end{verbatim}
-\begin{center} \begin{tabular}{|c||c|c||c|c||}
-\hline & \textbf{``major''} & \textbf{``minor''} & \textbf{``first''} & \textbf{``modified''} \\
-\hline ``A'' & ``1'' & ``2'' & ``2002-01-01'' & ``2002-03-01'' \\
-\hline ``B'' & ``1'' & ``7'' & ``2002-02-01'' & ``2002-04-01'' \\
-\hline
-\end{tabular} \end{center} \end{footnotesize}
-\caption{A set of attributed values displayed as a table} \label{Table}
-\end{figure}
-
-The above example gives a spatial idea of the geometry of an {\av} set ({\ie}
-a query result) which fits in 4 dimensions: namely we can extend independently
-the set of the head strings (dimension 1), the attributes in each group
-(dimension 2), the groups in each {\av} (dimension 3) and the contents of each
-attribute (dimension 4).
-The metadata defined in the table of \figref{Table} will be used in subsequent
-examples.
-For this purpose assume that ``first'' and ``modified'' are the components
-of a structured property ``date'' available for the resources ``A'' and ``B''.
-
-\item
-The value of an {\RDF} property is encoded in an {\av} distinguishing three
-cases:
-
-\begin{itemize}
-
-\item
-If the property is unstructured, its value is placed in the {\av} head
-string and no attributes are defined.
-
-\item
-If the property is structured and its value has a main component%
-\footnote{Which is set by the \emph{rdf:value} property or defined by a
-specific application.},
-the content of this component is placed in the {\av} head string and the
-other components are stored in the {\av} attributes as in the case 1.
-
-\item
-For the value of a structured property without a main component, the head
-string is empty and the components are stored in the attributes.
-
-\end{itemize}
-
-\begin{figure}
-\begin{footnotesize} \begin{verbatim}
-First example, one instance:
- "" attr {/"major" = "1"; /"minor" = "2"} no main component
- "1" attr {/"minor" = "2"} main component is "major"
- "2" attr {/"major" = "1"} main component is "minor"
-
-Second example: two separate instances:
- "" attr {/"major" = "1"; /"minor" = "2"},
- {/"major" = "1"; /"minor" = "7"} no main component
- "1" attr {/"minor" = "2"}, {/"minor" = "7"} main component is "major"
-
-Third example: two mixed instances:
- "" attr {/"major" = "3", "6"; /"minor" = {"4", "9"}} no main component
-\end{verbatim} \end{footnotesize}
-\vspace{-1pc}
-\caption{The representation of the structured value of a property}
-\label{AVTwo}
-\end{figure}
-
-\figref{AVTwo} (first example) shows three possible ways of representing in
-{\av}'s an instance of a structured property ``id'' whose value has two
-fields ({\ie} properties) ``major'' and ``minor''.
-In this instance, ``major'' is set to ``1'' and ``minor'' is set to ``2''.
-The representations depend on which component of ``id'' is chosen as the
-main component (none, ``major'' or ``minor'' respectively).
-Several structured property values sharing a common main component can be
-encodes in a single {\av} exploiting the grouping facility: in this case the
-attributes of every instance are enclosed in separate groups.
-\figref{AVTwo} (second example) shows the representations of two instances of
-``id'': the former and a new one for which ``major'' is ``1'' and ``minor'' is
-``7''.
-
-Note that if the attributes of the two groups are encoded in a single group,
-the notion of which components belong to the same property value can not be
-recovered in the general case because the values of an attribute form a set
-and thus are unordered.
-As an example think of two instances of ``id'' encoded as in \figref{AVTwo}
-(third example).
-
-\item
-A natural number is stored, using its decimal representation, in the head
-string of a single {\av} with no attributes.
-
-\item
-The boolean value \emph{false} is stored as an empty {\av} set, whereas
-an inhabited {\av} set may be interpreted as the boolean value \emph{true}.
-The default representation of \emph{true} is a single {\av} with an empty
-head string and no attributes.
-
-\end{enumerate}
-
-{\MathQL} defines five core binary operations on {\av} sets: two unions, two
-intersections and a difference. The first four are defined in terms of an
-operation, that we call \emph{addition}, involving two {\av}'s with the same
-head string.
-The result is an {\av} with the same head string of the operands but there are
-two ways to compose the attribute groups:
-
-\begin{itemize}
-
-\item
-with the \emph{set-theoretic} addition, the set of attribute groups in the
-resulting {\av} is the set-theoretic union of the sets of attribute groups in
-the operands;
-
-\item
-with the \emph{distributive} addition, the set of attribute groups in the
-resulting {\av} is the ``Cartesian product'' of the sets of attribute groups
-in the two operands.
-Here an element of the ``Cartesian product'' is not a pair of groups but it is
-the set-theoretic union of these groups where the contents of homonymous
-attributes are clustered together using set-theoretic unions.
-
-\end{itemize}
-
-\figref{Addition} shows an example of the two kinds of addition.
-
-\begin{figure}
-\begin{footnotesize} \begin{verbatim}
-Attributed values used as operands for the addition:
- "1" attr {/"A" = "a"}, {/"B" = "b1"}
- "1" attr {/"A" = "a"}, {/"B" = "b2"}
-
-Set-theoretic addition:
-" 1" attr {/"A" = "a"}, {/"B" = "b1"}, {/"B" = "b2"}
-
-Distributive addition:
- "1" attr {/"A" = "a"}, {/"A" = "a"; /"B" = "b2"},
- {/"B" = "b1"; /"A" = "a"}, {/"B" = {"b1", "b2"}}
-\end{verbatim} \end{footnotesize}
-\vspace{-1pc}
-\caption{The addition of attributed values}
-\label{Addition}
-\end{figure}
-
-Now we can discuss the five operations between {\av} sets:
-
-\begin{itemize}
-
-\item
-The two unions corresponds to the set-theoretic union of their operand where
-the {\av}'s sharing the head string are added either set-theoretically or
-distributively as explained above (thus we have a set-theoretic union and a
-distributive union in the two cases). In this context the empty {\av} set
-plays the role of the neutral element.
-These operations play a central role {\MathQL} architecture and allow to
-compose the attributes of the operands preserving their group structure.
-
-\item
-The two intersections are the dual of the above unions: they contain the
-{\av}'s whose head string appears in each argument where the {\av}'s sharing
-the head string are added either set-theoretically or distributively as before.
-
-The distributive intersection has the double benefit of filtering the
-common values of the given {\av} sets, and of merging their attribute groups
-in every possible way. This feature enables the possibility of performing
-additional filtering operations checking the content of the merged groups.
-
-\item
-The difference of two {\av} sets contains the {\av}'s of the first
-argument whose head string does not appear in the second argument.
-
-\end{itemize}
-
-\figref{Binary} shows how the above operations work in a simple example.
-
-\begin{figure}
-\begin{footnotesize} \begin{verbatim}
-Sets of attributed values used as operands for the operations:
- "1" attr {/"A" = "a"}; "2" attr {/"B" = "b1"}
- "2" attr {/"B" = "b2"}
-
-Set-theoretic union:
- "1" attr {/"A" = "a"}; "2" attr {/"B" = "b1"}, {/"B" = "b2"}
-
-Distributive union:
- "1" attr {/"A" = "a"}; "2" attr {/"B" = {"b1", "b2"}}
-
-Set-theoretic intersection:
- "2" attr {/"B" = "b1"}, {/"B" = "b2"}
-
-Distributive intersection:
- "2" attr {/"B" = {"b1", "b2"}}
-
-Difference:
- "1" attr {/"A" = "a"}
-\end{verbatim} \end{footnotesize}
-\vspace{-1pc}
-\caption{The binary operations on sets of attributed values}
-\label{Binary}
-\end{figure}
projects / helm.git / blobdiff