deletions | additions       diff --git a/the dilated triple model.tex b/the dilated triple model.tex  index 26f99dd..12e0e5d 100644  --- a/the dilated triple model.tex  +++ b/the dilated triple model.tex 
...
Definition 1. A Dilated Triple  Given a set of triples $R$ and a triple $\tau \in R$, a dilation of $\tau$ is a set of triples $T_\tau \subset R$ such that $\tau \in T_\tau$.  The dilated form of $\tau \in R$ is $T_\tau$. Informally, $T_\tau$ servers to elaborate the meaning of $\tau$. Formally, $T_\tau$ is a graph that at minimum contains only $\tau$ and at maximum contains all triples in $R$. The set of all non-$\tau$ triples in $T_\tau$ (i.e.~$T_\tau \setminus \tau$) are called \textit{supplementary triples} as they serve to contextualize, or supplement, the meaning of $\tau$. Finally, it is worth noting that every supplemental triple in $T_\tau$ has an associated dilated form, so that $T_\tau$ can be considered a set of nested sets.\footnote{The set of all dilated triples forms a \textit{dilated graph} denoted $\mca{T} = \bigcup_{\tau \in R} \{ T_\tau \}$.} An instance of $\tau$, its subject $s$, predicate $p$, object $o$, and its dilated form $T_\tau$, are diagrammed in Figure \ref{fig:dilated-triple}. \ref{fig:dilatedtriple}.  A dilated triple can be conveniently represented in RDF using a named graph \cite{named:carroll2005}. Statements using the named graph construct are not triples, but instead, are quads with the fourth component being denoted by a URI or blank node. Formally, $\tau = (s,p,o,g)$ and $g \in U \cup B$. The fourth component is considered the "graph" in which the triple is contained. Thus, multiple quads with the same fourth element are considered different triples in the same graph. Named graphs were developed as a more compact (in terms of space) way to reify a triple. The reification of a triple was originally presented in the specification of RDF with the \texttt{rdf:Statement} construct \cite{rdfcon:klyne2004}. RDF reification has historically been used to add specific metadata to a triple, such as provenance, pedigree, privacy, and copyright information. In this article, the purpose of reifying a triple is to supplement its meaning with those of additional triples. While it is possible to make additional statements about the dilated triple (i.e.~the named graph component $g$), the motivation behind the dilated triple is to encapsulate many triples within a single graph, not to make statements about the graph \textit{per se}.