\documentclass[11pt,twoside]{article}

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\authorfoot{E. Franconi, I. Horrocks}
\shorttitle{DLs with n-ary Relations}
\title{Description Logics with n-ary Relations}
\author{Enrico Franconi and Ian Horrocks\\
   Dept. of Computer Science, Univ. of Manchester \\
  Manchester M13 9PL, UK \\
  ${\tt \{{\tt franconi}\mid{\tt horrocks}\}@cs.man.ac.uk}$\\
  ${\tt http://www.cs.man.ac.uk/\{{\tt \sim\!franconi}\mid{\tt \sim\!horrocks}\}}$
}
\date{}

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\newcommand{\DLR}   {\mbox{${\cal DLR}$}}
 
\newcommand{\Ifact}{i\Fact}%                I-FaCT
\newcommand{\TopC}{\ensuremath{\top_\mathcal{C}}}
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\begin{document}

\maketitle

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In this abstract we will briefly mention the research work which has
been done at the University of Manchester on recent advances in
Description Logics (DLs), namely the \DLR\ generalisation of DLs
towards n-ary relations. \DLR\ was first introduced
by~\cite{lenzerini:kr-98} as a means for encoding expressive semantic
data models for information systems such as extended Entity
Relationship.  It turns out that \DLR\ elegantly generalises standard
DLs, by allowing formulae referring to n-ary relations, not just to
binary relations (i.e., roles). Still, the flavour of the obtained
language is a modal logic, in the sense that formulae state properties
holding in classes of worlds, determined by means of possibly complex
paths constructed from n-ary relations. As a special case of the
language, we again obtain a standard DL or a standard normal modal
logic.
%
We present here the first order fragment of the language introduced
in~\cite{lenzerini:kr-98}. We have chosen to limit the expressivity to
first order since we are looking for a language implementable with the
current technology.  In particular, we refer to the current academic
implementations of expressive DLs, namely the systems
\Fact~\cite{Horrocks98c} and \Ifact (\Fact extended with inverse
relations).  It has been recently demonstrated~\cite{Horrocks98i} that
the logic we are considering here allows for the implementation of
sound and complete reasoning algorithms that behave quite well both in
realistic applications and systematic tests.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The semantics of \DLR\ rely on a model theoretic {\em interpretation\/}
$\I = (\Delta^\I,\cdot^\I)$, consisting of a set $\Delta^\I$, called
the {\em domain\/}, and an interpretation function $\cdot^\I$ that
assigns to each concept $C$ a subset $C\ifunc$ of \deltai and to each
relationship \role{R} of arity $n$ a subset $\role{R}\ifunc$ of
$(\deltai)^n$, such that the conditions in column~2 of
Table~\ref{tab:concepts}{} are satisfied (where $A$ is an atomic
concept, \role{P} is an atomic relationship, $C$ and $D$ are concepts,
$R$ and $S$ are relationships, $\$i$ denotes the $i$th component of an
$n$-ary relationship, and $k$ is a non-negative
integer).


\begin{table}%[t!]
\begin{center}
{\setlength{\arraycolsep}{0em}
\begin{tabular}{|l|l|l|}
\hline
\DLR\ syntax & \DLR\ semantics & DL encoding\\
\hline
%
\TopC &
$\TopC\ifunc \subseteq \deltai$ &
$\TopC \quad (\TopC \sqsubseteq \top)$ \\
%
$A$ &
$A\ifunc \subseteq \TopC\ifunc$ &
$A \quad (A \sqsubseteq \TopC)$ \\
%
$\neg C$ &
$\TopC\ifunc \setminus C\ifunc$ &
$\neg C$ \\
%
$C \sqcap D$ &
$C\ifunc \cap D\ifunc$ &
$C \sqcap D$ \\
%
$\exists[\$i]\role{R}$ &
$\{d \in \TopC\ifunc \mid \exists\langle d_1,\ldots,d_n\rangle 
\in \role{R} . d_i = d\} \leqslant k\}$ &
\some{U_i^-}{C_R} \\
%
$\leqslant k [\$i]\role{R}$ &
$\{d \in \TopC\ifunc \mid \sharp\{\langle d_1,\ldots,d_n\rangle 
\in \role{R} \mid d_i = d\} \leqslant k\}$ &
\atmostq{k}{U_i^-}{C_R} \\
%
\hline
%
\Topn &
$\Topn\ifunc \subseteq (\deltai)^n$ &
$\Topn \quad (\Topn \sqsubseteq \top)$ \\
%
\role{P} &
$\role{P}\ifunc \subseteq \Topn\ifunc$ &
$C_P \quad (C_P \sqsubseteq \Topn)$ \\
%
$\neg\role{R}$ &
$\Topn\ifunc \setminus \role{R}\ifunc$ &
$\neg C_R$ \\
%
$\role{R} \sqcap \role{S}$ &
$\role{R}\ifunc \cap \role{S}\ifunc$ &
$C_R \sqcap C_S$ \\
%
$\$i_n:C$ &
$\{\langle d_1,\ldots,d_n\rangle \in \Topn\ifunc \mid d_i \in C\ifunc\}$ &
$\Topn \sqcap \all{U_i}{C}$ \\
%
\hline
\end{tabular}}
\end{center}
\caption{\DLR\ concepts and relationships, their semantics and 
  their DL encodings}\label{tab:concepts}
\end{table}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It turns out that first order \DLR\ theories can be encoded into an
expressive DL theory, equipped with generalised axioms, inverse
relationships, and graded modalities (i.e., qualified number
restrictions)~\cite{lenzerini:kr-98}. A \DLR\ theory consists of a
finite set of axioms of the form $C \sqsubseteq D$ and $R \sqsubseteq
S$, where $C$ and $D$ are concepts, and $R$ and $S$ are relationships
of equal arity. Such a theory can be encoded in an equisatisfiable DL
theory using the encodings described in column~3 of
Figure~\ref{tab:concepts} (where $C_R$ is a concept corresponding to
the \DLR\ relationship $R$, and $U_i$ is a functional role
representing the $i$th component of a \DLR\ relationship).  Note that
\DLR\ relations are reified as DL concepts, and the components of the
relationship represented as functional roles.  While the encoding
obviously relies on the ability of the DL to reason with inverse
roles, another less obvious consequence is that this DL does not have
the finite model property. New techniques have been studied which
extend the algorithm of the \Fact system with blocking strategies that
allow complete reasoning with such an expressive
DL~\cite{Horrocks98j}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The \DLR\ Description Logic was designed such that it is able to
capture database schemata expressed in the most interesting Semantic
Data Models and Object-Oriented Data Models
\cite{lenzerini:kr-98,franconi:sattler:aiia-99}. For that purpose, the
first order fragment suffices.  The most common conceptual data model
for database and information system design is the Entity-Relationship
(ER) semantic data model.  \cite{lenzerini:kr-98} refers to an
extended version of the basic ER model, called EER, including
taxonomic relationships, arbitrary boolean constructs, and entity
definitions by means of both necessary and sufficient conditions and,
more generally, by means of generalised axioms. It is shown how a
schema expressed in an EER conceptual data model can be expressed in a
\DLR\ theory -- whose models correspond with legal database states of
the EER schemata -- allowing for reasoning services such as
satisfiability of a schema or the computation of a logically implied
formula.

Various experiments have been done in the context of the European
ESPRIT-4 Long Term Research project ``Foundations of Data Warehouse
Quality'' (DWQ), using EER schemata as a source for testing the
efficiency of reasoning with \DLR\ theories in realistic domains.
Previous approaches~\cite{Calvanese98a} based on the translation of
first order \DLR\ theories into the basic \alc DL (in correspondence
with multi-modal \mbox{$\cal K$}) introduce very large numbers of
axioms, and have proved impractical.  Using \Ifact and a slightly
simplified version of the above encoding, promising initial results
have been obtained with a \DLR\ theory representing (fragments of)
extended real world database schemata.  Work is in progress to extend
the algorithm implemented in \Ifact in order to allow a complete
encoding of \DLR~\cite{Horrocks99b}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bibliography{aisb99-bib}

\begin{thebibliography}{CDR98}

\bibitem[CDR98]{Calvanese98a}
D.~Calvanese, G.~{De~Giacomo}, and R.~Rosati.
\newblock A note on encoding inverse roles and functional restrictions in
  $\mathcal{ALC}$ knowledge bases.
\newblock In E.~Franconi, G.~{De~Giacomo}, R.~M. MacGregor, W.~Nutt, C.~A.
  Welty, and F.~Sebastiani, editors, {\em Collected Papers from the
  International Description Logics Workshop (DL'98)}, pages 69--71. CEUR, May
  1998.

\bibitem[DL99]{lenzerini:kr-98}
G.~{De Giacomo} and M.~Lenzerini.
\newblock Description logic framework for information integration.
\newblock In {\em Proceedings of the 6th International Conference on the
  Principles of Knowledge Representation and Reasoning (KR-98)}, pages 2--13.
  Morgan Kaufmann, 1999.

\bibitem[FS99]{franconi:sattler:aiia-99}
Enrico Franconi and Ulrike Sattler.
\newblock A data warehouse conceptual data model for multidimensional
  aggregation: a preliminary report.
\newblock {\em {AI*IA} Italian Association for Artificial Intelligence
  journal}, 1999.
\newblock To appear.

\bibitem[Hor98]{Horrocks98c}
I.~Horrocks.
\newblock Using an expressive description logic: {FaCT} or fiction?
\newblock In A.~G. Cohn, L.~Schubert, and S.~C. Shapiro, editors, {\em
  Principles of Knowledge Representation and Reasoning: Proceedings of the
  Sixth International Conference (KR'98)}, pages 636--647. Morgan Kaufmann
  Publishers, San Francisco, CA, June 1998.

\bibitem[HPS99]{Horrocks98i}
I.~Horrocks and P.~F. Patel-Schneider.
\newblock Optimising description logic subsumption.
\newblock {\em Journal of Logic and Computation}, 1999.
\newblock To appear.

\bibitem[HS99]{Horrocks98j}
I.~Horrocks and U.~Sattler.
\newblock A description logic with transitive and inverse roles and role
  hierarchies.
\newblock {\em Journal of Logic and Computation}, 1999.
\newblock To appear.

\bibitem[HST99]{Horrocks99b}
I.~Horrocks, U.~Sattler, and S.~Tobies.
\newblock Practical reasoning for expressive description logics.
\newblock Submitted, 1999.

\end{thebibliography}

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\end{document}
