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\newcommand{\ALE}{{\cal ALE}}
\newcommand{\Dom}[1]{{\sf Dom(}#1{\sf )}}

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\semtitle{Description Logics}
\semsubject{Knowledge Bases in Description Logics}
\semauthor{Enrico Franconi}
\sememail{franconi@cs.man.ac.uk}
\semwebpage{http://www.cs.man.ac.uk/\~{}franconi}
\seminstitution{Department of Computer Science, University of Manchester}

\begin{document}

\makesemtitle

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

\begin{slide}{Understanding Knowledge Bases}

  \centerline{\fbox{\mbox{$\Sigma = \langle {\sf TBox},{\sf Abox} \rangle$}}}

\vspace{1em}

  \begin{itemize}

  \item {\bf Terminological Axioms:} $C\IsSubs D$

  \item {\bf Assertional Axioms:} $C(a), R(a,b)$

  \end{itemize}

\vspace{1em}

\begin{itemize}
\item An interpretation $\I = (\dom, \Int\cdot)$ {\em satisfies\/} the
  statement $C~ \IsSubs~ D$ if $\Int C \subseteq \Int D$.
\item $\I$ satisfies $C(a)$ if $\Int a \in \Int{C}$.
\item $\I$ satisfies $R(a,b)$ if $(\Int a,\Int b) \in \Int{R}$.
\end{itemize}

\vspace{1em}

An interpretation $\I = (\dom, \Int\cdot)$ is said to be a {\em model\/} of
$\Sigma$ if every axiom of $\Sigma$ is satisfied by $\I$. $\Sigma$ is said
to be {\em satisfiable\/} if it admits a model.

\end{slide}

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

\begin{slide}{TBox statements}

\noindent
$
\begin{array}{l@{\hspace{3em}}c@{\hspace{2em}}l}
\mbox{(1)} & A \IsSubs C  & \mbox{Primitive concept definition} \\
\mbox{(2)} & A \doteq C  & \mbox{Concept definition} \\
\mbox{(3)} & C \IsSubs D  & \mbox{Concept inclusion} \\
\mbox{(4)} & C \doteq D  & \mbox{Concept equation} \\
\end{array}
$

\end{slide}

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

\begin{slide}{Acyclic simple TBox}

{\it Simple TBox:} 
$
\begin{array}{l@{\hspace{3em}}c@{\hspace{2em}}l}
\mbox{(1)} & A \IsSubs C  & \mbox{Primitive concept definition} \\
\mbox{(2)} & A \doteq C  & \mbox{Concept definition} \\
\end{array}
$

\vspace{2em}

{\it Acyclic simple TBox:} well-founded definitions.

\vspace{1em}

A concept name $A$ {\it directly uses} a concept name $B$ in a TBox $\Sigma$
iff the definition of $A$ mentions $B$. A concept name $A$ {\it uses} a
concept name $B_n$ iff there is a chain of concept names $\langle A, B_1,
\ldots, B_n\rangle$ such that $B_i$ directly uses $B_{i+1}$. A TBox is {\em
  acyclic} iff no concept name uses itself.

\end{slide}

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

\begin{slide}{Acyclic simple TBox}

\fbox{\parbox[t]{\textwidth}{Subsumtpion in acyclic simple TBoxes
    ($\Sigma\models C\IsSubs D$) can be reduced in subsumtpion in an
    empty TBox ($\models \widehat C\IsSubs \widehat D$).}}

\vspace{1em}

In order to get $\widehat C$ (and $\widehat D$):

\vspace{1ex}

1) Transform the TBox $\Sigma$ into a new TBox $\Sigma'$, by replacing every
primitive concept definition in $\Sigma$ of the form $A \IsSubs C$ with
a concept definition $A \doteq C \And A^*$ -- where $A^*$ is a freshly new
generated concept name (called {\it primitive component} of $A$). Now
$\Sigma'$ contains only (acyclic) concept definitions.

\vspace{1ex}

2) Iteratively substitute every occurrence of any defined concept name in
$C$ (and $D$) by the corresponding definition in $\Sigma'$. Since $\Sigma'$
is still acyclic, the process terminates in a finite number of iterations.
This process is called {\em unfolding} or {\em expansion}.

\end{slide}

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

\begin{slide}{Theorems}

\begin{itemize}
\item For each interpretation of $\Sigma$ there exists an
  interpretation of $\Sigma'$ (and viceversa) such that $\Int C =
  C^{\I'}$ for each concept name $C$ in $\Sigma$. \vspace{1ex}

  \begin{minipage}[c]{.4\textwidth}
      \includegraphics[width=.9\textwidth]{primitive.eps}
  \end{minipage}\tab
  \begin{minipage}[c]{.4\textwidth}
    $A \IsSubs C \smalltab\leadsto\smalltab A \doteq C \And A^*$
  \end{minipage}  

  $A^*$ denotes the {\em unexpressed} part of meaning implicitly contained
  in the primitive concept definition.

\item $\Sigma\models C\IsSubs D$ \tab iff \tab $\Sigma'\models
  C\IsSubs D$
\item $\Sigma'\models C\IsSubs D$ \tab iff \tab 
  $\models \widehat C\IsSubs \widehat D$
\end{itemize}

\end{slide}


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

\begin{slide}{Necessary and Sufficient conditions}

\begin{itemize}

\item A primitive concept definition $A \IsSubs C$ states a necessary
  but not sufficient condition for membership in the class $A$. Having the
  property $C$ is necessary for an object in order to be in the class $A$;
  however, this condition alone is not sufficient in order to conclude that
  the object is in the class $A$.

\vspace{1em}

\item A concept definition $A \doteq C$ states necessary and sufficient
  condition for membership in the class $A$. Having the property $C$ is
  necessary for an object in order to be in the class $A$; moreover, this
  condition alone is sufficient in order to conclude that the object is in
  the class $A$.

\end{itemize}

\end{slide}

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

\begin{slide}{Necessary and Sufficient conditions}

When transforming primitive concept definitions into concept definitions we
get necessary and sufficient conditions for membership in the primitive
class $A$. However, the condition of being in the primitive component $A^*$
can never be satisfied, since the concept name $A^*$ can never be referred
to by any other concept.

\vspace{2em}

\fbox{\parbox[t]{\textwidth}{A concept is subsumed by a primitively defined
    concept if and only if it refers to its name in its (unfolded)
    definition.}}

\end{slide}

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

\begin{slide}{Inheritance}

Unfolding realizes what is usually called {\it inheritance} in
Object-Oriented frameworks.

%\vspace{1em}

${\tt Person \doteq \some{NAME}{String} \And \some{ADDRESS}{String}}$

${\tt Parent \doteq Person \And \some{CHILD}{Person}}$

%\vspace{1em}

${\tt \widehat{Parent} \doteq \parbox[t]{30em}{$\tt\some{NAME}{String} \And
    \some{ADDRESS}{String} \And \\ \some{CHILD}{(\some{NAME}{String} \And
      \some{ADDRESS}{String})}$}}$

\vspace{1em}

${\tt Female \doteq \Not Male}$

${\tt Man \doteq Person \And \all{SEX}{Male}}$

${\tt Woman \doteq Person \And \all{SEX}{Female}}$

${\tt Transexual \doteq Man \And Woman}$

%\vspace{1em}

${\tt \widehat{Transexual} \doteq 
  \parbox[t]{30em}{$\tt
    \some{NAME}{String} \And \some{ADDRESS}{String} \And
    \all{SEX}{\bot}$}}$

\end{slide}

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

\begin{slide}{Inheritance in O-O}

Problems in O-O frameworks: overriding strategies for multiple inheritance.

\vspace{3em}

      \includegraphics[width=.8\textwidth]{fig10.09.ps}
%\epsfxsize=.8\textwidth
%\epsfbox{../norvig-fig/fig10.09.ps}


\end{slide}

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

\begin{slide}{Complexity of Unfolding}

  $C_1 \doteq \all{R_1}{C_0}\And \all{R_2}{C_0}\And \ldots\And
  \all{R_m}{C_0}$

  $C_2 \doteq \all{R_1}{C_1}\And \all{R_2}{C_1}\And \ldots\And
  \all{R_m}{C_1}$

  $\cdots$

  $C_n \doteq \all{R_1}{C_{n-1}}\And \all{R_2}{C_{n-1}}\And \ldots\And
  \all{R_m}{C_{n-1}}$

\vspace{1em}

\begin{itemize}
\item The size of the TBox is ${\cal O}(n\times m)$.
\item The size of the unfolded concept $\widehat{C_n}$ is ${\cal O}(m^n)$.
\item The complexity of the subsumtpion problem in \FLm\ with empty TBox
  ($\models C\IsSubs D$) is P.
\item The complexity of the subsumtpion problem in \FLm\ with an acyclic
  simple TBox ($\Sigma\models C\IsSubs D$) is co-NP-complete.
\end{itemize}

\end{slide}

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

\begin{slide}{Efficiency of Subsumption in practice}

\vspace{-1em}

  The exponential worst case is unlikely to occurr in real knowledge bases.

\begin{itemize}

\item Let $n$ be the {\em depth} of a TBox, i.e., the max number of
  iterations while unfolding every concept definition.

\item Let $m$ be the size of the largest definition.

\item Let $s$ be the size of the TBox, i.e., $m$ times the number of concept
  definitions.

\end{itemize}

The size of an unfolded concept is ${\cal O}(m^n)$.

\vspace{1ex}

\fbox{\parbox[t]{\textwidth}{
If $n\leq\mbox{log}_m s$ the size of an unfolded concept becomes polynomial
${\cal O}(s)$ with respect to the size of the TBox.}}

\vspace{1ex}

This is a reasonable assumption, since the depth of concept definitions is
usually much smaller than the size of the knowledge base. This is why
systems behave well in practice.

\end{slide}

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

\begin{slide}{Definitions}

\begin{itemize}

\item Definitions are intended to provide an exact account for the concept
  name being defined.

\item Given an initial interpretation of the primitive concept names there
  exists a unique way determine the interpretation of defined concept names;
  indeed, that's why they are called {\em definitions}.

\item This justifies the correctness of unfolding: we can always replace a
  concept name with its definition, since it doesn't add anything to the
  theory.

\item However, if the (simple) TBox is cyclic, this is no more true.

\end{itemize}

\end{slide}

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

\begin{slide}{Example of recursive definition}

${\tt Bird \doteq Animal \And \all{SKIN}{Feather}}$

\vspace{1ex}
${\tt\dom=\{tweety,goofy,fea1, fur1\}}$\\
${\tt\Int{Animal}=\{tweety,goofy\}}$\\
${\tt\Int{Feather}=\{fea1\}}$\\
${\tt\Int{SKIN}=\{\langle tweety,fea1\rangle,\langle goofy,fur1\rangle\}}$\\
${\tt\Longrightarrow\smalltab\Int{Bird}=\{tweety\}}$ 

\vspace{1ex}

${\tt Quiet{-}Person \doteq Person \And \all{FRIEND}{Quiet{-}Person}}$

\vspace{1ex}
${\tt\dom=\{john,sue,andrea,bill\}}$ \\
${\tt\Int{Person}=\{john,sue,andrea,bill\}}$\\ 
${\tt\Int{FRIEND}=\{\langle john,sue\rangle,\langle
  andrea,bill\rangle,\langle bill,bill\rangle\}}$\\
${\tt\Longrightarrow\smalltab\Int{Quiet{-}Person}=\{john,sue\}}$ 
${\tt\Longrightarrow\smalltab\Int{Quiet{-}Person}=\{john,sue,andrea,bill\}}$ 

\end{slide}

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

\begin{slide}{Descriptive semantics}

(It is the one we have introduced before.)

\begin{itemize}

\item An interpretation $\I = (\dom, \Int\cdot)$ {\em satisfies\/} the
  concept definition $A\doteq C$ iff $\Int A = \Int C$.

\vspace{2em}

\item The definition is a {\em constraint} stating a restriction on the
  valid models of the knowledge base, and in particular on the possible
  interpretations of $A$, where $A$ is no better specified.

\vspace{2em}

\item It allows both models of the previous cyclic definition.

\end{itemize}

\end{slide}

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

\begin{slide}{Fixpoint Semantics}

We associate to a cyclic concept definition an operator $F:2^{\dom} \mapsto
2^{\dom}$, such that the interpretation of $A$ correspond to the fixpoints
of the operator $F$.

\vspace{3em}

Thus, we associate the equation $$A= F(A)$$ to a cyclic concept
definition of the type $$A\doteq C$$ where $C$ mentions $A$.


\end{slide}

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

\begin{slide}{Least Fixpoint Semantics}

The {\sc lfs} interprets a recursive definition $$A= F(A)$$ by assigning to
$A$ the {\em smallest} possible extension in each interpretation ${\cal I}$
-- if it exists~-- among those satisfying $$\Int A= \Int{F(A)}$$ i.e., the
least fixpoint of the corresponding operator.

\vspace{3em}

If the operator is monotonic, then the equation above singles out a {\em
  unique} interpretation (subset of $\dom$), hence it {\em defines} the
concept $A$.


\end{slide}

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

\begin{slide}{Example}


${\tt Quiet{-}Person \doteq Person \And \all{FRIEND}{Quiet{-}Person}}$

\vspace{2em}

$F~=~\lambda A\per\{\parbox[t]{\textwidth}
{$\tt x\in\dom\mid \\ \Int{Person}(x)\wedge\forall
y\per\Int{FRIEND}(x,y)\rightarrow \mbox{$A$}(y) \}$}$

\vspace{1em}

$A=F(A)$

\vspace{2em}

${\tt\dom=\{john,sue,andrea,bill\}}$ \\
${\tt\Int{Person}=\{john,sue,andrea,bill\}}$ \\
${\tt\Int{FRIEND}=\{\langle john,sue\rangle,\langle
  andrea,bill\rangle,\langle bill,bill\rangle\}}$

\vspace{1em}

${\tt\Longrightarrow\smalltab\Int{Quiet{-}Person}=\{john,sue\}}$ 

\end{slide}

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

\begin{slide}{Problems}

${\tt Human \doteq Mammal \And \usome {PARENT} \And \all{PARENT}{Human}}$

${\tt Horse \doteq Mammal \And \usome {PARENT} \And \all{PARENT}{Horse}}$

\vspace{2em}

Under the fixpoint semantics $${\tt Human \equiv Horse}$$ i.e., in any
interpretation ${\cal I}$ satisfying the above definitions $${\tt
  \Int{Human} = \Int{Horse}}$$

\end{slide}

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

\begin{slide}{Inductive definitions}

\vspace{-1ex}

\begin{itemize}
\item An {\tt Empty{-}List} is a {\tt List}.
\item A {\tt Node}, that has exactly one {\tt SUCCESSOR} that is a {\tt
    List}, is a {\tt List}.
\item Nothing else is a {\tt LIST}.
\end{itemize}

${\tt Node \doteq \Not Empty{-}List}$\\
${\tt List \doteq \parbox[t]{\textwidth}{$\tt
    Empty{-}List \Or (Node \And \atmost 1 {SUCCESSOR} \And
    \some {SUCCESSOR} {List}) $}}$

\vspace{1ex}

${\tt\dom=\{a,b,nil\}}$ \\
${\tt\Int{Node}=\{a,b\}}$ \\
${\tt\Int{Empty{-}List}=\{nil\}}$ \\
${\tt\Int{SUCCESSOR}=\{\langle a,nil\rangle,\langle b,b\rangle\}}$


\begin{tabular}{ll}

With descriptive semantics: &
${\tt\Int{List}=\{a,b,nil\}}$ \\
%
With least fixpoint semantics: &
${\tt\Int{List}=\{a,nil\}}$ 
\end{tabular}

\end{slide}

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

\begin{slide}{Inductive definitions}

\begin{itemize}
\item Compare with Logic Programming, where inductive definitions come for
  free.

\item Descriptive semantics is expressible in FOL.

\item Least fixpoint semantics (and inductive definitions) go beyond First
  Order.
\end{itemize}

\end{slide}

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

\begin{slide}{Free TBox}

$
\begin{array}{l@{\hspace{3em}}c@{\hspace{2em}}l}
\mbox{(3)} & C \IsSubs D  & \mbox{Concept inclusion} \\
\mbox{(4)} & C \doteq D  & \mbox{Concept equation} \\
\end{array}
$

\vspace{1ex}

(There is no syntactic constraint on the left hand side of the axiom).

\vspace{1em}

Concept inclusions make sense only with descriptive semantics -- we
will ignore here the extensions of Descritpion Logics where it is
possible to specify explicitly the semantics to be given to a
knowledge base.

\vspace{1em}

{\em Theorem:}

Descritpion logics with simple TBoxes (with general concept
definitions \psframebox*[fillcolor=lightgray]{$A\doteq D$}) and free
TBoxes (with concept inclusions \psframebox*[fillcolor=lightgray]{$C
  \IsSubs D$}) have the same expressive power.

\end{slide}

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

\begin{slide}{Simple TBoxes and Free TBoxes}
  
  Satisfiability in a knowledge base $\Sigma$ with a free TBox can be
  reduced into satisfiability in a knowledge base $\Sigma'$ with a
  simple TBox.

\vspace{1em}

$C_i\doteq D_i \smalltab \leadsto \tab A_i\IsSubs D_i ,\ 
C_i\IsSubs A_i$

\vspace{1em}

$C_j \IsSubs D_j \smalltab \leadsto$
\tab\parbox[t]{.7\textwidth}{%
$A\doteq
\parbox[t]{.9\textwidth}{$
(\Not C_1\Or D_1)\And\cdots\And(\Not C_n\Or D_n)\ \And\\
A^*\And\all{R_1}{A}\And\cdots\And\all{R_m}{A}$}$}

\vspace{1em}

where $A$ is a new concept name not appearing in $\Sigma$ and $R_i$ are all
the role names appearing in $\Sigma$.

\vspace{1ex}

The process of eliminating general axioms is called {\em internalization}.
The above simple TBox emulates the general axiom $$(\Not C_1\Or
D_1)\And\cdots\And(\Not C_n\Or D_n)\doteq\top$$

\end{slide}

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

\end{document}
