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\newcommand {\minitab}  {\hspace*{1ex}}
\newcommand {\smalltab} {\hspace*{5mm}}
\newcommand {\tab}      {\hspace*{10mm}}
\newcommand {\bigtab}   {\hspace*{15mm}}
\newcommand {\hugetab}  {\hspace*{20mm}}

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\newcommand{\Id}[1]{\mbox{\sf id}(#1)}
\newcommand{\ALCQ}{\mbox{${\cal ALCQ}$}}
%\newcommand{\ALCF}{\mbox{${\cal ALCF}$}}
\newcommand{\ALCIO}{\mbox{${\cal ALCIO}$}}
\newcommand{\ALCO}{\mbox{${\cal ALCO}$}}
\newcommand{\ALCreg}{\mbox{\ALC$_{\mbox{\it reg}}$}}
\newcommand{\ALCIreg}{\mbox{${\cal ALCI}_{\mbox{\it reg}}$}}
\newcommand{\ALCQIreg}{\mbox{${\cal ALCQI}_{\mbox{\it reg}}$}}
\newcommand{\ALCQITreg}{\mbox{${\cal ALCQIT}_{\mbox{\it reg}}$}}
\newcommand{\ALCQIOTreg}{\mbox{${\cal ALCQIOT}_{\mbox{\it reg}}$}}
\newcommand{\ALCFIreg}{\mbox{${\cal ALCFI}_{\mbox{\it reg}}$}}
\newcommand{\ALCQOreg}{\mbox{${\cal ALCQO}_{\mbox{\it reg}}$}}
\newcommand{\ALCIOreg}{\mbox{${\cal ALCIO}_{\mbox{\it reg}}$}}
\newcommand{\Km}{\mbox{${\cal K}_{(m)}$}}
\newcommand{\PDL}{\mbox{${\cal PD\!L}$}}
\newcommand{\CPDL}{\mbox{${\cal CPD\!L}$}}
\newcommand{\DPDL}{\mbox{${\cal DPD\!L}$}}
\newcommand{\DCPDL}{\mbox{${\cal DCPD\!L}$}}
\newcommand{\HS}{\mbox{${\cal HS}$}}
\newcommand{\ALCT}{\mbox{${\cal ALCT}$}}
%\newcommand{\TLALCF}{\mbox{${\cal T\!L\mbox{-}ALCF}$}}
%\newcommand{\TLF}{\mbox{${\cal T\!L{-}F}$}}
\newcommand{\DL}{\mbox{${\cal D\!L}$}}
\newcommand{\ALCK}{\mbox{${\cal ALCK}$}}

\newcommand{\Cthree}{\mbox{$\ddot{\mathcal{C}}^{3}$}}
\newcommand{\Ctwo}{\mbox{$\ddot{\mathcal{C}}^{2}$}}
\newcommand{\Lthree}{\mbox{$\ddot{\mathcal{L}}^{3}$}}
\newcommand{\Ltwo}{\mbox{$\ddot{\mathcal{L}}^{2}$}}
\newcommand{\Cn}{\mbox{$\ddot{\mathcal{C}}^{n}$}}
\newcommand{\Ln}{\mbox{$\ddot{\mathcal{L}}^{n}$}}

\def\NOW{\mbox{$\sharp$}}
\def\Parm{\mbox{$\star$}}
\def\SOMETIME{\Diamond}

\newcommand{\AutoK}{\mbox{\fontfamily{phv}\fontseries{b}\selectfont K}}

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

\begin{document}

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

\begin{slide*}
  
\slidetitle{\fbox{\parbox[t]{13em}{\centerline{Description Logics}
      \centerline{and Logics}}}}

\hspace{3cm}

\centerline{Enrico Franconi}

\hspace{3cm}

\centerline{\small\tt franconi@cs.man.ac.uk}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Summary}

  \begin{itemize}
  \item Description Logics
  \item \Km, multi-modal Normal Modal Logic ${\cal K}$
  \item Propositional Dynamic Modal Logics
  \item Propositional $\mu$-calculus
  \item Propositional Temporal Modal Logics
  \item \Ln\ FOL fragments
  \item Guarded Fragment of FOL
  \end{itemize}

\end{slide*}

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

\begin{slide*}

\slidetitle{What is a Description Logic}

A logical system, based on {\bf objects} (individuals), {\bf classes}
(concepts), and {\bf relationships} (roles), constituted by:

\begin{itemize}
\item a {\it description language}, which specifies how to construct
  concept and relationship expressions,
\item a {\it knowledge base language}, which specifies properties of
  objects, concepts, and relationships,
\item a set of (decidable) {\it reasoning services} over a knowledge
  base, with sound and complete procedures.
\end{itemize}

\end{slide*}

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

\begin{slide*}

\slidetitle{An example}

$\Sigma$\/:

\vspace{1em}

\smalltab{\it TBox}\\ 
\tab$\Some{\tt TEACHES}{\tt Course} \Subsumedby \Not{\tt Undergrad}\Or{\tt Prof}$

\vspace{1em}

\smalltab{\it ABox}\\
$\tab{\tt TEACHES}({\tt mary},{\tt cs415}),
{\tt Course}({\tt cs415}), $\\
$\tab{\tt Undergrad}({\tt mary})$

\vspace{2em}

$\Sigma\models{\tt Prof}({\tt mary})$


\end{slide*}

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

\begin{slide*}

\slidetitle{Description Logics: syntax}

\vspace{-1em}

{\scriptsize

    $\hspace{-2em}%
    \begin{array}{|rcl@{\hspace*{2ex}}|l@{\hspace*{2ex}}|p{8.8em}|}\hline
      C,D & \rightarrow\ 
        & A \mid & A & (primitive concept) \\
      & & \Not C  \mid & ({\tt not\ } C) & (complement) \\
      & & C\And D \mid & ({\tt and\ } C\ D\ \ldots) & (conjunction) \\
      & & C\Or D \mid & ({\tt or\ } C\ D\ \ldots) & (disjunction) \\
      & & \All {R}{C} \mid & ({\tt all\ } R\ C) & (universal quant.) \\
      & & \Some {R}{C} \mid & ({\tt some\ } R\ C) & (existential quant.) \\
      & & {\overlay{3}\ATLEAST n {R}{C} \mid} & {\overlay{3}({\tt atleast\ } n\ R\ C)} & {\overlay{3}\vspace{-2ex}(minimum cardin.)} \\
      & & {\overlay{3}\ATMOST n {R}{C} \mid} & {\overlay{3}({\tt atmost\ } n\ R\ C)} & {\overlay{3}\vspace{-2ex}(maximum cardin.)} \\
      & & {\overlay{4}\Selects {f}{C} \mid} & {\overlay{4}({\tt in\ } f\ C)} & {\overlay{4}\vspace{-2ex}(selection)} \\
      & & {\overlay{4}\Undef{f}\, \mid }& {\overlay{4}({\tt undefined\ } f) }& {\overlay{4}\vspace{-2ex}(undefinedness) }\\
      & & {\overlay{5}\Oneof{\!a, b \ldots\!}} &{\overlay{5}({\tt oneof}\ a\ b\ \ldots) }& {\overlay{5}\vspace{-2ex}(enumeration)} \\
      \hline
      {\overlay{1}R,S} & {\overlay{1}\rightarrow\ }
        & {\overlay{1}P \mid }& {\overlay{1}P }& {\overlay{1}\vspace{-2ex}(primitive role) }\\
      & & {\overlay{1}R\Or S \mid }& {\overlay{1}({\tt or\ } R\ S) }& {\overlay{1}\vspace{-2ex}(disjunction)}\\
      & & {\overlay{1}R \Compose S\mid }& {\overlay{1}({\tt compose\ } R\ S) }& {\overlay{1}\vspace{-2ex}(role chain)}\\
      & & {\overlay{1}R^* \mid }& {\overlay{1}({\tt trans\ } R) }& {\overlay{1}\vspace{-2ex}(transitive closure)}\\
      & & {\overlay{1}\Id{C} \mid }& {\overlay{1}({\tt self\ } C) }& {\overlay{1}\vspace{-2ex}(role identity)}\\
      & & {\overlay{2}\Inverse R \mid }& {\overlay{2}({\tt inverse\ } R) }& {\overlay{2}\vspace{-2ex}(inverse role)}\\
      & & {\overlay{4}f }& {\overlay{4}f }& {\overlay{4}\vspace{-2ex}(feature)} \\
      \hline
      {\overlay{4}f,g }& {\overlay{4}\rightarrow\ }
        & {\overlay{4}p \mid }& {\overlay{4}p }& {\overlay{4}\vspace{-2ex}(primitive feature) }\\
      & & {\overlay{4}f \Compose g }& {\overlay{4}({\tt compose\ } f\ g) }& {\overlay{4}\vspace{-2ex}(feature chain)}\\
    \hline
  \end{array}
    $

\vspace{2em}

\hspace{-2em}\begin{minipage}[c]{.5\textwidth}
TBox: $(C\ \dot\Subsumedby\ D),\ \  (R\ \dot\Subsumedby\ S)$

ABox: $C(a),\ \  R(a,b)$
\end{minipage}
%
\begin{minipage}[c]{.5\textwidth}
\fbox{\huge${\cal ALC{\overlay{2}I}{\overlay{3}Q}{\overlay{4}F}{\overlay{5}O}}{\overlay{1}_{\mbox{\it reg}}}$}
\end{minipage}

}


\end{slide*}

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

\begin{slide*}

\slidetitle{Description Logics: semantics}

\vspace{-2em}

{\scriptsize

\renewcommand{\arraystretch}{1.2}
$
\hspace{-4em}\begin{array}{|r@{\hspace{1ex}}l@{\hspace{.3ex}}|l|}\hline

  \Int A \subseteq &\Int\Delta  & A(\gamma) \\ 

  \Int{(\Not C)} = & \Int\Delta \setminus \Int C & \neg F_C(\gamma)\\ 

  \Int{(C\And D)} = & \Int C \cap \Int D & F_C(\gamma) \wedge F_D(\gamma)\\ 

  \Int{(C\Or D)} = & \Int C \cup \Int D & F_C(\gamma) \vee F_D(\gamma)\\ 

  \Int{(\All{R}{C})} = & \{\parbox[t]{11em}{$
i\in\Int\Delta \mid \\ \forall j\per 
\Int R(i,j)
    \Rightarrow \Int C(j)\}$} & \forall x\per F_R(\gamma,x) \Rightarrow
  F_C(x)\\ 

  \Int{(\Some{R}{C})} = & \{\parbox[t]{11em}{$i\in\Int\Delta \mid \\ \exists
    j\per \Int R(i,j)\wedge \Int C(j)\}$} & \exists x\per
  F_R(\gamma,x)\wedge F_C(x)\\ 

  {\overlay{1}\Int{(\ATLEAST n {R}{C})} = } & {\overlay{1} \{\parbox[t]{13em}{$i\in\Int\Delta\mid \\ 
    \sharp\set{j\mid \Int R(i,j)\wedge\Int C(j)} \geq n\}$} } & {\overlay{1} \exists^n x\per
  F_R(\gamma,x)\wedge F_C(x)} \\ 

  {\overlay{1}\Int{(\ATMOST n {R}{C})} = } & {\overlay{1} \{\parbox[t]{13em}{$i\in\Int\Delta\mid \\ 
    \sharp\set{j\mid \Int R(i,j)\wedge\Int C(j)} \leq n\}$} } & {\overlay{1} \neg\exists^{n+1} x\per
  F_R(\gamma,x)\wedge F_C(x)} \\ 

  {\overlay{1}\Int{(\Selects{f}{C})} = } & {\overlay{1} \set{ i\in\Dom{\Int f} \mid \Int C(\Int f(i))} } & {\overlay{1}
  \exists x\per F_f(\gamma,x) \wedge F_C(x)} \\ 

  {\overlay{1}\Int{(\Undef{f})} = } & {\overlay{1} \Int\Delta\setminus\Dom{\Int f} } & {\overlay{1} \neg \exists x\per
  F_f(\gamma,x)} \\ 

  {\overlay{1}\Int{\Oneof{a, b \ldots}} = } & {\overlay{1} \set{\Int a, \Int b \ldots} } & {\overlay{1}
  \gamma=a\vee\gamma=b\vee\ldots} \\ 

  \hline

  {\overlay{1}\Int P \subseteq } & {\overlay{1}\Int\Delta\times\Int\Delta  } & {\overlay{1} P(\alpha,\beta) } \\ 

  {\overlay{1}\Int{(R\Or S)} = } & {\overlay{1} \Int R\cup \Int S } & {\overlay{1} 
  F_R(\alpha,\beta) \vee F_S(\alpha,\beta)} \\

  {\overlay{1}\Int{(R\Compose S)} = } & {\overlay{1}\Int R{\scriptstyle\circ}\ \Int S } & {\overlay{1} \exists x\per
  F_R(\alpha,x) \wedge F_S(x,\beta)} \\

  {\overlay{1}\Int{(R^*)} = } & {\overlay{1} (\Int R)^*  } & {\overlay{1} \ldots} \\

  {\overlay{1}\Int{(\Id{C})} = } & {\overlay{1} \{(i,i)\in\Int\Delta\times\Int\Delta\mid \Int C(i)\}
} & {\overlay{1} F_C(\alpha)\wedge\alpha=\beta} \\ 

  {\overlay{1}\Int{(\Inverse R)} = } & {\overlay{1} \{\parbox[t]{11em}{$(i,j)\in\Int\Delta\times\Int\Delta\mid\\
    \Int R(j,i)\}$} } & {\overlay{1} F_R(\beta,\alpha)} \\ 

  {\overlay{1}\Int f = } & {\overlay{1} \{\parbox[t]{11em}{$(i,j)\in\Int\Delta\times\Int\Delta\mid \\ 
    \Int f(i)=j\}$} } & {\overlay{1} F_f(\alpha,\beta) } \\ 

  \hline

  {\overlay{1}\Int p = } & {\overlay{1} \Int\Delta\stackrel{\it partial}\longmapsto\Int\Delta } & {\overlay{1}
  p(\alpha)=\beta } \\ 

  {\overlay{1}\Int{(f\Compose g)} = } & {\overlay{1} \Int f{\scriptstyle\circ}\ \Int g } & {\overlay{1} \exists x\per
  F_f(\alpha)=x \wedge F_g(x)=\beta} \\

\hline
\end{array}$
\renewcommand{\arraystretch}{1}

}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Another Example}

{\footnotesize 

$
\begin{array}{rcl}
{\tt \Some{TEACHES}{Course}} & \dot\Subsumedby & 
  {\tt (Student\And\Some{DEGREE}{Bs})\Or Prof} \\
{\tt Prof} & \dot\Subsumedby & {\tt \Some{DEGREE}{Ms}} \\
{\tt \Some{DEGREE}{Ms}} & \dot\Subsumedby & {\tt \Some{DEGREE}{Bs}} \\
{\tt Ms\And Bs} & \dot\Subsumedby & {\tt \bot} \\
\end{array}
$

\vspace{3em}

{\tt TEACHES(paul,cs415), Course(cs415)}, \\
${\tt
\mbox{\large (}\Some{FRIEND}{(\Oneof{paul} \And\ \Atmost 1 {DEGREE})}\mbox{\large )}
(john)
}$

\vspace{5em}

{\tt
$\Sigma \models$ Student(paul)
}

}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{\ALC\ and \Km}

\begin{center}
$
%\hspace{-2.5em}
\begin{array}{r||l}
\ALC & \Km \\
\hline\hline
\Int{C}\ \mbox{\tiny is a set of individuals} & \Int{\alpha_C}\ \mbox{\tiny is a set of worlds} \\
\hline
\Int{R}\ \mbox{\tiny is a set of pairs of individuals} & R\ \mbox{\tiny is an accessibility relation}\\
\hline\hline
A & P_{A} \\
\hline
C \And D & \alpha_C \wedge \alpha_D \\
\hline
C \Or D & \alpha_C \vee \alpha_D \\
\hline
\Not C & \Not\alpha_C \\
\hline
\All R C & {\displaystyle\Box}_{\!_R}\ \alpha_C \\
\hline
\Some R C  & {\displaystyle\Diamond}_{\!_R}\ \alpha_C \\
\hline\hline
o\in\Int{C} & {\cal I},o\models\alpha_C\\
\hline\hline
{\tt
\Some{T}{C}\ \dot\Subsumedby\ \Not{U}\Or{P}}
& {\displaystyle\Diamond}_{\tt T}{\tt C}\rightarrow\Not{\tt U}\lor{\tt P}\\
\hline
{\tt
U(m),T(m,c),C(c)}
& \epsfysize=2.5em \epsfbox{km-struct.eps} \\
\hline
\Sigma\models{\tt P}({\tt m})
& \epsfysize=2.5em \epsfbox{km-struct-der.eps}
\end{array}
$
\end{center}

\end{slide*}

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

\begin{slide*}
  
\slidetitle{Some Results on satisfiability}

\vspace{-1em}$\begin{array}{|ccl|}
\hline
\ALC & \Longleftrightarrow & \Km \\
\cap & & \\
\ALC^+ &\Longleftrightarrow & \Km \cup S4 \\
\cap & & \\
\cdots &  &  \\ \hline
\ALC & \Longleftrightarrow & \Km \\
\cap & & \\
\ALCO & \Longleftrightarrow  & \Km\ \mbox{with nominals}\\ \hline
\ALC & \Longleftrightarrow & \Km \\
\cap & & \\
\ALC\ \mbox{with axioms} & \Longleftrightarrow  & \Km\ \mbox{theories}\\
\hline
\end{array}$

\vspace{1ex}

\begin{itemize}
\item $\ALC$: PSPACE-complete
\item $\ALC^+$: PSPACE-complete
\item $\ALCO$: PSPACE-complete
\item $\ALC$\ with axioms: EXPTIME-complete
\end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{\ALCreg\ and \PDL}

\begin{itemize}
\item The domain of the interpretation is to be read as a set of program
  states.
\item Concepts are to be interpreted as the set of states in which they
  hold.
\item Roles are to be interpreted as {\em nondeterministic programs}.
\end{itemize}


\end{slide*}

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

\begin{slide*}
  
  \slidetitle{\ALCreg\ and \PDL}

\begin{itemize}
\item $\All R C$\ as ${\mathbf [R]C}$: ``whenever program $R$ terminates,
  proposition $C$ holds on termination''.
\item ${\mathbf R_1 \Compose R_2}$\ as ``run $R_1$ and $R_2$
  consecutively''.
\item ${\mathbf R_1 \Or R_2}$\ as ``nondeterministically do $R_1$ or
  $R_2$''.
\item ${\mathbf R^\star}$\ as ``repeat program $R$ a nondeterministically
  chosen number of times $\geq 0$''.
\item ${\mathbf \Id C}$ as ``proceed without changing the program state iff
  proposition $C$ holds''.  

  {\overlay{1}\item ${\mathbf \Inverse R}$\ as ``run $R$ in reverse''
  (\ALCIreg\ and \CPDL).}
\end{itemize}


\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Internalization of axioms}

\begin{center}
$
\begin{array}{rcl}
 
  \psi\models\varphi & \leadsto & 
  \models\left[\nu\right]\psi\Rightarrow\varphi \\

&&\\
 
  C\ \dot\Subsumedby\ D\models\varphi & \leadsto &
  \models\left[\nu\right](\alpha_C \Rightarrow \alpha_D)
  \Rightarrow\alpha_\varphi 
 
\end{array}
$
 
\vspace{2em}
 
$\nu \doteq {(R_1\vee \Inverse{R_1}\vee \ldots \vee R_n\vee \Inverse{R_n})^\star}$
 
\end{center}

\vspace{2em}
 
\begin{itemize}

\item Reasoning with theories is reduced to satisfiability of single
formulas
\item The complexity does not change

\end{itemize}


\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Results: satisfiability}

\vspace{-1em}

$\begin{array}{ccl}

\ALCreg & \Longleftrightarrow & \PDL \\
\Uparrow & & \\
\ALCIreg &\Longleftrightarrow & \CPDL \\
\Uparrow & & \\
\ALCFIreg^- & \Longleftrightarrow & \DCPDL \\
\Uparrow & & \\
\ALCFIreg &  &  \\
\Uparrow & & \\
\ALCQIreg^- & \Longleftrightarrow & \CPDL + \mbox{graded modalities} \\
\Uparrow & & \\
\ALCQIreg &  & \\
\end{array}$

\vspace{1.5em}

\begin{itemize}
\item Satisfiability is EXPTIME-complete\\{\small(subsumption and logical
    implication {\em wrt a free TBox} can be reduced to satisfiability)}.
\item \CPDL\ has the finite model property, \DCPDL\ not.
\end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Results: satisfiability \\ with individuals}

$\hspace{-2.5em}\begin{array}{ccl}

\ALCreg & \Longleftrightarrow & \PDL \\
\Uparrow & & \\
\ALCQOreg &\Longleftrightarrow & \PDL + \parbox[t]{10em}
                                           {graded modalities and \\ nominals}\\
 & & \\
\ALCIreg & \Longleftrightarrow & \CPDL \\
\Uparrow & & \\
\ALCIOreg & \Longleftrightarrow & \CPDL + \mbox{nominals}\\
 & & \\
\ALCIreg & \Longleftrightarrow & \CPDL \\
\Uparrow & & \\
\ALCQIreg + \mbox{ABox} &  & \\
\end{array}$

\vspace{2em}

Satisfiability is EXPTIME-complete.

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{So what?}

  \begin{itemize}

  \item DL extend modal logic in interesting ways:

    \begin{itemize}

    \item Reasoning in DL is always reasoning with theories.

    \item Nominals.

    \item Graded modalities.

    \end{itemize}
  
  \item Results:

    \begin{itemize}

    \item \ALC\ theories are EXPTIME-complete.
      
    \item \ALCQIreg\ theory reduces to \ALCreg\ theory (\DCPDL\ to
      \PDL\/).
      
    \item \ALCIOreg\ theory reduces to \ALCreg\ theory (\CPDL\ +
      nominals to \PDL\/).

    \item \ALCQIreg\ + Abox theory reduces to \ALCreg\ theory.

    \end{itemize}

  \end{itemize}
  
\end{slide*}

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

\begin{slide*}
  
  \slidetitle{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}

\vspace{1ex}

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

\vspace{2em}

\begin{minipage}{.75\textwidth}
${\tt\dom=\{a,b,nil\}}$ \\
${\tt\Int{Node}=\{a,b\}}$ \\
${\tt\Int{Empty\mbox{-}List}=\{nil\}}$ \\
${\tt\Int{SUCCESSOR}=\{\langle a,nil\rangle,\langle b,b\rangle\}}$
\end{minipage}
\begin{minipage}{.3\textwidth}
\epsfxsize=\textwidth \epsfbox{list.eps}
\end{minipage}

\vspace{1em}

With descriptive semantics:\\
${\tt\Int{List}=\{a,nil\}}$; ${\tt\Int{List}=\{a,b,nil\}}$ 

\vspace{.4ex}

With least fixpoint semantics:\\
${\tt\Int{List}=\{a,nil\}}$ 

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Propositional $\mu$-calculus}

\hspace{-2em}%
${\tt Node \doteq \Not Empty{-}List}$

\hspace{-2em}\parbox[t]{\textwidth}{
${\tt List \doteq \mu \mbox{$X$}\per (\parbox[t]{\textwidth}{$\tt
    Empty{-}List \Or \\ (Node \And \Atmost 1 {SUCCESSOR} \And
    \Some {SUCCESSOR} {\mbox{$X$}})) $}}$}

\vspace{2em}

$C\ \rightarrow\ \ldots \mid \mu X\per C \mid \nu X\per C \mid X$

\vspace{2em}

$\begin{array}{ccl}

\mu\ALC & \Longleftrightarrow & \mbox{Propositional}\ \mu\mbox{-calculus} \\
\Uparrow & & \\
\mu\ALCQI & & \\
\end{array}$

\vspace{2em}

\begin{itemize}
\item Satisfiability is EXPTIME-complete.
\item Can express {\em well-foundedness} of relations, useful to
describe finite structures.
\item Open problems: individuals.
\end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Tense Logic \\ {\em (point ontology)}}

  \begin{itemize}

  \item Tense logic is a propositional modal logic, interpreted over
    temporal structure ${\cal T}~=~({\cal P}, <)$, where ${\cal P}$ is a set
    of time points and $<$ is a strict partial order on ${\cal P}$.

  \item ${\tt Mortal\Subsumedby \parbox[t]{\textwidth}
  {$\tt LivingBeing \AND \All{LIVES\mbox{-}IN}{Place}\AND\\
  ( LivingBeing\ {\cal U}\
      (\Box^+ {\neg LivingBeing})) $} }$
\vspace{.5ex}

\item Satisfiability in \ALCT\ -- the combination of tense logic with
  \Km\ -- over a linear, unbounded, and discrete temporal structure
  has the same complexity as its base (PSPACE-complete).
    
  \item Satisfiability in \ALCQITreg\ with ABox -- the combination of
    tense logic with \ALCQIreg\ with ABox -- over a linear, unbounded,
    and discrete temporal structure has the same complexity as its
    base (EXPTIME-complete).

  \end{itemize}
  
\end{slide*}

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

\begin{slide*}
  
  \slidetitle{\HS: Interval Temporal \\ Propositional Modal Logic}

\vspace{-2em}

  \begin{itemize}

  \item \HS\ is a propositional modal logic interpreted over an interval set
    ${\cal T_<^*}$, defined as the the set of all closed intervals $[u,v]
    \doteq \set{ x \in {\cal P} \mid u \leq x \leq v, u\neq v}$ in some
    temporal structure ${\cal T}$.

  \item \HS\ extends propositional logic with modal formul\ae\ $\langle R 
  \rangle\phi$ and $[ R ]\phi$ -- where $R$ is a basic
    Allen's algebra temporal relation:

{\scriptsize
   \begin{center}
    \setlength{\unitlength}{.6em}
    \begin{picture}(23,12)
      \linethickness{2pt}
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  \item ${\tt Mortal\doteq LivingBeing \wedge \langle{\sf after}\rangle\per
      \neg LivingBeing}$

  \item Satisfiability \HS\ is undecidable for the most interesting classes
    of temporal structures.

  \item Therefore, $\HS\cup\Km$ is undecidable.

  \end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Decidable Interval \\ Temporal Description Logics}

\vspace{-1em}

  The combination of \ALCF\ and $\HS^{*}$ :
  
\begin{itemize}
    \item $\HS^{*}$:
    \begin{itemize}
    \item No universal quantification, or restricted to homogeneous
      properties: \\
      $\Box({\sf =,starts,during,finishes})\per\ \psi$
      \item Allows for temporal variables: \\ 
        $\Diamond\stackrel{\rightarrow}{x}{\sf
          TN}(\stackrel{\rightarrow}{x})\per\psi$ \\ $\psi@x$
      \end{itemize}
    \item Global roles -- denoting temporal {\em independent}
    properties.
\end{itemize}

\vspace{1em}

\begin{itemize}
\item Logical implication in the combined language is decidable
  (PSPACE-hard); satisfiability is PSPACE-complete.
\item Logical implication in \TLF\ is NP-complete.
\item Useful for event representation and plan recognition.
\end{itemize}

\vspace{1em}


\end{slide*}

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

\begin{slide*}

\slidetitle{The Block World Domain}

\vspace{-3em}

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\put(16.5,4){\line(1,0){3}}
\put(16.5,4){\line(0,-1){0.5}}
\put(18,5.5){\line(0,-1){1.5}}
\put(19.5,4){\line(0,-1){0.5}}

\put(0,-1){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]Initial State}}
\put(15,-1){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]Approach}}
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\end{center}

%%%

\vspace{0.6em}

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\put(15,-1){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]Final State}}
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\end{center}

\vspace{2em}

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\end{picture}

\vspace{.5em}

{\footnotesize
\hspace{-1em}$
{\tt Stack} ~ ~ \doteq ~ ~ \\  \SOMETIME  (x\ y\ z\ v\ w)\ \ 
  \parbox[t]{75ex}{$      
       (\NOW\ {\sf finishes}\ x)
       (\NOW\ {\sf meets}\ y) 
       (\NOW\ {\sf meets}\ z)\\
       (v\ {\sf overlaps}\ \NOW)
       (w\ {\sf finishes}\ \NOW)
       (v\ {\sf meets}\ w) \per\\
       \mbox{\large (}
       \parbox[t]{70ex}{$
        ({\tt \Parm OBJECT2}: {\tt Clear\mbox{-}Block})@x \And \\
        (\Agree{{\tt \Parm OBJECT1}\!\Compose\!{\tt ON}}{\tt \Parm OBJECT2})@y \And\\
        ({\tt \Parm OBJECT1}:{\tt Clear\mbox{-}Block})@v \And \\
        ({\tt \Parm OBJECT1}:{\tt Holding\mbox{-}Block})@w \And\\
        ({\tt \Parm OBJECT1}:{\tt Clear\mbox{-}Block})@z      
      \mbox{\large )}
      $}
  $}
$
}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{\Ln\ FOL fragments}

  \begin{itemize}
    
  \item \Ln\ is the set of function-free FOL formulas with equality
    and constants, with only unary and binary predicates, and which
    can be expressed using at most $n$ variable symbols.

  \item Satisfiability of \Lthree\ formulas is undecidable.

  \item Satisfiability of \Ltwo\ formulas is NEXPTIME-complete.

  \end{itemize}
  
\end{slide*}

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

\begin{slide*}
  
  \slidetitle{The \DL\ description logic}

  \ALCQIreg\ ~\parbox[t]{\textwidth}{
    {\tt -~} the transitive closure operator, \\
    {\tt -~} number restriction operators, \\
    {\tt +~} propositional calculus on roles, \\
    {\tt +~} the concept $(R\subseteq S)$. \\
    }
  
  \begin{itemize}

  \item The \DL\ description logic and \Lthree\ are equally expressive.

  \item The \DL$^-$\ description logic (i.e., \DL\ without the composition
    operator) and \Ltwo\ are equally expressive.
  
  \item Open problem: relation between \DL\ including cardinalities
    and \Cn -- adding counting quantifiers to \Ln.
    
  \end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Guarded Fragments of FOL}

The {\it guarded fragment} GF of FOL is defined as:

\begin{enumerate}

\item Every relational atomic formula is in GF

\item GF is propositionally closed

\item If ${\mathbf x}$, ${\mathbf y}$ are tuples of variables,
$\alpha({\mathbf x},{\mathbf y})$ is atomic, and $\psi({\mathbf
x},{\mathbf y})$ is a formula in GF, such that {\sf
free}$(\psi)\subseteq\ ${\sf free}$(\alpha)=\{{\mathbf x},{\mathbf y}\}$, then the following formulae are in GF:

$\exists{\mathbf y}\per\ \alpha({\mathbf x},{\mathbf
y})\wedge\psi({\mathbf x},{\mathbf y})$

$\forall{\mathbf y}\per\ \alpha({\mathbf x},{\mathbf
y})\rightarrow\psi({\mathbf x},{\mathbf y})$

\end{enumerate}

\vspace{1em}

The guarded fragment contains the modal fragment of FOL (and
Description Logics); a weaker definition (LGF) is needed to include
temporal logics.

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{Properties of GF}

\begin{itemize}
\item GF has the finite model property
\item GF and LGF have the tree model property
\item Many important model theoretic properties which hold for FOL and
the modal fragment, do hold also for GF and LGF
\item Satisfiability is decidable for GF and LGF (deterministic double
exponential time complete)
\item Bounded-variable or bounded-arity fragments of GF and LGF (which
include Description Logics) are in EXPTIME.
\item GF with fix-points is decidable.
\end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{That's not all, folks$\ldots$}

\begin{itemize}
\item Defaults and non monotonic logic
\item Other modal extensions
\item Autoepistemic logic
\item Formal ontology
\item Decision procedures
\item Database theory
\item $\ldots$
\end{itemize}

\end{slide*}

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

\begin{slide*}
  
  \slidetitle{References}

\begin{itemize}

\item Donini, F., Lenzerini, M., Nardi, D., Schaerf, A., `Reasoning in
  Description Logics', Principles of Knowledge Representation and Reasoning,
  editor, G. Brewka, Studies in Logic, Language and Information, CLSI
  Publications, pp 193-238, 1996.

\item Schild, K., `A correspondence theory for terminological logics:
  preliminary report', IJCAI-91, pp 466-471, 1991.

\item De Giacomo, G., Decidability of Class-Based Knowledge Representation
  Formalisms, Ph.D. thesis, Dipartimento di Informatica e Sistemistica,
  Universita degli Studi di Roma "La Sapienza", 1995.

\item De Giacomo, G., Lenzerini, M., `TBox and ABox Reasoning in Expressive
  Description Logics', KR-96, pp 316-327, 1996.

\item De Giacomo, G., Lenzerini, M., `A Uniform Framework for Concept
  Definitions in Description Logics', Journal of Artificial Intelligence
  Research, Vol 6, pp 87-110, 1997.

\item Schild, K., `Combining terminological logics with tense logic', Proc.
  of the 6th Portuguese Conference on Artificial Intelligence - EPIA-93, pp
  105-120, 1993.

\item Artale, A., Franconi, E., `A Computational Account for a Description
  Logic of Time and Action', KR-94, pp 3-14, 1994.

\item Alex Borgida, `On the relative expressive power of Description Logics
  and Predicate Calculus', Artificial Intelligence 82(1996) 353-367.

\item Erich Gr\"adel, `Guarded Fragments of Fisrt-Order Logic: A
Perspective for New Description Logics?', DL-98, pp 5-7, 1998.

\end{itemize}

\end{slide*}

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

\end{document}

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