\documentclass[a4,semrot,dvips,slidesec]{seminar}
\usepackage{epsfig}
\usepackage{dl}
\usepackage{amssymb}
\usepackage{fancybox}
%\usepackage{multicol}
%\usepackage{euscript}
%\usepackage{pifont}

\renewcommand{\con}[1]{\textrm{#1}}
\newcommand{\FaCT}{FaCT\xspace}
\newcommand{\Sall}{21}
\newcommand{\Dlpstar}{DLP$^*$\xspace}%
\newcommand{\Srole}[1]{\ensuremath{\EuScript{S}}(#1)}
\newcommand{\Ucon}{\ensuremath{\EuScript{M}}\xspace}
%\newcommand{\Sat}{\emph{satisfiable}\xspace}
\newcommand{\N}{{\rm I\!N}} 

\input{cadslide.tex}

\hyphenation{se-man-tics con-cept pro-vi-ded back-jump-ing}

%%% declare the title:
\title{%
{\Large {\bf Reasoning}\\
with\\[1ex]
{\bf Expressive Description Logics}
}}
%\title{%
%{\Large\bf Practical Algorithms for Expressive\\[1ex]
%Description Logics}}

\author{
Ian Horrocks\\[1ex]
%\epsfig{file=../../miglogo.ps,height=1.0in}\\[-2ex]
%\epsfig{file=img.ps,height=0.5in}\\%[-2ex]
Department of Computer Science\\
University of Manchester, UK\\
{\small \texttt{horrocks@cs.man.ac.uk}}
}

\date{}

%%% set the 3-part running footer
\newcommand{\Rfleft}{\tiny Enrico Franconi and Ian Horrocks}
\newcommand{\Rfcent}{\footnotesize \theslide}
\newcommand{\Rfright}{\tiny Combining Modal Logics: the Description Logics perspective}
%\newcommand{\Rfleft}{\tiny Expressive DLs}
%\newcommand{\Rfcent}{\footnotesize \theslide}
%\newcommand{\Rfright}{\tiny Ian Horrocks}

\newcommand {\smalltab} {\hspace*{5mm}}
\newcommand {\tab}      {\hspace*{10mm}}
\def\And{\ensuremath{\mathbin\sqcap}}
\newcommand{\Or}{\ensuremath{\mathbin\sqcup}}
\newcommand{\Not}{\ensuremath{\neg}}
\newcommand{\Int}[1]{#1^{\cal I}\/}         
\renewcommand{\PDL}{\mbox{${\cal PD\!L}$}\xspace}
\renewcommand{\CPDL}{\mbox{${\cal CPD\!L}$}\xspace}
\newcommand{\DPDL}{\mbox{${\cal DPD\!L}$}\xspace}
\newcommand{\DCPDL}{\mbox{${\cal DCPD\!L}$}\xspace}
\renewcommand{\N}{{\rm I\!N}} 
\newcommand{\HR}{\ensuremath{\mathcal{H}}}

%*******************************************************************************
\begin{document}


%%% heres the title page with `empty' page style:
%%%  (also..set the `slide' count)
%\begin{cadslide}{}
%    \maketitle
%    \addtocounter{slide}{-1}
%    \slidepagestyle{empty}
%\end{cadslide}
%%% heres the title page with `empty' page style:

%%% heres the contents page:
%\begin{cadslide}{Contents}
%\mbox{}\vspace*{0.2in}
%\slidecontents
%\end{cadslide}

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

\begin{cadslide}{Description Logics in one example}

\vspace{1em}

$\Sigma$\/:

\vspace{1em}

\smalltab{\it TBox}\\ 
\tab$\some{\tt TEACHES}{\tt Course} \mbox{~$\dot\Issub$~} \Not{\tt Undergrad}\Or{\tt Prof}$

\vspace{1em}

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

\vspace{2em}

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


\end{cadslide}

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

\rotateheadersfalse

\begin{cadslideportrait}{Correspondence with Modal Logics}

\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\Issub\ \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{cadslideportrait}

\rotateheaderstrue

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

\begin{cadslide}{Combining Modal Logics \\ 
    in the Description Logics perspective}

  \vspace{1em}

  \begin{itemize}

  \item DLs extend modal logic in interesting ways:

    \begin{itemize}

    \item Reasoning in DLs is always reasoning with theories.

    \item Nominals.
      
    \end{itemize}

  \item Studying the effects of augmenting the expressivity is
    central:

    \begin{itemize}
        
    \item adding operators cab be seen as {\bf combining} modal logics
      with the basic \Km;
        
    \item if the basic logic is expressive enough (e.g., \PDL),
      possible reductions are studied;
        
    \item more {\it typical} combinations are also important, such as
      combinations with tense logic or modal logic with concrete
      domains.

    \end{itemize}
  
  \end{itemize}
  
\end{cadslide}

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

\begin{cadslide}{Combining Modal Logics vs Reductions}

  \vspace{1em}

  \begin{itemize}

  \item Combining the basic \Km with a modal logic having:

    \begin{itemize}
    \item inverse, or graded, or deterministic modalities, $\ldots$
    \end{itemize}
    
  \item Reducing a complex combination to the basic \PDL:

    \begin{itemize}
    \item \DCPDL to \PDL, \CPDL + nominals to \PDL, $\ldots$
    \end{itemize}
  
  \item The combination approach may be more interesting than the
    reduction approach; example:

    \begin{itemize}
    \item \PDL versus $\Km^{\HR} \cup \Sfour$,
    \item the combination can be seen as the FOL fragment of \PDL,
    \item same complexity class,
    \item different algorithmic properties ({\it cut rule\/}).
    \end{itemize}
  
  \end{itemize}
  
\end{cadslide}

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

\begin{cadslide}{Combining Modal Logics
    in the Description Logics perspective}

  \vspace{1em}

  \begin{itemize}

  \item Decidability,

  \item complexity class,

  \item (how to extend) algorithms,

  \item (how to re-adapt) strategies and optimisations.
  
  \end{itemize}
  
%\end{cadslide}

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

%*******************************************************************************

\continueslide

%\begin{cadslide}{Examples of Combining Modal Logics}

Some examples of how combining modalities with different properties can
affect:

\begin{itemize}
\item Complexity class
\item Algorithmic complexity
\item Decidability
\end{itemize}

%*******************************************************************************

\continueslide

\begin{itemize}
\item Decision problems for $\Km^\HR$ and \Sfourm known to
  be in \textsc{PSpace}
\item Combination allows use of \emph{universal modality} to
  \emph{internalise} arbitrary set of axioms:
  \begin{itemize}
  \item Define new transitive modality $u$ that includes all other modalities
  \item Satisfiability of $\phi$ w.r.t.\ $\psi_1 \rightarrow
    \varphi_1, \ldots, \psi_n \rightarrow \varphi_n$ equivalent to
    satisfiability of $\phi \land \square_u((\psi_1 \rightarrow
    \varphi_1) \land \ldots \land (\psi_n \rightarrow \varphi_n))$
  \end{itemize}
\item Decision problem w.r.t.\ arbitrary set of axioms known to be in
  \textsc{ExpTime} even for \Km
\end{itemize}


%*******************************************************************************

\continueslide

\begin{itemize}
\item Decision problem for $\Km^\HR \cup \Sfourm$ in \textsc{ExpTime}, but
  tableaux algorithm presents no special problems:
  \begin{itemize}
  \item For transitive modalities, propogate $\square_i\phi$ terms
    along $i$ modalities
  \item Use simple \emph{blocking} technique to check for cycles
    caused by e.g.\ $\square_i\lozenge_i\phi$
  \item Cycle in algorithm $\Rightarrow$ valid cyclical model
  \end{itemize}
\end{itemize}

%*******************************************************************************

\continueslide

\begin{itemize}
\item Decision problem for $\Km^\HR$ combined with deterministic
  and converse modalities still in \textsc{PSpace} and solvable with simple
  tableaux algorithm (no blocking)
\item Combination no longer has finite model property --- requires
  new blocking technique to detect cycles implying valid but
  non-finite models, e.g.\ for:
%$$\neg\phi \land \square_{R^\smile}\lozenge_{S^\smile}\phi$$
$$\neg\phi \land [R^\smile]\langle S^\smile\rangle\phi$$
where $R$ is transitive, $S$ is deterministic and $R$ includes $S$
\end{itemize}


%*******************************************************************************

\continueslide
\begin{itemize}
\item Decision problem for $\Km^\HR \cup \Sfourm$ in \textsc{ExpTime}
\item Decision problem for $\Km^\HR$ combined with graded modalities in
  \textsc{PSpace}
\item Decision problem for $\Km^\HR \cup \Sfourm$
  combined with graded modalities is undecidable --- shown by
  reduction of domino problem
\item Representing $\N \times \N$ grid (the tricky bit) uses
  combination of \HR, transitive and non-transitive modalities and
  graded modalities
\item Decidability restored by restricting the way transitive and
  graded modalities can be combined
\end{itemize}

\end{cadslide}

%*******************************************************************************


\end{document}
