\documentclass[pdf,plainsem,slideColor]{prosper}

\usepackage{xspace}
\usepackage{latexsym}
\usepackage{amssymb}

\usepackage{dl-macros}

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

\newcommand {\minitab}  {\hspace*{1ex}}
\newcommand {\smalltab} {\hspace*{5mm}}
\newcommand {\tab}      {\hspace*{10mm}}
\newcommand {\bigtab}   {\hspace*{15mm}}
\newcommand {\hugetab}  {\hspace*{20mm}}
\newcommand{\per}{\mathpunct{\mbox{\bf .}}}

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

\newcommand{\HS}{\mbox{${\cal HS}$}}
\newcommand{\ALCUS}{\mbox{${\cal ALC}_{\cal US}$}}
\newcommand{\ALCQIUS}{\mbox{${\cal ALCQI}_{\cal US}$}}
\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{\ALCI}{\mbox{${\cal ALCI}$}}


\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}}

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

\semtitle{Description Logics}
\semsubject{Description Logics and 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}{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 ${\tt Mortal\IsSubs \parbox[t]{\textwidth}
  {$\tt LivingBeing \And \all{LIVES\mbox{-}IN}{Place}\And\\
  ( LivingBeing\ {\cal U}\
      (\Box^+ {\neg LivingBeing})) $} }$
\vspace{.5ex}

\item Satisfiability in \ALCUS\ -- 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 \ALCQIUS\ with ABox -- the combination of
    tense logic with \ALCQI\ 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}{\HS: Interval Temporal Propositional Modal Logic}

\vspace{-1em}

  \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}
      \put(13, 0){\line( 1, 0){5}}
      \put(14, 2){\line( 1, 0){5}}
      \put(15, 4){\line( 1, 0){5}}
      \put(17, 6){\line( 1, 0){4}}
      \put(18, 8){\line( 1, 0){4}}
      \put(19,10){\line( 1, 0){4}}
      \put(15,11){\line( 1, 0){3}}
      \thinlines
      \put(15, 0){\line( 0, 1){11}}
      \put(18, 0){\line( 0, 1){11}}
      \put(16,12){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] {\it i}}}
      \put(20,12){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] {\it j}}}
        \put(-1,10){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] before $(i,j)$}}
        \put(-1,8){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] meets $(i,j)$}}
        \put(-1,6){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] overlaps $(i,j)$}}
        \put(-1,4){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] starts $(i,j)$}}
        \put(-1,2){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] during $(i,j)$}}
        \put(-1,0){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt] finishes $(i,j)$}}
    \end{picture}
  \end{center}
}

  \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, $\HS\cup\ALC$ is undecidable.

  \end{itemize}

\end{slide}

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

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

\vspace{-1em}

\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 $\HS^{*}\cup\ALC$
  is decidable (PSPACE-hard); satisfiability is PSPACE-complete.
\item Logical implication in $\HS^{*}\cup{\cal F}$ is NP-complete.
\item Useful for event representation and plan recognition.
\end{itemize}

\end{slide}

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

\begin{slide}{The Block World Domain}

\vspace{-5em}

\begin{center}
\setlength{\unitlength}{0.1in}
\begin{picture}(27,7)(0,0)
{\scriptsize
\thicklines
\put(1,0){\framebox(2,2){1}}
\put(5,0){\framebox(2,2){2}}

\put(16,0){\framebox(2,2){1}}
\put(20,0){\framebox(2,2){2}}

\put(0,0){\line(1,0){8}}
\put(15,0){\line(1,0){8}}

\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}}
}
\end{picture}
\end{center}

%%%

\vspace{0.6em}

\begin{center}
\setlength{\unitlength}{0.1in}
\begin{picture}(27,7)(0,0)
{\scriptsize
\thicklines
\put(1,3){\framebox(2,2){1}}
\put(5,0){\framebox(2,2){2}}

\put(20,0){\framebox(2,2){2}}
\put(20,2){\framebox(2,2){1}}

\put(0,0){\line(1,0){8}}
\put(15,0){\line(1,0){8}}

\put(0.5,5.5){\line(1,0){3}}
\put(2,7){\line(0,-1){1.5}}
\put(0.5,5.5){\line(0,-1){0.5}}
\put(3.5,5.5){\line(0,-1){0.5}}

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

\vspace{2em}

\setlength{\unitlength}{0.007in}
\begin{picture}(400,95)(20,670)
{\tiny
\thicklines
\put(120,760){\vector(-1, 0){  0}}
\put(120,760){\vector( 1, 0){180}}
\put(160,720){\vector(-1, 0){  0}}
\put(160,720){\vector( 1, 0){140}}
\put(420,720){\vector(-1, 0){120}}
\put( 20,720){\vector( 1, 0){140}}
\put( 20,680){\vector( 1, 0){280}}
\put(400,680){\vector(-1, 0){100}}
\put(160,725){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\tt Holding-Block(OBJ1)}}}
\put( 20,685){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\tt Clear-Block(OBJ2)}}}
\put(300,685){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\tt ON(OBJ1, OBJ2)}}}
\put( 20,725){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\tt Clear-Block(OBJ1)}}}
\put(300,725){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\tt Clear-Block(OBJ1)}}}
\put(120,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\tt Stack(OBJ1, OBJ2)}}}
\put(180,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\it w}}}
\put( 40,670){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\it x}}}
\put(320,670){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\it y}}}
\put( 40,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\it v}}}
\put(320,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\it z}}}
\put(140,750){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\mbox{$\sharp$}}}}
}
\end{picture}

\vspace{.5em}

{\scriptsize
\hspace{-1em}$
{\tt Stack} ~ ~ \doteq ~ ~$  $ \SOMETIME  (x\ y\ z\ v\ w)\ \ 
  \parbox[t]{.4\textwidth}{$      
       (\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\\
       (\parbox[t]{70ex}{$
        ({\tt \Parm OBJECT2}: {\tt Clear\mbox{-}Block})@x \And \\
         (\same{{\tt \Parm OBJECT1}\!\comp\!{\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)    
      $}
  $}
$
}

\end{slide}

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

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

  \ALCI ~\parbox[t]{\textwidth}{
    {\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}{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}{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}

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

\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 