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\usepackage{dl-macros}

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

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\newcommand{\DLRUSm}{\ALlang{DLR}{^-_{\cal US}}}
\newcommand{\ALCM}{\ALlang{ALC_{M}}{}}
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\newcommand{\CIQUS}{\ALlang{CIQ_{US}}{}}
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\newcommand{\SOMER}[2]{\exists[#1]#2}
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\newcommand{\EXACTLYR}[3]{\exists^{= #1} [#2]#3}
\newcommand{\PROJ}[3]{#1|_{#2,#3}}

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\newcommand{\ALCQIT}  {\mbox{${\cal ALCQIT}$}}
\def\NOW{\mbox{$\sharp$}}
\def\Parm{\mbox{$\star$}}
\def\SOMETIME{\Diamond}
\def\NEXTTIME{\mbox{\footnotesize\ensuremath{\bigcirc}}}
\def\ST#1#2{\SOMETIME #1 \per #2}
\def\SST#1#2#3{\SOMETIME #1\ #2\per #3}
\def\ALLTIME{\Box}
\def\ALLT#1#2{\ALLTIME #1 \per #2}
\def\ALLTE{\ALLTIME^*}
\def\ALLTP{\ALLTIME^+}
\def\ALLTM{\ALLTIME^-}
\def\STP{\SOMETIME^+}
\def\STM{\SOMETIME^-}
\def\STE{\SOMETIME^*}
\def\NTP{\mbox{\ensuremath{\oplus}}}
\def\NTM{\mbox{\ensuremath{\ominus}}}
\def\NTE{\mbox{\ensuremath{\otimes}}}

\def\IsSubsS {\mathbin{\IsSubs^\ast}}           %%% global inclusion axiom

\newcommand{\Intt}[2][\I(t)]{#2^{#1}}
\newcommand{\Bint}[1]{#1^{\cal B}\/}
\newcommand{\Bintt}[1]{#1^{{\cal B}(t)}\/}

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\def\TSS {\mbox{${\cal T}_{\!p}$}}
\def\P {\mbox{${\cal P}$}}
\def\Sc {\mbox{${\cal D}$}}
\def\B  {\mbox{${\cal B}$}}
\def\U {\mbox{${\cal U}$}}
\def\S {\mbox{${\cal S}$}}

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\def\temp{\mbox{\Large$\bullet$}}

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{\footnotesize\psset{unit=1.1cm}%
     \begin{pspicture}(0.5,2)(12,6)
        \rput(2,5){\rnode{employee}{\psframebox{Employee}}}
     \rput(2,5.8){\rnode{pslip}{\underline{\scriptsize%
     Name(String)}}}
     \rput(3.25,5.5){\rnode{salary}{\scriptsize Salary(Integer)}}
        \rput(5.25,3.75){\rnode{project}{\psframebox{Project}}}
     \rput(7,4.5){\rnode{projectcode}{\underline{\scriptsize%
     ProjectCode(String)}}}
        \rput(2,4){\rnode{manager}{\psframebox{Manager}}}
        \rput(2.9,2.5){\rnode{tmanager}{\psframebox{TopManager}}}
        \rput(1.1,2.5){\rnode{amanager}{\psframebox{AreaManager}}}
        \rput(9,2.25){\rnode{dept}{\psframebox{Department}}}
        \rput(11,2.25){\rnode{sig}{\psframebox{InterestGroup}}}
        \rput(10,3.75){\rnode{orgunit}{\psframebox{OrganisationalUnit}}}
        \rput(2,3.25){\rnode{xmanager}{X}}
        \rput(10,3){\rnode{xdept}{X}}
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        \pscircle(10,3){1.2ex}
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     \rput(3,5.25){\rnode{asalary}{\ }}
     \rput(6.25,4.25){\rnode{aprojcode}{\ }}
        \rput(5.25,5){\dianode{works}{Works\mbox{-}for}}
        \rput(5.25,2.5){\dianode{manages}{Manages}}
        \rput(7.25,3.75){\dianode{responsible2}{Resp\mbox{-}for}}
%         \rput(3.85,5.2){(0,n)}
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        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{project}{responsible1}
        \ncline{responsible1}{dept}
        \ncline{project}{responsible2}
        \ncline{responsible2}{orgunit}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{amanager}{xmanager}
        \ncline{tmanager}{xmanager}
        \ncline{->}{xmanager}{manager}
        \ncline[doubleline=true]{xmanager}{manager}
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        \ncline{->}{xdept}{orgunit}
        \ncline[doubleline=true]{xdept}{orgunit}
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\begin{tabular}{c}#1\end{tabular}\renewcommand{\arraystretch}{1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\semtitle{\parbox[t]{\semslidewidth}%
{The \DLRUS Temporal Description Logic}}

\semauthor{\textbf{Enrico Franconi}\\
  Department of Computer Science, University of Manchester \\ (joint
  work with \parbox[t]{\semslidewidth}{\textbf{Alessandro Artale,
    Milenko Mosurovi\'c,\\
    Frank Wolter, Michael Zakharyaschev)}}}

\semothertitle{FM Seminar, 12 October 2001}

\begin{document}

\makesemtitle

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

\begin{slide}{Summary}

\begin{itemize}
    \item Temporal conceptual modelling
    \item Decidability and complexity results
    \item Query containment under temporal constraints
    \item Decidability and complexity results
\end{itemize}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
\overlays{2}{%
\begin{slide}{A Temporal Database}    

{\footnotesize$\tt 
CompanyEmployee(\underline{name}, project, \underline{time})$} \\
{\footnotesize$\tt 
CompanyProject(\underline{project}, manager, department, \underline{time})$}

\vspace{1ex}

\MiniPage
{\scriptsize\begin{tabular}{|l|l|l|}
    \hline
    \multicolumn{3}{|c|}{\bf\tt CompanyEmployee}\\
    \hline
    {\tt name} & {\bf\tt project} & {\bf\tt time}  \\
    \hline
    gary & esprit-dwq & 1999 \\
    \hline
    $\cdots$ &  $\cdots$  &  $\cdots$ \\
    \hline
\end{tabular}}
{\scriptsize\begin{tabular}{|l|l|l|l|}
    \hline
    \multicolumn{4}{|c|}{\bf\tt CompanyProject} \\
    \hline
    {\bf\tt project} & {\bf\tt manager} & {\bf\tt department} & {\bf\tt time}\\
    \hline
    esprit-dwq & enrico & cs-uman & 1997 \\
    \hline
    esprit-dwq & enrico & cs-uman & 1998  \\
    \hline
    esprit-dwq & enrico & cs-uman & 1999  \\
    \hline
    $\cdots$ &  $\cdots$ &  $\cdots$ &  $\cdots$ \\
    \hline
\end{tabular}}

\vspace{1em}

\fromSlide{2}{\footnotesize Query = \\
``{\color{blue}For each time, tell me the
  projects in which Gary works, and their managers and departments}.''

\vspace{1em}

{\Large$\pi$}$_{\tt project,manager,dept.,time}${\Large$\sigma$}%
$\tt_{name=gary}~(CompanyEmployee$%
{\Large$~\bowtie$}%
$\tt_{project,time}~CompanyProject)$

\vspace{1em}

$\tt Query(x,y,z,t)\Leftrightarrow$
$\tt CompanyEmployee(gary,x,t)\land CompanyProject(x,y,z,t)$
}

\end{slide}}

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

\overlays{3}{%
\begin{slide}{The Temporal Semantics}

\begin{itemize}

\item We assume a linear, unbounded, and discrete {\em temporal
    structure\/}:

\smalltab {\color{blue}$\TS =({\TSS},<)$}

\vspace{1ex}

\smalltab where ${\color{blue}\TSS}$ is a set of time points
(or chronons), \\
\smalltab and ${\color{blue}<}$ is a strict linear order on ${\TSS}$.

\vspace{1em}

\fromSlide{2}{\item A {\em temporal interpretation} is a mapping from
  time points in ${\TSS}$ to standard DL interpretations, with a
  constant domain:}

\vspace{1ex}

\fromSlide{3}{\color{blue} 
\smalltab $\I \doteq \langle \TS,\ \dom,\ \Intt\cdot \rangle$

\item This is the TDB \emph{snapshot model}.}

\end{itemize}

\end{slide}}

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

\overlays{3}{
\begin{slide}{The \DLRUS Temporal Description Logics}
  
  \DLRUS is obtained by combining \DLR with the standard propositional
  linear temporal modal logic with {\bf since} and {\bf until}

\vspace{1ex}

\begin{center}
$\begin{array}{|rcl|}
  \hline
    R & \to & \top_{n}\ \mid\  R\!N
    \ \mid\ \Not R\ \mid\ R_{1}\And R_{2}\mid\ R_{1}\Or R_{2}\mid\
    \selects {i/n} C \fromSlide{3}{\mid} \\
    &&\fromSlide{3}{\color{blue}
    \STP R \mid\ \STM R \mid \ALLTP R \mid\ \ALLTM R
    \mid \NTP R \mid\ \NTM R \mid
    \ R_1{\cal U}R_2 \mid\ R_1{\cal S}R_2} \\
  &&\\
    %
    C & \to & \top\ \mid\  C\!N\ \mid\ \Not C\ \mid\  C_{1}\And
    C_{2}\  \mid\  C_{1}\Or
    C_{2}\  \mid\  \SOMER{i} R \mid\  \EXISTR{k}{i} R 
    \fromSlide{3}{\mid}\\
    &&\fromSlide{3}{\color{blue}
    \STP C \mid\ \STM C \mid\ALLTP C \mid\ \ALLTM C
    \mid\NTP C \mid\ \NTM C \mid\
    C_1{\cal U}C_2 \mid\ C_1{\cal S}C_2 }
  \\ \hline
\end{array}$
\end{center}

\vspace{1ex}

\begin{itemize}

\fromSlide*{2}{%
\item {\color{red}Note:\ }%
\parbox[t]{.8 \semslidewidth}{%
if $R$ is a binary relation (i.e., a \textit{role})\\
the standard \ALC expression $\some R C$ is equivalent to: \\
${\color{blue}\some R C} \equiv 
{\color{blue}\SOMER{1}{(R \And (\selects{2/2} C))}}$
}}

\fromSlide{3}{
\item \DLRUS is a fragment of the first-order temporal logic {\it
L}$^{\{\mbox{\footnotesize\bf since, until}\}}$
}

\end{itemize}

\end{slide}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
\overlays{2}{
\begin{slide}{\DLRUS Knowledge Bases}

\begin{itemize}
  
\item \textit{Atomic formulas:}
  \begin{itemize}
  
  \item $C_{1}\IsSubs C_{2}$
  
  \item $R_{1}\IsSubs R_{2}$ \hspace{6em} {\small($R_{1},R_{2}$ with
      same arity)}
  \end{itemize}

  \fromSlide{2}{
  \item \textit{General formulas:}\\
    if $\varphi$ and $\psi$ are formulas, then so are
    \begin{itemize}

    \item   $\Not\varphi$
      
    \item   $\varphi\land\psi$

    \item   $\varphi\lor\psi$

    \item   $\STP\varphi$

    \item   $\varphi\S\psi$

    \item   $\ldots$
  \end{itemize}}

\end{itemize}

\end{slide}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
\overlays{2}{
\begin{slide}{Reasoning}

\begin{itemize}
  
\item A formula $\varphi$ is (locally) {\em satisfiable} if there is
  $\I$ such that {\color{blue}$\I,t \models \varphi$}, for some $t$.
  
\item A formula $\varphi$ is globally {\em satisfiable} if there $\I$
  such that $\I,t \models \varphi$, for {\em every} $t$\\
  ({\color{blue}$\I\models\varphi$}).
  
\item $\psi$ globally {\em implies} $\varphi$ if we have
  $\I\models\varphi$ whenever $\I\models\psi$\\ ({\color{blue}$\psi
    \models\varphi$}).
  
\end{itemize}

\vspace{2em}

{\color{blue} \fromSlide{2}{All reasoning tasks can be reduced to
    (local) satisfiability of formulas.}}

\end{slide}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
\overlays{2}{
\begin{slide}{Conceptual Modelling}
  
\vspace{-1.5em}
\erfigure

    \fromSlide*{2}{\tiny
    
    ${\tt Works\mbox{-}for \IsSubsS
    \selects{emp/2}{Employee}\And\selects{act/2}{Project} 
    }$

${\tt Manages \IsSubsS
    \selects{man/2}{TopManager}\And\selects{prj/2}{Project} 
    }$

${\tt Employee \IsSubsS
\parbox[t]{\textwidth}{
${\tt 
\EXACTLYR{1}{worker}{(Name\And\selects{thename/2}{String})}\And}$
${\tt \EXACTLYR{1}{payee}{(Salary\And\selects{amount/2}{Integer})}}$}}$

${\tt \top \IsSubsS 
\ATMOSTR{1}{thename}{(Name\And\selects{worker/2}{Employee})}}$

${\tt Manager \IsSubsS Employee\And (AreaManager\Or TopManager)}$

${\tt AreaManager \IsSubsS Manager\And\Not TopManager}$

${\tt TopManager \IsSubsS Manager \And \EXACTLYR{1}{man} {Manages}}$

${\tt Project \IsSubsS
    \ATLEASTR{1}{act} {Works\mbox{-}for} \And
    \EXACTLYR{1}{prj} {Manages}}$
\\
    \vspace{-1ex}$\cdots$}

\end{slide}}

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

\begin{slide}{The \icom tool for Intelligent Conceptual Modelling}

\begin{itemize}
  
\item \icom is an advanced CASE tool which allows the user to design
  \textit{multiple} extended Entity-Relationship diagrams with inter-
  and intra-schema \textit{constraints}.
  
\item Complete logical reasoning is employed by the tool to:

\begin{itemize}
  \item verify the specification, 
  \item infer implicit facts, 
  \item devise stricter constraints, 
  \item and manifest any inconsistency.
  \end{itemize}

\end{itemize}

\vspace{3em}

\centerline{\large\texttt{http://www.cs.man.ac.uk/$\sim$franconi/icom/}}

\end{slide}

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

\overlays{2}{
\begin{slide}{Temporal Axioms}

\vspace{-1.5em}
\erfigure

{\footnotesize {\color{blue}A manager must be qualified, i.e. should
  have passed a period as an employee}:

${\footnotesize\tt Manager \IsSubsS Qualified ~{\cal S}~
  (Employee\And\Not Manager)}$

{\color{blue}Managers are the employees who do not work for
  a project (she/he just manages it)}:
  

\parbox[t]{\semslidewidth}{%
${\footnotesize\tt Employee \And\Not (\ATLEASTR{1} {emp}
  {Works\mbox{-}for}) \IsSubsS Manager}$
~{\color{red}\Large /}~
${\footnotesize\tt Manager \IsSubsS \Not (\ATLEASTR{1} {emp}
  {Works\mbox{-}for})}$ }

} 

%\vspace{1ex}

\fromSlide*{2}{ $\leadsto~$ {\color{blue}A top manager worked in a
    project before managing some (possibly different) project:}

${\tt \Sigma\models TopManager\IsSubsS  \STM \ATLEASTR{1}
      {emp} {(Works\mbox{-}for \And \selects{act}
      Project)}}$
      }

\end{slide}}

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

\begin{slide}{Timestamps}

\vspace{-1.5em}
{\scriptsize\psset{unit=1.1cm}%
     \begin{pspicture}(0.5,2)(12,6)
        \rput(2,5){\rnode{employee}{\psframebox{Employee \snap}}}
     \rput(2,5.8){\rnode{pslip}{\underline{\scriptsize%
     Name(String)}}}
     \rput(3.25,5.5){\rnode{salary}{\scriptsize Salary(Integer)}}
        \rput(5.25,3.75){\rnode{project}{\psframebox{Project}}}
     \rput(7,4.5){\rnode{projectcode}{\underline{\scriptsize%
     ProjectCode(String)}}}
        \rput(2,4){\rnode{manager}{\psframebox{Manager\temp}}}
        \rput(2.9,2.5){\rnode{tmanager}{\psframebox{TopManager}}}
        \rput(1.1,2.5){\rnode{amanager}{\psframebox{AreaManager}}}
        \rput(9,2.25){\rnode{dept}{\psframebox{Department \snap}}}
        \rput(11,2.25){\rnode{sig}{\psframebox{InterestGroup}}}
        \rput(10,3.75){\rnode{orgunit}{\psframebox{OrganisationalUnit}}}
        \rput(2,3.25){\rnode{xmanager}{X}}
        \rput(10,3){\rnode{xdept}{X}}
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        \pscircle(10,3){1.2ex}
     \rput(2,5.55){\rnode{apslip}{\ }}
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     \rput(6.25,4.25){\rnode{aprojcode}{\ }}
        \rput(5.25,5){\dianode{works}{Works\mbox{-}for\temp}}
        \rput(5.25,2.5){\dianode{manages}{Manages}}
        \rput(7.25,3.75){\dianode{responsible2}{Resp\mbox{-}for \snap}}
%         \rput(3.85,5.2){(0,n)}
        \rput(5.55,4.5){(1,n)}
        \rput(4.1,2.2){(1,1)}
        \rput(5.55,3){(1,1)}
        \ncline{employee}{works}
        \ncline{works}{project}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{project}{responsible1}
        \ncline{responsible1}{dept}
        \ncline{project}{responsible2}
        \ncline{responsible2}{orgunit}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{amanager}{xmanager}
        \ncline{tmanager}{xmanager}
        \ncline{->}{xmanager}{manager}
        \ncline[doubleline=true]{xmanager}{manager}
        \ncline{dept}{xdept}
        \ncline{sig}{xdept}
        \ncline{->}{xdept}{orgunit}
        \ncline[doubleline=true]{xdept}{orgunit}
        \ncline{->}{manager}{employee}
       \ncline{employee}{apslip}
       \ncline{employee}{asalary}
       \ncline{project}{aprojcode}
       \pscircle[fillstyle=solid,fillcolor=white](2,5.55){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](3,5.25){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](6.25,4.25){.7ex}
    \end{pspicture}}

{\small
\begin{itemize}
\item ${\tt{\color{red} Employee} \IsSubsS (\ALLTP Employee) \And
    (\ALLTM Employee)}$

\item ${\tt{\color{red} Department} \IsSubsS (\ALLTP Department) \And
    (\ALLTM Department)}$

\item ${\tt{\color{blue} Manager} \IsSubsS (\STP\Not Manager)\Or(\STM\Not
    Manager)}$

\item ${\tt{\color{red} Responsible\mbox{-}for} \IsSubsS
      (\ALLTP Responsible\mbox{-}for) \And
      (\ALLTM Responsible\mbox{-}for)
  }$

\item ${\tt{\color{blue} Works\mbox{-}for} \IsSubsS
    (\STP\Not Works\mbox{-}for)\Or(\STM\Not
    Works\mbox{-}for)}$


  \end{itemize}
}

\end{slide}

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

\overlays{5}{
\begin{slide}{Inference with Timestamps}

\def\snapd{\mbox{\color{red}$\clubsuit$}}
\def\tempd{\mbox{\color{blue}$\clubsuit$}}

\vspace{-1.5em}
{\scriptsize\psset{unit=1.1cm}%
     \begin{pspicture}(0.5,2)(12,6)
        \rput(2,5){\rnode{employee}{\psframebox{Employee \snap}}}
     \rput(2,5.8){\rnode{pslip}{\underline{\scriptsize%
     Name(String)}}}
     \rput(3.25,5.5){\rnode{salary}{\scriptsize Salary(Integer)}}
        \rput(5.25,3.75){\rnode{project}{\psframebox{Project
            \fromSlide{5}{\snapd}}}}
     \rput(7,4.5){\rnode{projectcode}{\underline{\scriptsize%
     ProjectCode(String)}}}
        \rput(2,4){\rnode{manager}{\psframebox{Manager\temp}}}
        \rput(2.9,2.5){\rnode{tmanager}{\psframebox{TopManager
            \fromSlide{2}{\tempd}}}}
        \rput(1.1,2.5){\rnode{amanager}{\psframebox{AreaManager
            \fromSlide{2}{\tempd}}}}
        \rput(9,2.25){\rnode{dept}{\psframebox{Department \snap}}}
        \rput(11,2.25){\rnode{sig}{\psframebox{InterestGroup
            \fromSlide{4}{\snapd}}}}
        \rput(10,3.75){\rnode{orgunit}{\psframebox{OrganisationalUnit
            \fromSlide{3}{\snapd}}}}
        \rput(2,3.25){\rnode{xmanager}{X}}
        \rput(10,3){\rnode{xdept}{X}}
        \pscircle(2,3.25){1.2ex}
        \pscircle(10,3){1.2ex}
     \rput(2,5.55){\rnode{apslip}{\ }}
     \rput(3,5.25){\rnode{asalary}{\ }}
     \rput(6.25,4.25){\rnode{aprojcode}{\ }}
        \rput(5.25,5){\dianode{works}{Works\mbox{-}for\temp}}
        \rput(5.25,2.5){\dianode{manages}{Manages}}
        \rput(7.25,3.75){\dianode{responsible2}{Resp\mbox{-}for \snap}}
%         \rput(3.85,5.2){(0,n)}
        \rput(5.55,4.5){(1,n)}
        \rput(4.1,2.2){(1,1)}
        \rput(5.55,3){(1,1)}
        \ncline{employee}{works}
        \ncline{works}{project}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{project}{responsible1}
        \ncline{responsible1}{dept}
        \ncline{project}{responsible2}
        \ncline{responsible2}{orgunit}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{amanager}{xmanager}
        \ncline{tmanager}{xmanager}
        \ncline{->}{xmanager}{manager}
        \ncline[doubleline=true]{xmanager}{manager}
        \ncline{dept}{xdept}
        \ncline{sig}{xdept}
        \ncline{->}{xdept}{orgunit}
        \ncline[doubleline=true]{xdept}{orgunit}
        \ncline{->}{manager}{employee}
       \ncline{employee}{apslip}
       \ncline{employee}{asalary}
       \ncline{project}{aprojcode}
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       \pscircle[fillstyle=solid,fillcolor=white](6.25,4.25){.7ex}
    \end{pspicture}}

{\small  

\fromSlide*{2}{\begin{itemize}

\item Sub-entities of temporary entities must be temporary.

\fromSlide*{3}{\item Super-entities of snapshot entities must be snapshot.}

\fromSlide*{4}{\item A schema is inconsistent if exactly one of a whole
    set of snapshot partitioning sub-entities is temporary.}

\fromSlide*{5}{\item Participants of snapshot relationships must be
    snapshot entities.}

\end{itemize}}}

\end{slide}}

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

\overlays{2}{
\begin{slide}{Timestamping Attributes}

\vspace{-1.5em}
{\footnotesize\psset{unit=1.1cm}%
     \begin{pspicture}(0.5,2)(12,6)
        \rput(2,5){\rnode{employee}{\psframebox{Employee}}}
     \rput(2,5.8){\rnode{pslip}{\underline{\scriptsize%
     Name(String)}\fromSlide*{2}{\snap}}}
     \rput(3.25,5.5){\rnode{salary}{\scriptsize Salary(Integer)}}
        \rput(5.25,3.75){\rnode{project}{\psframebox{Project}}}
     \rput(7,4.5){\rnode{projectcode}{\underline{\scriptsize%
     ProjectCode(String)}}}
        \rput(2,4){\rnode{manager}{\psframebox{Manager}}}
        \rput(2.9,2.5){\rnode{tmanager}{\psframebox{TopManager}}}
        \rput(1.1,2.5){\rnode{amanager}{\psframebox{AreaManager}}}
        \rput(9,2.25){\rnode{dept}{\psframebox{Department}}}
        \rput(11,2.25){\rnode{sig}{\psframebox{InterestGroup}}}
        \rput(10,3.75){\rnode{orgunit}{\psframebox{OrganisationalUnit}}}
        \rput(2,3.25){\rnode{xmanager}{X}}
        \rput(10,3){\rnode{xdept}{X}}
        \pscircle(2,3.25){1.2ex}
        \pscircle(10,3){1.2ex}
     \rput(2,5.55){\rnode{apslip}{\ }}
     \rput(3,5.25){\rnode{asalary}{\ }}
     \rput(6.25,4.25){\rnode{aprojcode}{\ }}
        \rput(5.25,5){\dianode{works}{Works\mbox{-}for}}
        \rput(5.25,2.5){\dianode{manages}{Manages}}
        \rput(7.25,3.75){\dianode{responsible2}{Resp\mbox{-}for}}
%         \rput(3.85,5.2){(0,n)}
        \rput(5.55,4.5){(1,n)}
        \rput(4.1,2.2){(1,1)}
        \rput(5.55,3){(1,1)}
        \ncline{employee}{works}
        \ncline{works}{project}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{project}{responsible1}
        \ncline{responsible1}{dept}
        \ncline{project}{responsible2}
        \ncline{responsible2}{orgunit}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{amanager}{xmanager}
        \ncline{tmanager}{xmanager}
        \ncline{->}{xmanager}{manager}
        \ncline[doubleline=true]{xmanager}{manager}
        \ncline{dept}{xdept}
        \ncline{sig}{xdept}
        \ncline{->}{xdept}{orgunit}
        \ncline[doubleline=true]{xdept}{orgunit}
        \ncline{->}{manager}{employee}
       \ncline{employee}{apslip}
       \ncline{employee}{asalary}
       \ncline{project}{aprojcode}
       \pscircle[fillstyle=solid,fillcolor=white](2,5.55){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](3,5.25){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](6.25,4.25){.7ex}
    \end{pspicture}}

\begin{itemize}

\item ${\tt Employee \IsSubsS
\EXACTLYR{1}{worker}{(Name\And\selects{nam/2}{String})}}$

\item ${\tt \top \IsSubsS
\ATMOSTR{1}{nam}{(Name\And\selects{worker/2}{Employee})}}$

\fromSlide*{2}{\item ${\tt Employee  \IsSubsS
\ATLEASTR{1}{worker}{\ALLTE {\color{red} Name}}}$
\hfill {\it(snapshot attribute\/)}}


  \end{itemize}


\end{slide}}

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

\overlays{2}{
\begin{slide}{Temporal Cardinalities}

\vspace{-1.5em}
{\footnotesize\psset{unit=1.1cm}%
     \begin{pspicture}(0.5,2)(12,6)
        \rput(2,5){\rnode{employee}{\psframebox{Employee}}}
     \rput(2,5.8){\rnode{pslip}{\underline{\scriptsize%
     Name(String)}}}
     \rput(3.25,5.5){\rnode{salary}{\scriptsize Salary(Integer)}}
        \rput(5.25,3.75){\rnode{project}{\psframebox{Project}}}
     \rput(7,4.5){\rnode{projectcode}{\underline{\scriptsize%
     ProjectCode(String)}}}
        \rput(2,4){\rnode{manager}{\psframebox{Manager}}}
        \rput(2.9,2.5){\rnode{tmanager}{\psframebox{TopManager}}}
        \rput(1.1,2.5){\rnode{amanager}{\psframebox{AreaManager}}}
        \rput(9,2.25){\rnode{dept}{\psframebox{Department}}}
        \rput(11,2.25){\rnode{sig}{\psframebox{InterestGroup}}}
        \rput(10,3.75){\rnode{orgunit}{\psframebox{OrganisationalUnit}}}
        \rput(2,3.25){\rnode{xmanager}{X}}
        \rput(10,3){\rnode{xdept}{X}}
        \pscircle(2,3.25){1.2ex}
        \pscircle(10,3){1.2ex}
     \rput(2,5.55){\rnode{apslip}{\ }}
     \rput(3,5.25){\rnode{asalary}{\ }}
     \rput(6.25,4.25){\rnode{aprojcode}{\ }}
        \rput(5.25,5){\dianode{works}{Works\mbox{-}for}}
        \rput(5.25,2.5){\dianode{manages}{Manages}}
        \rput(7.25,3.75){\dianode{responsible2}{Resp\mbox{-}for}}
%         \rput(3.85,5.2){(0,n)}
        \rput(5.55,4.5){(1,n)}
        \rput(4.1,2.2){(1,1)}
        \fromSlide*{2}{{\color{red}\rput(4.1,2.8){[0,5]}}}
        \rput(5.55,3){(1,1)}
        \ncline{employee}{works}
        \ncline{works}{project}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{project}{responsible1}
        \ncline{responsible1}{dept}
        \ncline{project}{responsible2}
        \ncline{responsible2}{orgunit}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{amanager}{xmanager}
        \ncline{tmanager}{xmanager}
        \ncline{->}{xmanager}{manager}
        \ncline[doubleline=true]{xmanager}{manager}
        \ncline{dept}{xdept}
        \ncline{sig}{xdept}
        \ncline{->}{xdept}{orgunit}
        \ncline[doubleline=true]{xdept}{orgunit}
        \ncline{->}{manager}{employee}
       \ncline{employee}{apslip}
       \ncline{employee}{asalary}
       \ncline{project}{aprojcode}
       \pscircle[fillstyle=solid,fillcolor=white](2,5.55){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](3,5.25){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](6.25,4.25){.7ex}
    \end{pspicture}}

\vspace{\topsep}

\begin{itemize}

\item ${\tt TopManager \IsSubsS \EXACTLYR{1}{man} {Manages}}$

\fromSlide*{2}{\item ${\tt TopManager \IsSubsS \ATMOSTR{5}{man} {\STE Manages}}$
\hfill{\it ({\color{red}max. temporal cardinality}\/)}}


\end{itemize}

\end{slide}}

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

\overlays{5}{
\begin{slide}{Dynamic Entities}

\vspace{-1.5em}
\erfigure

\vspace{3em}

\begin{itemize}

\fromSlide{2}{
\item ${\tt TopManager \IsSubsS \STM AreaManager}$ \hfill
{\it (Dynamic Extension\/)}}

\fromSlide{3}{
\item ${\tt AreaManager \IsSubsS \ALLTM \Not TopManager}$ \hfill
{\it (Uni-directional Dynamic Evolution\/)}}

\fromSlide{4}{
\item ${\tt Department \IsSubsS \ALLTE \Not Interest\mbox{-}Group}$ \hfill
{\it (Incompatible Dynamic Entities\/)}}

\fromSlide{5}{
\item ${\tt \ALLTE \Not(Employee \IsSubs \bot)}$ \hfill {\it (Global
    Consistency\/)}}

\end{itemize}


\end{slide}}

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

\overlays{2}{
\begin{slide}{Schema Evolution}

\vspace{-1.5em}
{\footnotesize\psset{unit=1.1cm}%
     \begin{pspicture}(0.5,2)(12,6)
        \rput(2,5){\rnode{employee}{\psframebox{Employee}}}
     \rput(2,5.8){\rnode{pslip}{\underline{\scriptsize%
     Name(String)}}}
     \rput(3.25,5.5){\rnode{salary}{\scriptsize 
         \onlySlide*{1}{Salary(Integer)}%
         \onlySlide*{2}{\psovalbox[boxsep=false,linecolor=red]{Salary(Integer)}}
       }}
        \rput(5.25,3.75){\rnode{project}{\psframebox{Project}}}
     \rput(7,4.5){\rnode{projectcode}{\underline{\scriptsize%
     ProjectCode(String)}}}
        \rput(2,4){\rnode{manager}{\psframebox{Manager}}}
        \rput(2.9,2.5){\rnode{tmanager}{\psframebox{TopManager}}}
        \rput(1.1,2.5){\rnode{amanager}{\psframebox{AreaManager}}}
        \rput(9,2.25){\rnode{dept}{\psframebox{Department}}}
        \rput(11,2.25){\rnode{sig}{\psframebox{InterestGroup}}}
        \rput(10,3.75){\rnode{orgunit}{\psframebox{OrganisationalUnit}}}
        \rput(2,3.25){\rnode{xmanager}{X}}
        \rput(10,3){\rnode{xdept}{X}}
        \pscircle(2,3.25){1.2ex}
        \pscircle(10,3){1.2ex}
     \rput(2,5.55){\rnode{apslip}{\ }}
     \rput(3,5.25){\rnode{asalary}{\ }}
     \rput(6.25,4.25){\rnode{aprojcode}{\ }}
        \rput(5.25,5){\dianode{works}{Works\mbox{-}for}}
        \rput(5.25,2.5){\dianode{manages}{Manages}}
        \rput(7.25,3.75){\dianode{responsible2}{Resp\mbox{-}for}}
%         \rput(3.85,5.2){(0,n)}
        \rput(5.55,4.5){(1,n)}
        \rput(4.1,2.2){(1,1)}
        \rput(5.55,3){(1,1)}
        \ncline{employee}{works}
        \ncline{works}{project}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{project}{responsible1}
        \ncline{responsible1}{dept}
        \ncline{project}{responsible2}
        \ncline{responsible2}{orgunit}
        \ncline{tmanager}{manages}
        \ncline{manages}{project}
        \ncline{amanager}{xmanager}
        \ncline{tmanager}{xmanager}
        \ncline{->}{xmanager}{manager}
        \ncline[doubleline=true]{xmanager}{manager}
        \ncline{dept}{xdept}
        \ncline{sig}{xdept}
        \ncline{->}{xdept}{orgunit}
        \ncline[doubleline=true]{xdept}{orgunit}
        \ncline{->}{manager}{employee}
       \ncline{employee}{apslip}
       \ncline{employee}{asalary}
       \ncline{project}{aprojcode}
       \pscircle[fillstyle=solid,fillcolor=white](2,5.55){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](3,5.25){.7ex}
       \pscircle[fillstyle=solid,fillcolor=white](6.25,4.25){.7ex}
    \end{pspicture}}

\vspace{2em}

{\color{blue}As soon as the organisation does not
    include interest groups anymore, the salary of top managers should
    be in the ``low amount'' class}:

\fromSlide{2}{  
  $\tt\ALLTE($\parbox[t]{\semslidewidth}{$\tt{\color{red}\Not}(InterestGroup
    ~{\color{blue}\IsSubs}~\bot)
    {\color{red}\lor}$ \\ $\tt\ALLTP(\ATLEASTR{1}{amount}{(Salary\And
      \selects{payee/2}{TopManager})~{\color{blue}\IsSubs}~ LowAmount))}$}
}

\end{slide}}

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

\overlays{4}{
\begin{slide}{Computational Properties of \DLRUS}

\begin{itemize}

\fromSlide{2}{
\item Global satisfiability of \DLRUS theories with only atomic
  formulas is {\it undecidable}.  }

\fromSlide{3}{
\item The {\it maximal} decidable fragment is the monodic
fragment \DLRUSm:
%
\begin{center}
$\begin{array}{rcl}
    R & \to & \top_{n}\ \mid\  R\!N
    \ \mid\ \Not R\ \mid\ R_{1}\And R_{2}\mid\ R_{1}\Or R_{2}\mid\
    \selects {i/n} E \mid\\
    &&\parbox[t]{30em}{$
    \STP R \mid\ \STM R \mid \ALLTP R \mid\ \ALLTM R
    \mid \NTP R \mid\ \NTM R \mid
    \ R_1{\cal U}R_2 \mid\ R_1{\cal S}R_2$}
    \hspace{-30em}{\color{blue}\rule[4pt]{30em}{1pt}}
    \\
    %
    E & \to & \top\ \mid\  E\!N\ \mid\ \Not E\ \mid\  E_{1}\And
    E_{2}\  \mid\  E_{1}\Or
    E_{2}\  \mid\  \EXISTR{k}{i} R \mid\\
    &&{
    \STP E \mid\ \STM E \mid\ALLTP E \mid\ \ALLTM E
    \mid\NTP E \mid\ \NTM E \mid\
    E_1{\cal U}E_2 \mid\ E_1{\cal S}E_2 }
\end{array}$
\end{center}
}

\fromSlide{4}{%
\item Formula satisfiability in \DLRUSm  is {\it EXPSPACE-complete};\\
  logical implication of \DLRUSm theories with only atomic formulas is
  {\it EXPTIME-complete}.}

\end{itemize}

\end{slide}}

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

\begin{slide}{Queries under \DLRUS constraints}
  
  Disjunction of conjunctive queries referring to predicates
  constrained by a \DLRUS theory.

% \fromSlide{2}{\item \smalltab $
% \mathcal{Q}(\overrightarrow{x}) \longleftarrow
% \bigvee_{j}\mathcal{Q}_{j}%
% (\overrightarrow{x},\overrightarrow{y}_{j},\overrightarrow{c}_{j})
% $

%     \vspace{1ex}

% \noindent where $\mathcal{Q}_{j}$ is a conjunctive query

%     \vspace{1ex}

% \smalltab $
% \mathcal{Q}_{j}%
% (\overrightarrow{x},\overrightarrow{y}_{j},\overrightarrow{c}_{j}) \equiv
% \bigwedge_{i} P_{j}^{i}%
% (\overrightarrow{x}_{j}^{i},\overrightarrow{y}_{j}^{i},\overrightarrow{c}_{j}^{i})
% $

%     \vspace{1ex}

% \noindent where $P_{j}^{i}$ are \DLRUS entity or relationship names.}

\vspace{1em}

Example:

${\tt Q_{1}(x,y) \ \mbox{:-}\ \parbox[t]{\textwidth}%
{$\tt {\Large(}\STM \ATLEASTR{1} {1}
{Works\mbox{-}for}{\Large)}(x) \wedge Manages(x,z)\wedge \\
Resp\mbox{-}for(y,z)\wedge\Not InterestGroup(y) $}}$

\end{slide}

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

\overlays{5}{
\begin{slide}{Semantics of Evaluation and Containment}
  
  The \textit{\color{blue} evaluation} at moment $t$ of a query
  $\mathcal{Q}$ of arity $n$
  given a \DLRUS theory $\Sigma$ \\
  over a temporal model $\I$ satisfying $\Sigma$


%\vspace{1em}

\fromSlide{2}{
is the set {\sc ans}$(\mathcal{Q},\I(t))$ 
composed by  $n$-tuples $\overrightarrow{o}$
such that}

\fromSlide{3}{\tab
{\sc ans}$(\mathcal{Q},\I(t)) =
\{
  \overrightarrow{o}\mid %\\
  {\I},t\models
  \bigvee_{j}\exists\overrightarrow{y_{j}}\per \mathcal{Q}_{j}
  (\overrightarrow{o},\overrightarrow{y_{j}},\overrightarrow{c_{j}})
\}
$
}

\vspace{4em}

\fromSlide{4}{$\mathcal{Q}_1$ \textit{\color{blue} is contained} in
  $\mathcal{Q}_2$ with respect to a \DLRUS theory $\Sigma$

  \fromSlide{5}{%
    if and only if  for every model $\I$ satisfying $\Sigma$ and every $t$

\tab
{\sc ans}$(\mathcal{Q}_1,\I(t))\subseteq\ ${\sc ans}$(\mathcal{Q}_2,\I(t))$%
} }


\end{slide}}

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

\overlays{2}{
\begin{slide}{Query Containment}
  
  {\bf Theorem:} Containment of disjunctions of conjunctive queries
  under \DLRUSm constraints is decidable in 2EXPTIME and it is
  EXPSPACE-hard.

  \vspace{1em}

  Example:

  ${\tt Q_{1}(x,y) \ \mbox{:-}}$\\
\smalltab  ${\tt (\STM \ATLEASTR{1} {1}
  {Works\mbox{-}for})(x) \wedge Manages(x,z)\wedge }$ \\
\smalltab  ${\tt   Resp\mbox{-}for(y,z)\wedge\Not InterestGroup(y) }$

${\tt Q_{2}(x,y) \ \mbox{:-}}$\\
\smalltab  ${\tt \Not AreaManager(x) \wedge
Manages(x,z)\wedge Project(z)\wedge }$\\
\smalltab  ${\tt Resp\mbox{-}for(y,z)\wedge
Department(y)
}$

\vspace{1em}

\fromSlide{2}{\color{red}\begin{center} ${\tt Q_{1} \subseteq Q_{2} \tab
Q_{2} \subseteq Q_{1}}$ \end{center}}

\end{slide}}

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

\begin{slide}{\fbox{Conclusions}}

\begin{itemize}

\item We have introduced \DLRUS, a novel temporal logic for temporal
conceptual modelling, and its monodic fragment.

\item Computational properties for the logical implication problem and
for the query containment problem are proved.

This the first decidability and complexity result we are aware
of on containment of temporal conjunctive queries under expressive
constraints.

\item The \DLRUS logic has the ability to express standard temporal
  conceptual models and a large class of temporal integrity
  constraints.

\end{itemize}


\vspace{5em}

%~\hfill\fbox{\it Made with \LaTeXe}

\end{slide}

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\end{document}

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