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  \vspace{0.25cm}
  if swift(Tim) and tall(Tim) then good\_player(Tim) \\
  ~~~~~~~~~~ with priority\_and\_credibility (1.0,~0.2)\\
  if swift and tall then good\_player \\
  ~~~~~~~~~~ with priority\_and\_credibility (0.8,~0.7)\\
  \vspace{0.25cm}
}

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\title[Introducing Priorities in RFuzzy]{Introducing Priorities in RFuzzy: \\ Syntax and Semantics}
\author[]{Victor Pablos-Ceruelo and Susana Muñoz-Hernandez ~vpablos, susana~@~fi.upm.es }
\institute[Babel Group, UPM, Spain]{Babel Group, Facultad de Informática \\ Universidad Politécnica de Madrid (T.U. of Madrid), Spain \\ http://babel.ls.fi.upm.es/ }
\date[CMMSE'2011, Alicante, Spain]{CMMSE'2011, Alicante, Spain}


\pgfdeclareimage[height=1.4cm]{logo-left}{logos/logo-left}
\pgfdeclareimage[height=1cm]{logo-right}{logos/logo-right}

\input{presentation_beamer_template}


\newcommand{\neck}{\leftarrow}
\begin{document}
\frame{\titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% motivation

% 0 slide: Content
% \begin{frame}
%   \frametitle{Content}
%   \framesubtitle{RFuzzy Approach}
%   \begin{itemize}
%   \item Motivation
%   \item RFuzzy {\bf Syntax}
%     \begin{itemize}
%     \item Rules
%     \item Defaults
%     \item Types
%     \end{itemize}
%   \item RFuzzy Operational {\bf Semantics}
%   \item Example
%   \item Conclusions
%   \end{itemize}
% \end{frame}

% \begin{frame}
%   \frametitle{Overview}
%   \tableofcontents
% \end{frame}

\begin{frame}
  \frametitle{Introduction}

It was Lotfi Zadeh in 1965 who introduced fuzzy set theory 
in the paper ``Fuzzy Sets'' %\cite{DBLP:journals/iandc/Zadeh65}, 
and its existence was justified in the paper 
"Is there a need for fuzzy logic?" % \cite{Zadeh20082751}.
We just outline here some ideas needed to understand 
our contribution.

\pgfdeclareimage[height=4cm]{lofti-a-zadeh}{Lotfi_A._Zadeh_2004.jpg}
\begin{center}
\pgfuseimage{lofti-a-zadeh}
\end{center}
\end{frame}

\begin{frame}
  \frametitle{Introduction}
  \only<1>{\framesubtitle{Linguistic Variables}}
  \only<2>{\framesubtitle{Inference}}
  \only<3>{\framesubtitle{Defuzzification}}
\begin{columns}
\column{.5\textwidth}

  \begin{figure}
    \centering
    \includegraphics[height=1.5cm]{Fuzzy_logic_temperature_en} 
    \caption{Temperature is a linguistic var} 
    % variable and (here) takes the values cold, warm and hot.}
    \label{fig:temperature}
    \vspace{-0.5cm}
  \end{figure}

  \visible<2->{
    \underline{Rules:} \\
    if temperature(warm) then fan\_speed(normal) \\
  }

 \visible<3->{
   \vspace{0.5cm}
   \begin{tabular}{|c|c|}
     \hline
     Fan\_speed & Voltage \\
     \hline
     fast & 10 volts \\
     normal & 5 volts \\
     slow & 2.5 volts \\
     stop & 0 volts \\
     \hline
   \end{tabular}
}

  \column{.5\textwidth}
  \vspace{2cm}
  \begin{tikzpicture}[every node/.style=draw, rounded corners, fill=gray!15] % , node distance=10mm   
    \visible<1->{
    \node (first) at (0,0) [text width=2.5cm]{\scriptsize temperature is 25 degrees };
  }
%
     \visible<1->{
       \node (second) at (0, -1.5) {\scriptsize temperature(warm)};
       \draw [->] (first) to (second) node [right, midway, draw=none] {fuzzification};
       %\arc{first}{second}
     }
%
     \visible<2->{
       \node (third) at (0, -2.5) {\scriptsize fan\_speed(normal)};
       \draw [->] (second) to (third) node [right, midway, draw=none] {inference};
       %\arc{first}{second}
     }

     \visible<3->{
       \node (fourth) at (0, -3.5) {\scriptsize feed the fan with 5 volts};
       \draw [->] (third) to (fourth) node [right, midway, draw=none] {defuzzification};
       %\arc{first}{second}
     }

    \end{tikzpicture}
\end{columns}
\end{frame}

\begin{frame}
  \frametitle{Introduction}
  \framesubtitle{Type-1 and Type-2 fuzzy sets and systems}

\begin{columns}
  \column{.5\textwidth}
  \vspace{0.25cm}
  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & 0.9 & 0.9 & 0.2 \\
    \hline
    John & 0.8 & 0.7 & 0.7 \\
    \hline
    Sarah & 0.7 & 0.8 & 0.7 \\
  \end{tabular}

  \column{.5\textwidth}
  \vspace{0.25cm}
  \only<1>{\underline{Type-1 fuzzy Rules:}}
  \only<2>{\underline{Type-2 fuzzy Rules:}}
  \\
  \vspace{0.25cm}
  if swift and tall then good\_player \\
  \only<2>{with credibility 0.2}
  ~~~\\
  %\vspace{0.25cm}
  aggregator: minimum\\
  \only<2>{conjunctor: minimum (G\"odel Logic)}
  ~\\
%  \only<2>{implicator:  (G\"odel Logic)}

  % Suppose we use minimum to aggregate the truth values from
  % the fuzzy predicates swift and tall. In this case:

\end{columns}

  \begin{eqnarray}
    \only<1>{Tim: &min(0.9, 0.9) = 0.9 \neq 0.2 ~~~~ ERROR !! \nonumber \\ }
    \only<1>{John: &min(0.8, 0.7) = 0.7 = 0.7 \nonumber  \\}
    \only<1>{Sarah: &min(0.7, 0.8) = 0.7 = 0.7 \nonumber }
    \only<2>{Tim: &min(0.2,~min(0.9, 0.9)) = 0.2 \leq 0.2 \nonumber \\ }
    \only<2>{John: &min(0.2,~min(0.8, 0.7)) = 0.2 \leq 0.7 \nonumber \\}
    \only<2>{Sarah: &min(0.2,~min(0.7, 0.8)) = 0.2 \leq 0.7 \nonumber }
  \end{eqnarray}
  
  % \underline{Type-1 Rules:} \\
  % \vspace{0.25cm}
  % if temperature(warm) then fan\_speed(normal) 

\end{frame}

\begin{frame}
  \frametitle{Introduction}
  \framesubtitle{Multi-Adjoint Semantics}

  Is there a way of relating real-world data and rule's credibility? 
  Yes, by means of adjoint pairs. (Works by Ojeda-Aciego et al)
% any structure allowing us to determine credibility from real-world data?
%  $\rightarrow$ multi-adjoint algebra. \\
\vspace{0.25cm}
  % Basically it defines the properties that two connectives, $\&$ and $\leftarrow$,
  % must obey to be able to obtain credibility from real world data and 
  % vice versa:

Departing from a Poset (a partially ordered set) $<~P,~\leq~>$
and introducing a pair of operations $(~\&,~\leftarrow~)$,
we say that the operations form an adjoint pair if 

\begin{enumerate}
\item[(1)] $\&$ is increasing in both arguments, 
\item[(2)] $\leftarrow$ is increasing in its first argument and decreasing in the second one and
\item[(3)] (\define{the adjoint property})
for any $x, y, z \in P$ we have that 
$z \leq (x \leftarrow y)$ holds if and only if $z~\&~y~\leq~x$.
\end{enumerate}
\end{frame}

\begin{frame}
  \frametitle{Introduction}
  \framesubtitle{Multi-Adjoint Semantics II}

  The interesting point about the adjoint property is that for a fuzzy rule 
  % 
  \begin{center}
    if B then A with credibility z
  \end{center}
  %
  where truth values for B and A are y and x respectively
  we can evaluate x from y and z or z from x and y:

  $$z~\leq~(x~\leftarrow~y)~~~~~~\iff~~~~~~z~\&~y~\leq~x$$

  %defines the inference rule
  % evaluation \inlineFixme{vpc}{Look for a better word}
  %$\frac{(B, y)~~~~(A \leftarrow B, z)}{(A, x)}$ 

\end{frame}

\begin{frame}
  \frametitle{Introduction}
  \framesubtitle{Multi-Adjoint Semantics: Example of usage}

\begin{columns}
  \column{.5\textwidth}
  \vspace{0.25cm}
  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & 0.9 & 0.9 & 0.2 \\
    \hline
    John & 0.8 & 0.7 & 0.7 \\
    \hline
    Sarah & 0.7 & 0.8 & 0.7 \\
  \end{tabular}

  \column{.5\textwidth}
%  \vspace{0.25cm}
  if swift and tall then good\_player \\
  with credibility z \\
  ~~~\\
  aggregator: minimum\\
  conjunctor: minimum (G\"odel Logic) \\
  implicator:  $\hat{\leftarrow}(x,y) = $ $x~\text{if}~x < y ~\text{else}~1$ (G\"odel Logic)

  % Suppose we use minimum to aggregate the truth values from
  % the fuzzy predicates swift and tall. In this case:

\end{columns}

  \begin{eqnarray}
    Tim: & min(0.9, 0.9) = 0.9 ; \text{~~since~~} 0.9 > 0.2 \text{~~then~~} 0.2 \nonumber \\ 
    John: & min(0.8, 0.7) = 0.7 ; \text{~~since~~} 0.7 = 0.7 \text{~~then~~} 1 \nonumber \\
    Sarah: & min(0.7, 0.8) = 0.7 ; \text{~~since~~} 0.7 = 0.7 \text{~~then~~} 1 \nonumber \\
   % \nonumber \\
    & z = infimum(0.2,~1,~1) = 0.2 \nonumber
  \end{eqnarray}
  
  % \underline{Type-1 Rules:} \\
  % \vspace{0.25cm}
  % if temperature(warm) then fan\_speed(normal) 

\end{frame}

\begin{frame}
  \frametitle{Motivation}
  \framesubtitle{Credibility inferred is not good to recover the original values}
  % makes the rule less accurate}

\begin{columns}
  \column{.5\textwidth}
  \underline{Real-world values:}
  \vspace{0.25cm}

  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & 0.9 & 0.9 & 0.2 \\
    \hline
    John & 0.8 & 0.7 & 0.7 \\
    \hline
    Sarah & 0.7 & 0.8 & 0.7 \\
  \end{tabular}
  % 
  \column{.5\textwidth}
  \underline{Computed values:}
  \vspace{0.25cm}

  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & 0.9 & 0.9 & 0.2 \\
    \hline
    John & 0.8 & 0.7 & 0.2 \\
    \hline
    Sarah & 0.7 & 0.8 & 0.2 \\
  \end{tabular}
\end{columns}  

  \vspace{0.5cm}

  \underline{Rule \& Credibility computed:}\\
  ~~~\\
  if swift and tall then good\_player with credibility z \\
  ~~~\\
1) We compute the credibility for the rule $\rightarrow$  z = 0.2 \\
2) We compute the value for good\_player as if we do not know it. \\
\vspace{1.5cm}


\end{frame}

\begin{frame}
  \frametitle{The problem}
  \framesubtitle{How to obtain a higher credibility for the rules?}

  Proposal 1: two rules. \\
  \vspace{0.25cm}

  if swift(Tim) and tall(Tim) then good\_player(Tim) with credibility 0.2 \\
  if swift and tall then good\_player with credibility 0.7 \\
  \vspace{0.25cm}

  Problem:
%  \vspace{0.25cm}
  \begin{eqnarray}
    Tim: & sup (min(0.2,~min(0.9, 0.9)), min(0.7,~min(0.9, 0.9))) = 0.7  >  0.2 \nonumber
  \end{eqnarray}

  So the solution is not valid because 0.7 $>$ 0.2 while our answer needs to be 
  smaller or equal to the real-world one.

\end{frame}

\begin{frame}
  \frametitle{The problem}
  \framesubtitle{How to obtain a higher credibility for the rules?}

  Proposal 2: two rules where one of them has more priority than the other one. \\
  \vspace{0.25cm}

  if swift(Tim) and tall(Tim) then good\_player(Tim) \\
  ~~~~~~~~~~ with credibility 0.2 and priority 1.0\\
  if swift and tall then good\_player \\
  ~~~~~~~~~~ with credibility 0.7 and priority 0.8\\
  \vspace{0.25cm}

  Problem:
  \vspace{0.25cm}

  How to relate the credibility, the priority and the real-world values?

\end{frame}

\begin{frame}
  \frametitle{Our Proposal}
  \framesubtitle{General Idea}

  Since the adjoint property is not restricted to the credibility, 
  we can attach the priority value to the credibility value: \\

  \vspace{0.5cm}
  we attach to the usual truth value 
  $\truthVarV \in [0, 1]$ a real number 
  $\prioVarP \in [0, 1]$ denoting the (accumulated) priority,
  resulting in the tuple of real numbers between 0 and 1
  symbolized by $\latticeValue{\prioVarP}{\truthVarV} \in \Omega^{(0)}$.\footnote{
    $\Omega^{(0)}$ is the set of connectives of arity 0 \\in our program.}

  \vspace{0.5cm}
  Our previous program is then seen as:

  \exampleCode  

\end{frame}

\begin{frame}
  \frametitle{Our Proposal}
  \framesubtitle{Prerequisites}

  \exampleCode  
  \vspace{0.5cm}

% \begin{center}
% \underline{we need}
% \end{center}
\begin{columns}
  \column{.5\textwidth}
  An ordering between our new set of values %elements 
%  \vspace{0.25cm}
%
\begin{align}
\bot & \latticeLEQ \bot \nonumber \\
\bot & \latticeLEQ \latticeValue{\prioVarP}{\truthVarV} \nonumber \\
\latticeValue{\prioVarP_1}{\truthVarV_1} & 
\latticeLEQ \latticeValue{\prioVarP_2}{\truthVarV_2} ~~\leftrightarrow~~ \nonumber \\
&~~~~(~~\prioVarP_1 < \prioVarP_2~~) ~~ or~~ \nonumber \\
&~~~~(~~\prioVarP_1 = \prioVarP_2 ~~ and ~~ \truthVarV_1 \leq \truthVarV_2~~) \nonumber 
\end{align}
% where $<$ is defined as usually 
% (note that $\truthVarV_i$ and $\prioVarP_j$ are just real numbers between 0 and 1).
% It is obvious that the pair $(~\latticeValues , \latticeLEQ)$ forms a complete lattice.
  \vspace{3cm}
  
  \column{.5\textwidth}
  Extend the connectives $\&$, $\leftarrow$, $\vee$, $@$:
  \vspace{0.25cm}
 
  $\&$, $\leftarrow$ and $\vee$ depend on the adjoint property 
  and the relation between t-norms and t-conorms.
  \vspace{0.25cm}
 
  $@$ has no restrictions but 
  $\hat{@}(0,~\ldots,~0)~=~0$ and $\hat{@}(1,~\ldots,~1)~=~1$.
  \vspace{4cm}

\end{columns}
\end{frame}

\begin{frame}
  \frametitle{Our Proposal}
  \framesubtitle{Example I}

  \exampleCode  

  \begin{columns}
    \column{.75\textwidth}
    \underline{Real-world values:}
    \vspace{0.25cm}
    
    \begin{tabular}{c|c|c|c}
      Name & Swift & Tall & Good\_Player\\
      \hline
      Tim & (1.0,~0.9) & (1.0,~0.9) & (??,~0.2) \\
      \hline
      John & (1.0,~0.8) & (1.0,~0.7) & (??,~0.7) \\
      \hline
      Sarah & (1.0,~0.7) & (1.0,~0.8) & (??,~0.7) \\
    \end{tabular}
    % 
    \column{.25\textwidth}
    
    ?? = 1.0 when real world data
  \end{columns}
  
  \vspace{0.25cm}
  conjunctor: minimum (G\"odel Logic) \\
  implicator:  $\hat{\leftarrow}(x,y) = $ $x~\text{if}~x < y ~\text{else}~1$ (G\"odel Logic) \\
  aggregator: minimum ??
  \vspace{2cm}
  
\end{frame}

\begin{frame}
  \frametitle{Our Proposal}
  \framesubtitle{Example \only<1-3>{II}\only<2>{I}\only<3>{II}\only<4->{V}:
  \only<1>{Computing the credibility for each rule}
  \only<2>{The credibility for each individual}
  \only<3>{Grouping individuals by the credibility value}
  \only<4>{Testing the computed credibilities}
}

%  \exampleCode  

\begin{columns}
  \column{.75\textwidth}
%   \underline{Real-world values:}
  \vspace{0.25cm}

  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & (1.0,~0.9) & (1.0,~0.9) & (\only<1-4>{??},~0.2) \\
    \hline
    John & (1.0,~0.8) & (1.0,~0.7) & (\only<1-4>{??},~0.7) \\
    \hline
    Sarah & (1.0,~0.7) & (1.0,~0.8) & (\only<1-4>{??},~0.7) 
    % 
%    \only<1-2>{\hline ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ }
    % \only<3>{ \\
    %   \hline
    %   Terry & unknown & (1.0,~0.9) & ???? \\
    %   \hline
    %   default & (0.1,~0.2) & (0.1,~0.6) & (0.1,~0.3) 
    % }
  \end{tabular}
  
  \column{.25\textwidth}

  \only<1-4>{
    ?? = 1.0 when real world data \\
    @ = min }
%  \only<5>{@ = (min, min) }
\end{columns}

\vspace{0.25cm}
  \only<1>{
    \begin{eqnarray}
      Tim: & min((1.0,~0.9), (1.0,~0.9)) = (1.0,~0.9) ; \nonumber \\
      & \text{~~since~~} (1.0,~0.9) > (1.0,~0.2) \text{~~then~~} (1.0,~0.2) \nonumber \\ 
      John: & min((1.0,~0.8), (1.0,~0.7)) = (1.0,~0.7) ; \nonumber \\
      & \text{~~since~~} (1.0,~0.7) = (1.0,~0.7) \text{~~then~~} (1.0,~1.0) \nonumber \\
      Sarah: & min((1.0,~0.7), (1.0,~0.8)) = (1.0,~0.7) ; \nonumber \\
      & \text{~~since~~} (1.0,~0.7) = (1.0,~0.7) \text{~~then~~} (1.0,~1.0) \nonumber 
      % \nonumber \\
    \end{eqnarray}
  }

  \only<2-3>{
    \begin{columns}
      \column{.3\textwidth}
      \begin{eqnarray}
        Tim: & (1.0,~0.2) \nonumber \\ 
        John: & (1.0,~1.0) \nonumber \\
        Sarah: & (1.0,~1.0) \nonumber 
        % \nonumber \\
      \end{eqnarray}
  }

  \only<2>{
    \column{.7\textwidth}
    ~~
    \end{columns}
  }

  \only<3>{
    \column{.7\textwidth}
    \begin{eqnarray}
      ~ &z_{Tim} = infimum((1.0,~0.2)) = (1.0,~0.2) \nonumber \\
      ~ &z = infimum((1.0,~1.0),~(1.0,~1.0)) = (1.0,~1.0) \nonumber
    \end{eqnarray}
  \end{columns}
}

  \only<4>{
    \begin{eqnarray}
      Tim: & sup(~min((1.0,~0.2), min((1.0,~0.9),~(1.0,~0.9))), ~~~~~\nonumber \\ 
      &      ~~~~~min((1.0,~1.0), min((1.0,~0.9),~(1.0,~0.9))))  \nonumber \\
      &      ~~~~~= (1.0,~0.9) \rightarrow WRONG~~~AGAIN !! \nonumber \\
      John: & sup(~min((1.0,~1.0), min((1.0,~0.8),~(1.0,~0.7)))~) = (1.0,~0.7) ~~~~~\nonumber \\
      Sarah: & sup(~min((1.0,~1.0), min((1.0,~0.7),~(1.0,~0.8)))~) = (1.0,~0.7) ~~~~~\nonumber 
      % \nonumber \\
    \end{eqnarray}
  }


%    \column{.5\textwidth}
    ~~


  \vspace{2cm}

\end{frame}


\begin{frame}
  \frametitle{Our Proposal}
  \framesubtitle{Example VI\only<2->{I}\only<3->{I}\only<4->{I}:
  \only<1>{Computing the credibility of each rule for a different aggregator}
  \only<2>{The credibility for each individual}
  \only<3>{Grouping individuals by the credibility value}
  \only<4>{Testing the computed credibilities}

}

%  \exampleCode  

\begin{columns}
  \column{.75\textwidth}
%  \underline{Real-world values:}
  \vspace{0.25cm}

  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & (1.0,~0.9) & (1.0,~0.9) & (\only<1->{1.0},~0.2) \\
    \hline
    John & (1.0,~0.8) & (1.0,~0.7) & (\only<1->{0.8},~0.7) \\
    \hline
    Sarah & (1.0,~0.7) & (1.0,~0.8) & (\only<1->{0.8},~0.7) 
    % 
%    \only<1-2>{\hline ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ }
    % \only<3>{ \\
    %   \hline
    %   Terry & unknown & (1.0,~0.9) & ???? \\
    %   \hline
    %   default & (0.1,~0.2) & (0.1,~0.6) & (0.1,~0.3) 
    % }
  \end{tabular}
  
  \column{.25\textwidth}

  \only<1-4>{
    @ = (min, min) 
}
  \only<1>{\\
    Note that we have changed the priority value for John and Sarah.
}
%  \only<5>{@ = (min, min) }
\end{columns}

\vspace{0.25cm}
  \only<1>{
    \begin{eqnarray}
      Tim: & (min,~min)~((1.0,~0.9), (1.0,~0.9)) = (1.0,~0.9) ; \nonumber \\
      & \text{~~since~~} (1.0,~0.9) > (1.0,~0.2) \text{~~then~~} (1.0,~0.2) \nonumber \\ 
      John: & (min,~min)~((1.0,~0.8), (1.0,~0.7)) = (1.0,~0.7) ; \nonumber \\
      & \text{~~since~~} (1.0,~0.7) > (0.8,~0.7) \text{~~then~~} (0.8,~0.7) \nonumber \\
      Sarah: & (min,~min)~((1.0,~0.7), (1.0,~0.8)) = (1.0,~0.7) ; \nonumber \\
      & \text{~~since~~} (1.0,~0.7) > (0.8,~0.7) \text{~~then~~} (0.8,~0.7) \nonumber 
      % \nonumber \\
    \end{eqnarray}
  }

  \only<2-3>{
    \begin{columns}
      \column{.3\textwidth}
      \begin{eqnarray}
        Tim: & (1.0,~0.2) \nonumber \\ 
        John: & (0.8,~0.7) \nonumber \\
        Sarah: & (0.8,~0.7) \nonumber 
        % \nonumber \\
      \end{eqnarray}
  }

  \only<2>{
    \column{.7\textwidth}
    ~~
    \end{columns}
  }

  \only<3>{
    \column{.7\textwidth}
    \begin{eqnarray}
      ~ &z_{Tim} = infimum((1.0,~0.2)) = (1.0,~0.2) \nonumber \\
      ~ &z = infimum((0.8,~0.7),~(0.8,~0.7)) = (0.8,~0.7) \nonumber
    \end{eqnarray}
  \end{columns}
}

  \only<4>{
    \begin{eqnarray}
      Tim: & sup(~min((1.0,~0.2), min((1.0,~0.9),~(1.0,~0.9))), ~~~~~\nonumber \\ 
      &      ~~~~~min((0.8,~0.7), min((1.0,~0.9),~(1.0,~0.9))))  \nonumber \\
      &      ~~~~~= (1.0,~0.2) \rightarrow OK !! \nonumber \\
      John: & sup(~min((0.8,~0.7), min((1.0,~0.8),~(1.0,~0.7)))~) = (0.8,~0.7) ~~~~~\nonumber \\
      Sarah: & sup(~min((0.8,~0.7), min((1.0,~0.7),~(1.0,~0.8)))~) = (0.8,~0.7) ~~~~~\nonumber 
      % \nonumber \\
    \end{eqnarray}
  }


%    \column{.5\textwidth}
    ~~


  \vspace{2cm}

\end{frame}


\begin{frame}
  \frametitle{Our Proposal}
  \framesubtitle{Example X\only<2->{I}\only<3->{I}:
    \only<1>{Adding an individual with an unknown value.}
    \only<2>{How defaults are managed.}
    \only<3>{Giving more priority to the rules using less default information.}
}
% \only<4->{I}
%  \exampleCode  

\begin{columns}
  \column{.75\textwidth}
%  \underline{Real-world values:}
  \vspace{0.25cm}

  \begin{tabular}{c|c|c|c}
    Name & Swift & Tall & Good\_Player\\
    \hline
    Tim & (1.0,~0.9) & (1.0,~0.9) & (\only<1->{1.0},~0.2) \\
    \hline
    John & (1.0,~0.8) & (1.0,~0.7) & (\only<1->{0.8},~0.7) \\
    \hline
    Sarah & (1.0,~0.7) & (1.0,~0.8) & (\only<1->{0.8},~0.7) 
    % 
%    \only<1-2>{\hline ~ & ~ & ~ & ~ \\ \hline ~ & ~ & ~ & ~ }
    \only<1->{ \\
      \hline
      Terry & unknown & (1.0,~0.9) & ???? \\
      \hline
      default & (0.1,~0.2) & (0.1,~0.6) & (0.1,~0.3) 
    }
  \end{tabular}
  
  \column{.25\textwidth}

  \only<1-2>{
    @ = (min, min) \\
  }
  \only<3->{
    @ = (mean, min) \\
  }

  \only<1->{
    unknown = $\bot$ \\
    \begin{eqnarray}
      z_{Tim} = (1.0,~0.2) \nonumber \\
      z = (0.8,~0.7) \nonumber
    \end{eqnarray}

  }
%  \only<5>{@ = (min, min) }
\end{columns}

\vspace{0.25cm}
  \only<2>{
    \begin{eqnarray}
      Terry: swift: & sup(~\bot,~(0.1,~0.2)~) = (0.1,~0.2) \nonumber \\
      Terry: good\_player: & sup(~min((0.8,~0.7), @~((0.1,~0.2),~(1.0,~0.9))), \nonumber \\
      & (0.1,~0.3)~)~=~(0.1,~0.3) ~~~~~~~~~~~~~~~~~~~\nonumber 
    \end{eqnarray}
    We are giving too much importance to defaults !!
  }
  \only<3>{
    \begin{eqnarray}
      Terry: swift: & sup(~\bot,~(0.1,~0.2)~) = (0.1,~0.2) \nonumber \\
      Terry: good\_player: & sup(~min((0.8,~0.7), @~((0.1,~0.2),~(1.0,~0.9)))~) \nonumber \\
      & (0.1,~0.3)~)~=~(0.55,~0.2) ~~~~~~~~~~~~~~~~~~~\nonumber 
    \end{eqnarray}
  }
%    \column{.5\textwidth}
    ~~
  \vspace{2cm}

\end{frame}


\begin{frame}
  \frametitle{Conclusions}
  \framesubtitle{RFuzzy with Priorities}
  \begin{itemize}
  \item We can group individuals to get rules where credibility is not as small as before, 
    getting results more accurate than before. \\
\vspace{1cm}
  \item We can give more importance to those rules using less default information.
  \end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% the end

\begin{frame}%[fragile]
  \frametitle{Questions}
  \vfill
  \begin{center}
    % \includegraphics[height=5.5cm]{questions} 
    { \huge Thank you}
  \end{center}
  \vfill
\end{frame}

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

% \frame{\titlepage}

\end{document}


% \begin{frame}
%   \frametitle{}
%   \vfill
%   \begin{itemize}
%   \item 
%   \end{itemize}
%   \vfill
% \end{frame}
