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\title{Weak Cardinality Theorems for First-Order Logic}
\author{Till Tantau}
\institute[Technische Universit\"at Berlin]{
  Fakultät für Elektrotechnik und Informatik\\
  Technische Universit\"at Berlin}
\date{Fundamentals of Computation Theory 2003}

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\AtBeginSection[]{\frame{\frametitle{Outline}\tableofcontents[current]}}

\begin{document}

\frame{\titlepage}

\section*{Outline}
\frame{\frametitle{Outline}\tableofcontents[part=1]}

\part{Main Part}
\section{History}

\subsection{Enumerability in Recursion and Automata Theory}

\frame
{
  \frametitle{Motivation of Enumerability}

  \begin{block}{Problem}
    Many functions are not computable or not efficiently computable.
  \end{block}
  \vskip-1em
  \begin{overprint}
  \onslide<1-2>
  \begin{example}
    \begin{overprint}
    \onslide<1>
      \vskip0.5em
      \begin{itemize}
      \item
        $\NumSAT$:\\
        How many satisfying assignments does a formula have?
      \end{itemize}

    \onslide<2>
      \vskip0.5em
        For difficult languages~$A$:
        \begin{itemize}
        \item
          Cardinality function $\NumA^n$:\\
          \alert{How many} input words are in~$A$?
        \item
          Characteristic function $\chi_A^n$:\\
          \alert{Which} input words are in~$A$?
        \end{itemize}
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      \end{pgfpicture}
    \end{overprint}
  \end{example}

  \onslide<3>
    \begin{block}{Solutions}
      Difficult functions can be 
      \begin{itemize}
      \item
        computed using probabilistic algorithms,
      \item
        computed efficiently on average,
      \item
        approximated, or
      \item<alert@1->
        enumerated.
      \end{itemize}
    \end{block}
  \end{overprint}
}

\frame
{
  \frametitle{Enumerators Output Sets of Possible Function Values}
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      \begin{definition}[1987, 1989, 1994, 2001]
        An \alert{$m$-enumerator} for a function~$f$
        \begin{enumerate}
        \item<alert@1-4>
          reads $n$ input words $w_1$, \dots, $w_n$,
        \item<alert@5>
          does a computation,
        \item<alert@6-8>
          outputs at most $m$ values,
        \item<alert@9>
          one of which is $f(w_1,\dots,w_n)$.
        \end{enumerate}
      \end{definition}
    \end{column}
  \end{columns}
}

\subsection{Known Weak Cardinality Theorem}

\frame
{
  \frametitle{How Well Can the Cardinality Function Be Enumerated?}

  \begin{block}{Observation}
    For fixed~$n$, the cardinality function $\NumA^n$
    \begin{itemize}
    \item
      can be \alert{$1$}-enumerated by Turing machines only for \alert{recursive}~$A$,~but\hskip-0.5cm\hbox{}
    \item
      can be \alert{$(n+1)$}-enumerated for \alert{every} language~$A$.
    \end{itemize}
  \end{block}

  \begin{alertblock}{Question}<2->
    What about $2$-, $3$-, $4$-, \dots, $n$-enumerability?
  \end{alertblock}
}

\newtheorem{card}{Cardinality Theorem}[theorem]
\newtheorem{weakcard}{Weak Cardinality Theorems}[theorem]

\frame
{
  \frametitle{How Well Can the Cardinality Function\\ Be Enumerated
    by Turing Machines?}

  \begin{card}[Kummer, 1992]
    If $\NumA^n$ is $n$-enumerable by a Turing machine, then $A$ is
    recursive.
  \end{card}

  \begin{weakcard}[\uncover<2->{\alert<1-2>{1987},} \uncover<3->{\alert<3>{1989},}
      \uncover<4->{\alert<4>{1992}}]<2-> 
    \begin{enumerate}
    \item<2-| alert@2>
      If $\chi_A^n$ is $n$-enumerable by a Turing machine, then $A$ is
        recursive.
    \item<3-| alert@3>
      If $\NumA^2$ is $2$-enumerable by a Turing machine, then $A$ is
        recursive.
    \item<4-| alert@4>
      If $\NumA^n$ is $n$-enumerable by a Turing machine that never
        enumerates both $0$ and~$n$, then $A$ is recursive.
    \end{enumerate}
  \end{weakcard}
}


\frame
{
  \frametitle{How Well Can the Cardinality Function\\ Be Enumerated
    by Finite Automata?}

  \begin{alertblock}{Conjecture}
    If $\NumA^n$ is $n$-enumerable by a \alert{finite automaton}, then $A$ is
    \alert{regular}.
  \end{alertblock}

  \begin{weakcard}[2001, 2002]
    \begin{enumerate}
    \item
      If $\chi_A^n$ is $n$-enumerable by a \alert{finite automaton}, then $A$ is
      \alert{regular}.
    \item
      If $\NumA^2$ is $2$-enumerable by a \alert{finite automaton}, then $A$ is
      \alert{regular}.
    \item
      If $\NumA^n$ is $n$-enumerable by a \alert{finite automaton} that never
      enumerates both $0$ and~$n$, then $A$ is \alert{regular}.
    \end{enumerate}
  \end{weakcard}
}


\subsection{Why Do Cardinality Theorems Hold Only for Certain Models?}

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\frame
{
  \frametitle{Why?}

  \begin{block}{First Explanation}<1>
    The weak cardinality theorems hold both for recursion and automata
    theory \alert{by coincidence}.
  \end{block}

  \begin{block}{Second Explanation}<1-2>
    The weak cardinality theorems hold both for
    recursion and automata theory, \alert{because they are
      instantiations of\\ single, unifying theorems}.
  \end{block}
  
  \vskip1em
  \invisible<1>{
    The second explanation is correct.\\
    The theorems can (almost) be unified using first-order logic.
    }
}
   
    

\section[Unification by Logic]{Unification by First-Order Logic}

\subsection{Elementary Definitions}

\frame
{
  \frametitle{What Are Elementary Definitions?}

  \begin{definition}
    A relation~$R$ is \alert{elementarily definable in a
    logical structure~$\mathcal S$} if
    \begin{enumerate}
    \item
      there exists a first-order formula~$\phi$,
    \item
      that is true exactly for the elements of~$R$.
    \end{enumerate}    
  \end{definition}

  \begin{example}
    The set of even numbers is elementarily definable in $(\Nat, +)$
    via the formula $\phi(x) \equiv \exists z \centerdot z+z=x$.
  \end{example}

  \begin{example}
    The set of powers of 2 is not elementarily definable in $(\Nat, +)$.
  \end{example}
}


\frame
{
  \frametitle{Characterisation of Classes by Elementary Definitions}

  \begin{theorem}[B\"uchi, 1960]
    There exists a logical structure~$(\Nat, +, \mathrm e_2)$
    such that a set $A \subseteq \Nat$ is\\ \alert{regular} iff it is 
    \alert{elementarily definable in~$(\Nat, +, \mathrm e_2)$}.
  \end{theorem}

  \begin{theorem}
    There exists a logical structure~$\mathcal R$ such that a set $A
    \subseteq \Nat$ is \alert{recursively enumerable} iff it is \alert{positively
    elementarily definable in~$\mathcal R$}.\hskip-0.5cm\hbox{}
  \end{theorem}
}



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\subsection{Enumerability for First-Order Logic}

\frame
{
  \frametitle{Elementary Enumerability is a Generalisation of\\ Elementary Definability}

  \begin{columns}
    \begin{column}{3.25cm}
      \begin{pgfpicture}{-0.25cm}{0cm}{3cm}{4cm}

        \color{shaded}
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        \pgfputat{\pgfxy(2.6,0)}{\pgfbox[left,center]{$x$}}
        \pgfputat{\pgfxy(0,3.2)}{\pgfbox[center,base]{$f(x)$}}
        \pgfputat{\pgfxy(2.55,2)}{\pgfbox[left,center]{$f$}}
      \end{pgfpicture}
    \end{column}
    \begin{column}{7.5cm}
      \begin{definition}
        A function~$f$ is\\
        \alert{elementarily $m$-enumerable in a structure~$\mathcal S$} if
        \begin{enumerate}
        \item
          its graph is contained in an\\
          \alert{elementarily definable} relation~$R$,
        \item
          which is \alert{$m$-bounded}, i.\kern1pt e., for each~$x$
          there are at most~$m$ different~$y$ with $(x,y) \in R$.
        \end{enumerate}
      \end{definition}
    \end{column}
  \end{columns}
}

\frame
{
  \frametitle{The Original Notions of Enumerability are Instantiations}

  \begin{theorem}
    A function is $m$-enumerable by a \alert{finite automaton} iff\\
    it is elementarily $m$-enumerable in \alert{$(\Nat, +, \mathrm e_2)$}.
  \end{theorem} 

  \begin{theorem}
    A function is $m$-enumerable by a \alert{Turing machine} iff\\
    it is positively elementarily $m$-enumerable in \alert{$\mathcal R$}.
  \end{theorem} 
}

%\subsection{Cross Product Theorem for First-Order Logic}

\subsection{Weak Cardinality Theorems for First-Order Logic}

\frame
{
  \frametitle{The First Weak Cardinality Theorem}
  
  \begin{theorem}
    Let $\mathcal S$ be a logical structure with universe~$U$ and let
    $A \subseteq U$. If
    
    \begin{enumerate}
    \item
      $\mathcal S$ is well-orderable and
    \item
      \alert{$\chi_A^n$} is elementarily \alert{$n$}-enumerable in~$\mathcal S$,
    \end{enumerate}
    
    then \alert{$A$ is elementarily definable} in~$\mathcal S$.    
  \end{theorem}
  \begin{overprint}
    \onslide<2>
      \begin{corollary}
        If $\chi_A^n$ is $n$-enumerable by a finite automaton, then
        $A$ is regular.
      \end{corollary}

    \onslide<3>
      \begin{corollary}[with more effort]
        If $\chi_A^n$ is $n$-enumerable by a Turing machine, then $A$
        is recursive.
      \end{corollary}
  \end{overprint}  
}

\frame
{
  \frametitle{The Second Weak Cardinality Theorem}
  
  \begin{theorem}
    Let $\mathcal S$ be a logical structure with universe~$U$ and let
    $A \subseteq U$. If
    
    \begin{enumerate}
    \item
      $\mathcal S$ is well-orderable,
    \item
      every finite relation on~$U$ is elementarily definable
      in~$\mathcal S$, and
    \item
      \alert{$\NumA^2$} is elementarily \alert{$2$}-enumerable in~$\mathcal S$,
    \end{enumerate}
    
    then \alert{$A$ is elementarily definable} in~$\mathcal S$.    
  \end{theorem}
%  \begin{overlayarea}{\textwidth}{2cm}
%    \only<2>{
%      \begin{corollary}
%        If $\NumA^2$ is $2$-enumerable by a finite automaton, then
%        $A$ is regular.
%      \end{corollary}}%
%    \only<3>{
%      \begin{block}{Corollary}
%        If $\NumA^2$ is $2$-enumerable by a Turing machine, then $A$
%        is recursive in the halting problem.
%      \end{block}
%      }
%  \end{overlayarea}
}

\frame
{
  \frametitle{The Third Weak Cardinality Theorem}
  
  \begin{theorem}
    Let $\mathcal S$ be a logical structure with universe~$U$ and let
    $A \subseteq U$. If
    
    \begin{enumerate}
    \item
      $\mathcal S$ is well-orderable,
    \item
      every finite relation on~$U$ is elementarily definable
      in~$\mathcal S$, and
    \item
      \alert{$\NumA^n$} is elementarily \alert{$n$}-enumerable in~$\mathcal S$ via a
      relation that \alert{never `enumerates' both $0$ and~$n$},
    \end{enumerate}
    
    then \alert{$A$ is elementarily definable} in~$\mathcal S$.    
  \end{theorem}
%  \begin{overlayarea}{\textwidth}{2cm}
%    \only<2>{
%      \begin{corollary}
%        If $\NumA^n$ is $n$-enumerable by a finite automaton that
%        never enumerates both $0$ and~$n$, then $A$ is regular.
%      \end{corollary}}%
%    \only<3>{
%      \begin{block}{Corollary}
%        If $\NumA^n$ is $n$-enumerable by a Turing machine that never
%        enumerates both $0$ and~$n$, then $A$ is recursive in the
%        halting problem.
%      \end{block}
%      }
%  \end{overlayarea}
}



\frame
{
  \frametitle{Relationships Between Cardinality Theorems (CT)}

  \begin{pgfpicture}{0cm}{0cm}{10cm}{5cm}
    \only<2>{%
    \color{alert}
    \pgfnodebox{autX}[virtual]{\pgfxy(2.2,4)}{CT}{2pt}{2pt}
    \color{black}}%
    \pgfnodebox{autA}[virtual]{\pgfxy(1,3)}{1st Weak CT}{2pt}{2pt}
    \pgfnodebox{autB}[virtual]{\pgfxy(1,2)}{2nd Weak CT}{2pt}{2pt}
    \pgfnodebox{autC}[virtual]{\pgfxy(1,1)}{3rd Weak CT}{2pt}{2pt}

    \only<2>{%
    \color{alert}
    \pgfnodebox{logX}[virtual]{\pgfxy(6.2,4.5)}{CT}{2pt}{2pt}%
    \color{black}}%
    \pgfnodebox{logA}[virtual]{\pgfxy(5,3.5)}{1st Weak CT}{2pt}{2pt}
    \pgfnodebox{logB}[virtual]{\pgfxy(5,2.5)}{2nd Weak CT}{2pt}{2pt}
    \pgfnodebox{logC}[virtual]{\pgfxy(5,1.5)}{3rd Weak CT}{2pt}{2pt}

    \pgfnodebox{recX}[virtual]{\pgfxy(10.2,4)}{CT}{2pt}{2pt}
    \pgfnodebox{recA}[virtual]{\pgfxy(9,3)}{1st Weak CT}{2pt}{2pt}
    \pgfnodebox{recB}[virtual]{\pgfxy(9,2)}{2nd Weak CT}{2pt}{2pt}
    \pgfnodebox{recC}[virtual]{\pgfxy(9,1)}{3rd Weak CT}{2pt}{2pt}

    \pgfputat{\pgfxy(1,4.5)}{\pgfbox[center,base]{\structure{automata theory}}}
    \pgfputat{\pgfxy(5,5)}{\pgfbox[center,base]{\structure{first-order logic}}}
    \pgfputat{\pgfxy(9,4.5)}{\pgfbox[center,base]{\structure{recursion
          theory}}}

    {%
    \color{structure}%
    \pgfxyline(3,0)(3,5)
    \pgfxyline(7,0)(7,5)
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    \pgfsetendarrow{\pgfarrowto}
    \pgfnodeconnline{logA}{autA}
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    \alert{\pgfnodeconnline{logX}{recX}}}
  \end{pgfpicture}
}


\section{Applications}

\subsection{A Separability Result for First-Order Logic}

%\frame
%{
%  \begin{columns}
%    \begin{column}{2.4cm}
%      \begin{pgfpicture}{-1.2cm}{-1.2cm}{1cm}{1cm}
%        \color{shaded}
%        \pgfrect[fill]{\pgfxy(-1.4,-1)}{\pgfxy(2.8,2)}

%        \color{white}
%        \pgfcircle[fill]{\pgfxy(-0.6,0)}{0.5cm}
%        \pgfcircle[fill]{\pgfxy(0.6,0)}{0.5cm}
%        \only<2->{%      
%          \color{softred}
%          \pgfcircle[fill]{\pgfxy(-0.6,0)}{0.6cm}}%
%        %
%        \color{black}
%        \pgfcircle[stroke]{\pgfxy(-0.6,0)}{0.5cm}
%        \pgfcircle[stroke]{\pgfxy(0.6,0)}{0.5cm}

%        \pgfputat{\pgfxy(-0.6,0)}{\pgfbox[center,center]{$A^{(n)}$}}
%        \pgfputat{\pgfxy(0.6,0)}{\pgfbox[center,center]{$\barA{}^{(n)}$}}     
%      \end{pgfpicture}
%    \end{column}
%    \begin{column}{8cm}
%      \begin{block}{Notation}
%        Let $A^{(n)}$ contain all $n$ tuples of\\
%        distinct elements of~$A$.
%      \end{block}
      
%      \begin{block}{Theorem}
%        Let $\mathcal S$ be a well-orderable logical structure in which
%        all finite relations are elementarily definable.\\[0.5em]
%        If $A^{(n)}$ and $\barA{}^{(n)}$ are \alert<2>{elementarily separable}
%        in~$\mathcal S$, then~so~are~$A$~and~$\barA$.
%      \end{block}

%      \uncover<3>{
%        \begin{alertblock}{Note}
%          The theorem is no longer true if $\barA$ is replaced by an
%          arbitrary set~$B$.
%        \end{alertblock}
%      }
%    \end{column}
%  \end{columns}
%}


\frame
{
  \begin{columns}
    \begin{column}{4cm}
      \begin{pgfpicture}{-2cm}{-1.75cm}{2cm}{2.25cm}
        \color{shaded}
        \pgfrect[fill]{\pgfxy(-2,-1.75)}{\pgfxy(4,4)}
                                %\pgfcircle[fill]{\pgforigin}{2cm}

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        \color{softred}
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        \pgfcircle[fill]{\pgfpolar{210}{1cm}}{\radius}
        \color{softblue}
        \pgfcircle[fill]{\pgfpolar{330}{1cm}}{\radius}}%
        %
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          \begin{pgftranslate}{\pgfpolar{30}{1cm}}
            \color{softgb}
            \pgfcircle[fill]{\pgfpolar{330}{1cm}}{\radius}
          \end{pgftranslate}
        \end{pgftranslate}}%
        %
        \color{black}
        \pgfcircle[stroke]{\pgfpolar{90}{1cm}}{\innerradius}
        \pgfcircle[stroke]{\pgfpolar{210}{1cm}}{\innerradius}
        \pgfcircle[stroke]{\pgfpolar{330}{1cm}}{\innerradius}

        \pgfputat{\pgfrelative{\pgfpolar{90}{1cm}}%
          {\pgfpoint{0pt}{-.5ex}}}%
        {\pgfbox[center,base]{$A\times \barA$}}
        \pgfputat{\pgfrelative{\pgfpolar{210}{1cm}}%
          {\pgfpoint{0pt}{-.5ex}}}%
        {\pgfbox[center,base]{$A\times A$}}
        \pgfputat{\pgfrelative{\pgfpolar{330}{1cm}}%
          {\pgfpoint{0pt}{-.5ex}}}%
        {\pgfbox[center,base]{$\barA\times \barA$}}

      \end{pgfpicture}
    \end{column}
    \begin{column}{6.8cm}
      \begin{theorem}
        Let $\mathcal S$ be a well-orderable logical structure in which
        all finite relations are elementarily definable.\\[0.5em]
        If there exist elementarily definable supersets of
        {\color<2>{darkgreen}$A \times A$},
        {\color<2>{darkred}$A \times \barA$}, and
        {\color<2>{darkblue}$\barA \times \barA$} whose
        intersection is empty,\\
        then $A$ is elementarily definable in~$\mathcal S$. 
      \end{theorem}  
      \begin{alertblock}{Note}<3>
        The theorem is no longer true\\
        if we add $\barA \times A$ to the list. 
      \end{alertblock}%
    \end{column}
  \end{columns}
}


\section*{Summary}

\frame
{
  \frametitle{Summary}

  \begin{block}{Summary}
  \begin{itemize}
  \item
    The weak cardinality theorems for first-order logic \alert{unify}\\
    the weak cardinality theorems of automata and recursion theory.
  \item
    The logical approach yields
    weak cardinality theorems for\\ \alert{other computational models}.
  \item
    Cardinality theorems are \alert{separability theorems} in disguise.
  \end{itemize}
  \end{block}{}

  \begin{block}{Open Problems}
    \begin{itemize}
    \item
      Does a cardinality theorem for first-order logic hold?
    \item
      What about non-well-orderable structures like $(\mathbb R, +,
      \cdot)$? 
    \end{itemize}
  \end{block}
}

\end{document}


