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\title{Typechecking Diderot}
\author{
  Gordon Kindlmann \\
  University of Chicago \\
  {\small\tt{}glk@cs.uchicago.edu} \\
\and
  John Reppy \\
  University of Chicago \\
  {\small\tt{}jhr@cs.uchicago.edu} \\
}
\date{\today}

\begin{document}

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\section{Introduction}
This document looks at the rules for typechecking Diderot.

\section{Types}
\begin{displaymath}
  \begin{array}{rclr}
    \nu & ::= & n & \text{dimension ($n > 0$)} \\
        & \mid & \phi & \text{dimension variable} \\
    \sigma & ::= & \epsilon & \text{empty sequence of dimensions} \\[1em]
        & \mid & \rho & \text{shape variable} \\
        & \mid & \sigma,\nu & \text{shape extension} \\[1em]
    \kappa & ::= & n & \text{$n$ levels of differentiation ($n \geq 0$)} \\
        & \mid & \kappa-n & \text{}
  \end{array}
\end{displaymath}%

The syntax of Diderot types is given in \figref{fig:types}.
\begin{figure}[t]
  \begin{displaymath}
    \begin{array}{rclr}
      \tau & ::= & \TYconst & \text{type constants} \\
           & \mid & \theta \\
%           & \mid & \TYmatrix{n}{m} & \text{$n\times{}m$ matrix} \\
           & \mid & \TYimage{\nu}{\sigma} & \text{$d$-dimension image of $\mu$ values}\\
           & \mid & \TYkern{k} & \text{convolution kernel with $k$ derivatives} \\
           & \mid & \TYfield{k}{\nu}{\sigma} & \text{field} \\
      \sigma & ::= \
    \end{array}%
  \end{displaymath}%
  where $o\in\Nat$ is the tensor order, $d,n,m\in\SET{2,3}$ are dimensions,
  and $k\in\Nat$ is the differentiability of a field.
  \caption{Diderot types}
  \label{fig:types}
\end{figure}%

\end{document}