\documentclass[11pt]{article} \input{defs} \setlength{\textwidth}{6in} \setlength{\oddsidemargin}{0.25in} \setlength{\evensidemargin}{0.25in} \setlength{\parskip}{5pt} \title{Diderot design} \author{ Gordon Kindlmann \\ University of Chicago \\ {\small\tt{}glk@cs.uchicago.edu} \\ \and John Reppy \\ University of Chicago \\ {\small\tt{}jhr@cs.uchicago.edu} \\ \and Thomas Schultz \\ University of Chicago \\ {\small\tt{}t.schultz@uchicago.edu} \\ } \date{\today} \begin{document} \maketitle \thispagestyle{empty} \section{Introduction} This document is a semi-formal design of Diderot. \section{Types} The syntax of Diderot types is given in \figref{fig:types}. \begin{figure}[t] \begin{displaymath} \begin{array}{rclr} \rho & ::= & $\ldots$ & \text{NRRD scalar types} \\[1em] \iota & ::= & \TYbool & \text{booleans} \\ & \mid & \TYint & \text{integers} \\[1em] % \mu for memory, where images of rawtensors will live (or on disk) \mu & ::= & \TYrawten{\Seq{d}}{\rho} & \begin{minipage}[l]{3in}\begin{flushright} tensors of order $|\Seq{d}|$ and dimensions $\Seq{d}$,\\with coefficients of type $\rho$ \end{flushright}\end{minipage}\\[1em] \theta & ::= & \TYtensor{\Seq{d}} & \begin{minipage}[l]{3in}\begin{flushright} tensors of order $|\Seq{d}|$ and dimensions $\Seq{d}$,\\with real coefficients \end{flushright}\end{minipage}\\[1em] \tau & ::= & \iota \\ & \mid & \theta \\ & \mid & \TYmatrix{n}{m} & \text{$n\times{}m$ matrix} \\ & \mid & \TYimage{d}{\mu} & \text{$d$-dimension image of $\mu$ values}\\ & \mid & \TYkern{k} & \text{convolution kernel with $k$ derivatives} \\ & \mid & \TYfield{k}{d}{\theta} & \text{$d$-dimension field of $\theta$ values with $k$ derivatives} \\ \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}% We use some type abbreviations for common cases: \begin{eqnarray*} \TYreal & = & \TYtensor{\epsilon} \quad\text{for any $d$} \\ \TYvec{d} & = & \TYtensor{d} \end{eqnarray*}% \section{Operations} \subsection{Scalar operations} \noindent{}\point Arithmetic: \begin{displaymath} \begin{array}{cl} \BinopTy{{\odot}}{\tau}{\tau}{\tau} & \text{for $\odot\in\SET{\OP{+},\OP{-},\OP{*},\OP{/}}$ and $\tau\in\SET{\TYint,\TYreal}$} \\ \UnopTy{\OP{-}}{\tau}{\tau} & \text{for $\tau\in\SET{\TYint,\TYreal}$} \end{array}% \end{displaymath}% \noindent{}\point Comparisons: \begin{displaymath} \BinopTy{{\odot}}{\tau}{\tau}{\TYbool} \qquad\text{for $\odot\in\SET{\OP{<},{\leq},\OP{=},{\neq}\OP{>},{\geq}}$ and $\tau\in\SET{\TYint,\TYreal}$} \end{displaymath}% \subsection{Matrix operations} \subsection{Tensor operations} \noindent{}\point Addition: \begin{displaymath} \BinopTy{{\odot}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}} \qquad\text{for $\odot\in\SET{\OP{+},\OP{-}}$} \end{displaymath}% \noindent{}\point Negation: \begin{displaymath} \UnopTy{\OP{-}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}} \end{displaymath}% \noindent{}\point Scalar division: \begin{displaymath} \BinopTy{\OP{/}}{\TYtensor{\Seq{d}}}{\TYreal}{\TYtensor{\Seq{d}}} \end{displaymath}% \noindent{}\point Scalar multiplication (scalar times order-N): \begin{displaymath} \begin{array}{c} \BinopTy{\OP{*}}{\TYreal}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}} \\ \BinopTy{\OP{*}}{\TYtensor{\Seq{d}}}{\TYreal}{\TYtensor{\Seq{d}}} \end{array}% \end{displaymath}% \noindent{}\point Tensor scalar multiplication (contraction of two order-N tensors down to a scalar, \eg{} dot product of vectors, double dot product of 2nd-order tensors) \begin{displaymath} \begin{array}{c} \BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}{\TYreal} \end{array}% \end{displaymath}% Defined by $c = A_{\vec{i}}B_{\vec{i}}$ ~(\eg{} $\mathbf{a}\cdot\mathbf{b} = a_ib_i$ or $\mathbf{A:B} = A_{ij}B_{ij}$) NOTE: In this and subsequent index notation expressions, a vector over the index variable (\eg{} $\vec{i}$) means that the variable is in fact standing for a sequence of contiguous index variables \\ Possible direct notation syntax: ~~ {\tt .} (period) ~~ {\tt :} (colon) ~~ {\tt dot} ~~ {\tt o} \noindent{}\point Tensor product (aka outer product; order output is sum of orders) \begin{displaymath} \begin{array}{c} \BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d'}}}{\TYtensor{\Seq{d}\Seq{d'}}} \end{array}% \end{displaymath}% Defined by $C_{\vec{i}\vec{j}} = A_{\vec{i}}B_{\vec{j}}$ \\ Possible direct notation syntax: ~~ {\tt x} ~~ {\tt (x)} ~~ {\tt out} \noindent{}\point Matrix Multiply \begin{displaymath} \begin{array}{c} \BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{d_1 d_2}}{\TYtensor{\Seq{d}}} \\ \BinopTy{{?}}{\TYtensor{d_1 d_2}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}} \end{array}% \end{displaymath}% Defined by $C_{i\vec{k}} = A_{ij}B_{j\vec{k}}$ and $C_{\vec{i}k} = B_{\vec{i}j}A_{jk}$ ~(\eg{} $u_i = M_{ij}v_j$ or $T_{ijl} = S_{ijk}M_{kl}$) \\ Possible direct notation syntax: ~~ ? \noindent{}\point Contracting out last or first index of tensor (order $o \geq 1$) with vector \begin{displaymath} \begin{array}{c} \BinopTy{{?}}{\TYtensor{d'\Seq{d}}}{\TYvec{d}}{\TYtensor{\Seq{d}}} \\ \BinopTy{{?}}{\TYvec{d}}{\TYtensor{d'\Seq{d}}}{\TYtensor{\Seq{d}}} \end{array}% \end{displaymath}% Defined by $C_{\vec{i}} = A_{\vec{i}j}v_j$ and $C_{\vec{j}} = v_iA_{j\vec{j}}$ \\ Possible direct notation syntax: ~~ ? \noindent{}\point Arbitrary tensor comprehension''. The product can also be expressed in general index notation, and which may increase, preserve, or decrease the tensor order. \begin{displaymath} \begin{array}{c} \BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{p}{d}}{\TYtensor{q}{d}} \end{array}% \end{displaymath}% Defined by the conventions of Einstein summation notation. \\ Possible syntax: {\tt } ~~ {\tt } \subsection{Field operations} \noindent{}\point Creation from an image: \begin{displaymath} \BinopTy{\OPconvolve}{\TYkern{k}}{\TYimage{d'}{\TYrawten{\Seq{d}}{\rho}}}{\TYfield{k}{d'}{\TYtensor{\Seq{d}}}} \end{displaymath}% \noindent{}\point Scalar multiplication: \begin{displaymath} \begin{array}{c} \BinopTy{\OP{*}}{\TYreal}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \\ \BinopTy{\OP{*}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}} \end{array}% \end{displaymath}% \noindent{}\point Scalar division: \begin{displaymath} \BinopTy{\OP{/}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}} \end{displaymath}% \noindent{}\point Negation: \begin{displaymath} \UnopTy{\OP{-}}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \end{displaymath}% \noindent{}\point Addition: \begin{displaymath} \begin{array}{c} \BinopTy{{\odot}}{\TYfield{k}{d}{\theta}}{\theta}{\TYfield{k}{d}{\theta}} \\ \BinopTy{{\odot}}{\theta}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \\ \BinopTy{{\odot}}{\TYfield{k_1}{d}{\theta}}{\TYfield{k_2}{d}{\theta}}{\TYfield{\min(k_1,k_2)}{d}{\theta}} \end{array}% \qquad\text{for $\odot\in\SET{\OP{+},\OP{-}}$} \end{displaymath}% \noindent{}\point Differentiation: \begin{displaymath} \UnopTy{\OPdiff}{\TYfield{k}{d'}{\TYtensor{\Seq{d}}}}{\TYfield{k-1}{d'}{\TYtensor{d'\Seq{d}}}} \qquad\text{for $k > 0$} \end{displaymath}% \noindent{}\point Probing: \begin{displaymath} \BinopTy{\OP{@}}{\TYfield{k}{d}{\theta}}{\TYvec{d}}{\theta} \end{displaymath}% \end{document}
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The output has ended thus: th} \BinopTy{\OP{@}}{\TYfield{k}{d}{\theta}}{\TYvec{d}}{\theta} \end{displaymath}% \end{document}