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1 :	jhr	16	\documentclass[11pt]{article}
2 :
3 :			\input{defs}
4 :
5 :			\setlength{\textwidth}{6in}
6 :			\setlength{\oddsidemargin}{0.25in}
7 :			\setlength{\evensidemargin}{0.25in}
8 :			\setlength{\parskip}{5pt}
9 :
10 :	jhr	17	\title{Diderot design}
11 :	jhr	16	\author{
12 :			Gordon Kindlmann \\
13 :			University of Chicago \\
14 :			{\small\tt{}glk@cs.uchicago.edu} \\
15 :			\and
16 :			John Reppy \\
17 :			University of Chicago \\
18 :			{\small\tt{}jhr@cs.uchicago.edu} \\
19 :	jhr	17	\and
20 :			Thomas Schultz \\
21 :			University of Chicago \\
22 :			{\small\tt{}t.schultz@uchicago.edu} \\
23 :	jhr	16	}
24 :			\date{\today}
25 :
26 :			\begin{document}
27 :
28 :			\maketitle
29 :			\thispagestyle{empty}
30 :
31 :			\section{Introduction}
32 :			This document is a semi-formal design of Dedierot.
33 :
34 :			\section{Types}
35 :	jhr	17	The syntax of Diderot types is given in \figref{fig:types}.
36 :			\begin{figure}[t]
37 :			\begin{displaymath}
38 :			\begin{array}{rclr}
39 :			\rho & ::= & $\ldots$ & \text{NNRD types} \\[1em]
40 :			\iota & ::= & \TYbool & \text{booleans} \\
41 :			& \mid & \TYint & \text{integers} \\[1em]
42 :			\theta & ::= & \TYtensor{o}{d} & \text{tensors of order $o$ and dimension $d$} \\[1em]
43 :			\tau & ::= & \iota \\
44 :			& \mid & \theta \\
45 :			& \mid & \TYmatrix{n}{m} & \text{$n\times{}m$ matrix} \\
46 :			& \mid & \TYimage{d}{\rho} & \text{image of dimension $d$ and $\rho$ elements}\\
47 :			& \mid & \TYkern{k} & \text{convolution kernel with $k$ derivatives} \\
48 :			& \mid & \TYfield{k}{d}{\theta} & \text{$d$-dimension field of $\theta$ values and $k$ derivatives} \\
49 :			\end{array}%
50 :			\end{displaymath}%
51 :			where $o\in\Nat$ is the tensor order, $d,n,m\in\SET{2,3}$ are dimensions,
52 :			and $k\in\Nat$ is the differentiability of a field.
53 :			\caption{Diderot types}
54 :			\label{fig:types}
55 :			\end{figure}%
56 :			We use some type abbreviations for common cases:
57 :	jhr	16	\begin{eqnarray*}
58 :			\TYreal & = & \TYtensor{0}{d} \quad\text{for any $d$} \\
59 :			\TYvec{d} & = & \TYtensor{1}{d}
60 :			\end{eqnarray*}%
61 :
62 :			\section{Operations}
63 :
64 :			\subsection{Scalar operations}
65 :			\noindent{}Arithmetic:
66 :			\begin{displaymath}
67 :	jhr	17	\begin{array}{cl}
68 :			\BinopTy{{\odot}}{\tau}{\tau}{\tau}
69 :			& \text{for $\odot\in\SET{{+},{-},{*},{/}}$ and $\tau\in\SET{\TYint,\TYreal}$} \\
70 :			\UnopTy{{-}}{\tau}{\tau}
71 :			& \text{for $\tau\in\SET{\TYint,\TYreal}$}
72 :			\end{array}%
73 :	jhr	16	\end{displaymath}%
74 :
75 :			\noindent{}Comparisons:
76 :			\begin{displaymath}
77 :			\BinopTy{{\odot}}{\tau}{\tau}{\TYbool}
78 :			\qquad\text{for $\odot\in\SET{{<},{\leq},{=},{\neq}{>},{\geq}}$ and $\tau\in\SET{\TYint,\TYreal}$}
79 :			\end{displaymath}%
80 :
81 :
82 :			\subsection{Matrix operations}
83 :
84 :			\subsection{Tensor operations}
85 :
86 :			\noindent{}Scalar multiplication:
87 :			\begin{displaymath}
88 :			\begin{array}{c}
89 :			\BinopTy{{*}}{\TYreal}{\TYtensor{o}{d}}{\TYtensor{o}{d}} \\
90 :			\BinopTy{{*}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
91 :			\end{array}%
92 :			\end{displaymath}%
93 :
94 :	jhr	17	\noindent{}Scalar division:
95 :			\begin{displaymath}
96 :			\BinopTy{{/}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
97 :			\end{displaymath}%
98 :
99 :	jhr	16	\noindent{}Addition:
100 :			\begin{displaymath}
101 :			\BinopTy{{\odot}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
102 :			\qquad\text{for $\odot\in\SET{{+},{-}}$}
103 :			\end{displaymath}%
104 :
105 :	jhr	17	\noindent{}Negation:
106 :			\begin{displaymath}
107 :			\UnopTy{-}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
108 :			\end{displaymath}%
109 :
110 :	jhr	16	\subsection{Field operations}
111 :
112 :			\noindent{}Creation from an image:
113 :			\begin{displaymath}
114 :	jhr	17	\BinopTy{\OPsample}{\TYkern{k}}{\TYimage{d}{\rho}}{\TYfield{k}{d}{\theta}}
115 :			\qquad\text{where $\theta$ is the real conversion of $\rho$.}
116 :	jhr	16	\end{displaymath}%
117 :
118 :	jhr	17	\noindent{}Scalar multiplication:
119 :			\begin{displaymath}
120 :			\begin{array}{c}
121 :			\BinopTy{{*}}{\TYreal}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \\
122 :			\BinopTy{{*}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}}
123 :			\end{array}%
124 :			\end{displaymath}%
125 :
126 :			\noindent{}Scalar division:
127 :			\begin{displaymath}
128 :			\BinopTy{{/}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}}
129 :			\end{displaymath}%
130 :
131 :	jhr	16	\noindent{}Addition:
132 :			\begin{displaymath}
133 :	jhr	17	\BinopTy{{\odot}}{\TYfield{k_1}{d}{\theta}}{\TYfield{k_2}{d}{\theta}}{\TYfield{\min(k_1,k_2)}{d}{\theta}}
134 :	jhr	16	\qquad\text{for $\odot\in\SET{{+},{-}}$}
135 :			\end{displaymath}%
136 :
137 :			\noindent{}Differentiation:
138 :			\begin{displaymath}
139 :	jhr	17	\UnopTy{\OPdiff}{\TYfield{k}{d}{\TYtensor{o}{d}}}{\TYfield{k-1}{d}{\TYtensor{o+1}{d}}}
140 :	jhr	16	\qquad\text{for $k > 0$}
141 :			\end{displaymath}%
142 :
143 :			\noindent{}Probing:
144 :			\begin{displaymath}
145 :	jhr	17	\BinopTy{@}{\TYfield{k}{d}{\theta}}{\TYvec{d}}{\theta}
146 :	jhr	16	\end{displaymath}%
147 :
148 :			\end{document}

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