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	revision 16, Wed Jan 13 23:03:09 2010 UTC	revision 17, Fri Jan 15 02:35:26 2010 UTC
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7	\setlength{\evensidemargin}{0.25in}	\setlength{\evensidemargin}{0.25in}
8	\setlength{\parskip}{5pt}	\setlength{\parskip}{5pt}
9
10	\title{Diderot}	\title{Diderot design}
11	\author{	\author{
12	Gordon Kindlmann \\	Gordon Kindlmann \\
13	University of Chicago \\	University of Chicago \\
#	Line 16	Line 16
16	John Reppy \\	John Reppy \\
17	University of Chicago \\	University of Chicago \\
18	{\small\tt{}jhr@cs.uchicago.edu} \\	{\small\tt{}jhr@cs.uchicago.edu} \\
19		\and
20		Thomas Schultz \\
21		University of Chicago \\
22		{\small\tt{}t.schultz@uchicago.edu} \\
23	}	}
24	\date{\today}	\date{\today}
25
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32	This document is a semi-formal design of Dedierot.	This document is a semi-formal design of Dedierot.
33
34	\section{Types}	\section{Types}
35	The syntax of Diderot types is as follows:	The syntax of Diderot types is given in \figref{fig:types}.
36		\begin{figure}[t]
37	\begin{displaymath}	\begin{displaymath}
38	\begin{array}{rclr}	\begin{array}{rclr}
39	\tau & ::=  & \TYbool & \text{booleans} \\	\rho  & ::= & $\ldots$ & \text{NNRD types} \\[1em]
40	& \mid & \TYint & \text{integers} \\	\iota & ::= & \TYbool & \text{booleans} \\
41	& \mid & \TYtensor{o}{d} & \text{tensors of order $o$ and dimension $d$} \\	& \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} \\	& \mid & \TYmatrix{n}{m} & \text{$n\times{}m$ matrix} \\
46	& \mid & \TYimage{d}{\tau} & \text{image of dimension $d$ and $\tau$ elements}\\	& \mid & \TYimage{d}{\rho} & \text{image of dimension $d$ and $\rho$ elements}\\
47	& \mid & \TYkern{k} & \text{convolution kernel with $k$ derivatives} \\	& \mid & \TYkern{k} & \text{convolution kernel with $k$ derivatives} \\
48	& \mid & \TYfield{k}{d}{\tau} & \text{$d$-dimension field of $\tau$ values and $k$ derivatives} \\	& \mid & \TYfield{k}{d}{\theta} & \text{$d$-dimension field of $\theta$ values and $k$ derivatives} \\
49	\end{array}%	\end{array}%
50	\end{displaymath}%	\end{displaymath}%
51	where $o\in\Nat$ is the tensor order, $d,n,m\in\SET{2,3}$ are dimensions,	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.	and $k\in\Nat$ is the differentiability of a field.
53		\caption{Diderot types}
54	Some type abbreviations:	\label{fig:types}
55		\end{figure}%
56		We use some type abbreviations for common cases:
57	\begin{eqnarray*}	\begin{eqnarray*}
58	\TYreal & = & \TYtensor{0}{d} \quad\text{for any $d$} \\	\TYreal & = & \TYtensor{0}{d} \quad\text{for any $d$} \\
59	\TYvec{d} & = & \TYtensor{1}{d}	\TYvec{d} & = & \TYtensor{1}{d}
#	Line 54	Line 64
64	\subsection{Scalar operations}	\subsection{Scalar operations}
65	\noindent{}Arithmetic:	\noindent{}Arithmetic:
66	\begin{displaymath}	\begin{displaymath}
67		\begin{array}{cl}
68	\BinopTy{{\odot}}{\tau}{\tau}{\tau}	\BinopTy{{\odot}}{\tau}{\tau}{\tau}
69	\qquad\text{for $\odot\in\SET{{+},{-},{*},{/}}$ and $\tau\in\SET{\TYint,\TYreal}$}	& \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	\end{displaymath}%	\end{displaymath}%
74
75	\noindent{}Comparisons:	\noindent{}Comparisons:
#	Line 77	Line 91
91	\end{array}%	\end{array}%
92	\end{displaymath}%	\end{displaymath}%
93
94		\noindent{}Scalar division:
95		\begin{displaymath}
96		\BinopTy{{/}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
97		\end{displaymath}%
98
99	\noindent{}Addition:	\noindent{}Addition:
100	\begin{displaymath}	\begin{displaymath}
101	\BinopTy{{\odot}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}	\BinopTy{{\odot}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
102	\qquad\text{for $\odot\in\SET{{+},{-}}$}	\qquad\text{for $\odot\in\SET{{+},{-}}$}
103	\end{displaymath}%	\end{displaymath}%
104
105		\noindent{}Negation:
106		\begin{displaymath}
107		\UnopTy{-}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
108		\end{displaymath}%
109
110	\subsection{Field operations}	\subsection{Field operations}
111
112	\noindent{}Creation from an image:	\noindent{}Creation from an image:
113	\begin{displaymath}	\begin{displaymath}
114	\BinopTy{\OPsample}{\TYkern{k}}{\TYimage{d}{\tau}}{\TYfield{k}{d}{\hat{\tau}}}	\BinopTy{\OPsample}{\TYkern{k}}{\TYimage{d}{\rho}}{\TYfield{k}{d}{\theta}}
115	\qquad\text{where $\hat{\tau}$ is the real conversion of $\tau$.}	\qquad\text{where $\theta$ is the real conversion of $\rho$.}
116		\end{displaymath}%
117
118		\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}%	\end{displaymath}%
130
131	\noindent{}Addition:	\noindent{}Addition:
132	\begin{displaymath}	\begin{displaymath}
133	\BinopTy{{\odot}}{\TYfield{k_1}{d}{\tau}}{\TYfield{k_2}{d}{\tau}}{\TYfield{\min(k_1,k_2)}{d}{\tau}}	\BinopTy{{\odot}}{\TYfield{k_1}{d}{\theta}}{\TYfield{k_2}{d}{\theta}}{\TYfield{\min(k_1,k_2)}{d}{\theta}}
134	\qquad\text{for $\odot\in\SET{{+},{-}}$}	\qquad\text{for $\odot\in\SET{{+},{-}}$}
135	\end{displaymath}%	\end{displaymath}%
136
137	\noindent{}Differentiation:	\noindent{}Differentiation:
138	\begin{displaymath}	\begin{displaymath}
139	\UnopTy{\OPdiff}{\TYfield{k}{d}{\tau}}{\TYfield{k-1}{d}{\tau}}	\UnopTy{\OPdiff}{\TYfield{k}{d}{\TYtensor{o}{d}}}{\TYfield{k-1}{d}{\TYtensor{o+1}{d}}}
140	\qquad\text{for $k > 0$}	\qquad\text{for $k > 0$}
141	\end{displaymath}%	\end{displaymath}%
142
143	\noindent{}Probing:	\noindent{}Probing:
144	\begin{displaymath}	\begin{displaymath}
145	\BinopTy{@}{\TYfield{k}{d}{\tau}}{\TYvec{d}}{\tau}	\BinopTy{@}{\TYfield{k}{d}{\theta}}{\TYvec{d}}{\theta}
146	\end{displaymath}%	\end{displaymath}%
147
148	\end{document}	\end{document}

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