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% my attempt to define a restricted tabular environment for BNF
% descriptions. Only two columns are allowed, \rightarrow is used
% between them, everything is aligned down the center, and the default
% type style is italic (so you have to explicitly use roman for
% terminals). Also defines \pipe and \term. 
% Shamelessly stolen from the definition of tabular

\def\bnf{\def\@halignto{}\@bnf}

\def\endbnf{\crcr\egroup\egroup $\egroup}

\def\@bnf{\leavevmode \hbox \bgroup $\let\@acol\@tabacol 
   \let\@classz\@tabclassz
   \let\@classiv\@tabclassiv 
   \let\\\cr
   \def\arraystretch{1.5}
   % make italic the default
   \it 
   \@array[c]{r@{\ $\longrightarrow $\ }l}
%   \@array[c]{r@{\ $\longrightarrow $\ }p{}}
}

% these are handy to have
\def\term#1{{\tt #1}}
\def\|{ $\mid$\ }
\newcommand{\code}[1]{{\tt #1}}

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\begin{document}
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The following BNF describes the syntax of the Common Answer
Specification for the ATIS domain:
\begin{center}
\begin{bnf}
answer 		& scalar-value \| relation \| \term{NO\_ANSWER} \\
boolean-value	& \term{yes \| true \| no \| false} \\
number-value	& integer \| real-number \\
relation	& \term{(} tuple$^\star$ \term{)} \\
scalar-value	& boolean-value \| number-value \| string \\
tuple		& \term{(} value$^+$ \term{)} \\
value		& scalar-value \| \term{NIL} \\
\end{bnf}
\end{center}

We assume as primitives the values {\it integer}, {\it real-number},
and {\it string}. Integers and reals are not distinguished, and only
non-exponential real numbers are allowed. Strings must always be
enclosed in double quotes (i.e., \term{"DFW"}), are case-sensitive,
and should be upper-case (since strings in the ATIS database are).
The special tokens \term{yes}, \term{no}, \term{true},
\term{false}, \term{no_answer} and \term{nil} are not case-sensitive. 

Answer relations must be derived from the existing relations in the
database, either by subsetting and combining relations or by
operations like averaging, summation, etc.  NIL as the representation
of missing data is allowed as a special case for any value, so a legal
answer indicating the costs of ground transportation in Boston would be

\begin{quote}
\term{(("L"   5.00 ) ("R"    nil ) ("A"    nil ) ("R"    nil ))}
\end{quote}


Empty tuples are not allowed (but empty relations are).  All the
tuples in a relation must have the same number of values, those values
must be of the same respective types (boolean, string, or number), and
the types in the answer must be the same as the types in the database
(i.e., database values like \term{"1355"} cannot be converted from strings
to numbers in answer expressions).

\end{document}