\end{quote}
+\subsection{Parentheses}
+
+Parentheses are used for grouping of alternatives within the right-hand side
+of a production rule. Specifically, a right-hand side
+\begin{quote}
+ \syntax{$\alpha$ @($\beta_1$ | $\beta_2$ | $\cdots$ | $\beta_n$@) $\gamma$}
+\end{quote}
+where $\alpha$, $\beta_i$, and $\gamma$ are any sequence of nonterminal
+symbols or parenthesized groups, is equivalent to the right-hand side
+\begin{quote}
+ \syntax{$\alpha$ $b$ $\gamma$}
+\end{quote}
+together with the new production
+\begin{quote}
+ \syntax{$b$ ::= $\beta_1$ | $\beta_2$ | $\cdots$ | $\beta_n$}
+\end{quote}
+where $b$ is a new nonterminal symbol.
+
+Given the indexed-nonterminal notation described below, one might consider a
+group \syntax{@($\beta_1$ | $\beta_2$ | $\cdots$ | $\beta_n$@)} equivalent to
+\syntax{<group>@[$\beta_1$ | $\beta_2$ | $\cdots$ | $\beta_n$@]}, where
+\begin{quote}
+ \syntax{<group>@[$x$@] ::= $x$}
+\end{quote}
+
+
\subsection{Indexed nonterminals}
Anywhere a simple nonterminal name $x$ may appear in the grammar, an
right-hand side of a production rule, each actual argument may be a sequence
of alternative right-hand sides, separated by `$|$', rather than a a simple
terminal or nonterminal symbol. A complex indexing of this form, say
-\syntax{$x$@[$\alpha_1^1$ @! $\alpha_1^2$ $| \cdots |$ $\alpha_1^{m_1},
-\ldots,$ $\alpha_n^1$ @! $\alpha_n^2$ $| \cdots |$ $\alpha_n^{m_n}$@]}
-means exactly the same as \syntax{$x$@[$a_1, \ldots, a_n$@]} with the
-additional rules
+\syntax{$x$@[$\alpha_1^1$ | $\alpha_1^2$ | $\cdots$ | $\alpha_1^{m_1},
+\ldots,$ $\alpha_n^1$ | $\alpha_n^2$ | $\cdots$ | $\alpha_n^{m_n}$@]} means
+exactly the same as \syntax{$x$@[$a_1, \ldots, a_n$@]} with the additional
+rules
\begin{quote}
- \syntax{$a_1$ ::= $\alpha_1^1$ @! $\alpha_1^2$ $| \cdots |$ $\alpha_1^{m_1}$} \\*
+ \syntax{$a_1$ ::= $\alpha_1^1$ | $\alpha_1^2$ | $\cdots$ | $\alpha_1^{m_1}$} \\*
\hbox{}\qquad $\vdots$ \\*
- \syntax{$a_n$ ::= $\alpha_n^1$ @! $\alpha_n^2$ $| \cdots |$ $\alpha_1^{m_n}$}
+ \syntax{$a_n$ ::= $\alpha_n^1$ | $\alpha_n^2$ | $\cdots$ | $\alpha_1^{m_n}$}
\end{quote}
where $a_1$, \ldots, $a_n$ are new nonterminal symbols.
\item @[$x$@] abbreviates @<optional>$[x]$, denoting an optional occurrence
of $x$:
\begin{quote}
- \syntax{@[$x$@] ::= <optional>$[x]$ ::= $\epsilon$ @! $x$}
+ \syntax{@[$x$@] ::= <optional>$[x]$ ::= $\epsilon$ | $x$}
\end{quote}
\item $x^*$ abbreviates @<zero-or-more>$[x]$, denoting a sequence of zero or
more occurrences of $x$:
\begin{quote}
\syntax{$x^*$ ::= <zero-or-more>$[x]$ ::=
- $\epsilon$ @! <zero-or-more>$[x]$ $x$}
+ $\epsilon$ | <zero-or-more>$[x]$ $x$}
\end{quote}
\item $x^+$ abbreviates @<one-or-more>$[x]$, denoting a sequence of one or
more occurrences of $x$:
\item @<list>$[x]$ denotes a sequence of one or more occurrences of $x$
separated by commas:
\begin{quote}
- \syntax{<list>$[x]$ ::= $x$ @! <list>$[x]$ "," $x$}
+ \syntax{<list>$[x]$ ::= $x$ | <list>$[x]$ "," $x$}
\end{quote}
\end{itemize}