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commit b5210e637ce32ae99d92c12d683a8288527b53f7 1 parent 67ff01b
authored February 27, 2014

Showing 1 changed file with 34 additions and 1 deletion. Show diff stats Hide diff stats

  1. 35  recce.ltx
35  recce.ltx
@@ -2060,7 +2060,7 @@ in the input, at most some finite number
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 of rightmost derivations.
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2062 2062
 \begin{definition}
2063  
-\Var{g} is
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+\var{g} is
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 a \dfn{finite derivation grammar} if \var{g}
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 is an unambiguous grammar,
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 and if,
@@ -2095,6 +2095,39 @@ where \var{c} is a finite constant,
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 which may be a function of the grammar.
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 \end{theorem}
2097 2097
 
  2098
+\begin{proof}
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+A medial YIM must take the form
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+\begin{equation*}
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+ \Veim{medial} = [ \Vdr{medial}, \Vloc{medial-origin} ]
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+\end{equation*}
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+where
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+\begin{equation*}
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+ \Vdr{medial} = [\Vsym{A} \de \Vsf{pre-dot} \mydot \Vsym{post-dot} ].
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+\end{equation*}
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+
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+We claim that that number of medial dotted rules
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+\begin{equation}
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+    \label{e:finite-medial-claim} \var{c} \le \size{\var{dotted-rules}}
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+\end{equation}
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+where \size{\var{dotted-rules}} is the cardinality
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+of the set of dotted rules in \var{g}.
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+To show \eqref{e:finite-medial-claim},
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+we assume, for a reductio, that there are
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+more medial Earley items at \var{cut} then there are
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+medial dotted rules.
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+This means that at least two of the medial Earley items,
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+call them \Veim{eim1} and \Veim{eim2} must share the
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+same dotted rule, call it \Vdr{shared}.
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+We can write these two Earley items
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+\begin{gather*}
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+ \Veim{eim1} = [ \Vdr{shared}, \Vloc{origin1} ] \\
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+ \text{and} \quad \Veim{eim2} = [ \Vdr{shared}, \Vloc{origin2} ]
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+\end{gather*}
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+Since by the assumption for the reduction,
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+\Veim{eim1} and \Veim{eim2} differ, we have
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+$\Vloc{origin1} \neq \Vloc{origin2}$.
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+\end{proof}
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+
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 \begin{theorem}
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 Let \var{g} be a finite derivation grammar.
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 The number of completed YIM's for it in the Earley set \var{cut},

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