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Jeffrey Kegler Jeffrey Kegler
Jeffrey Kegler authored and Jeffrey Kegler committed Feb 28, 2014
1 parent 67ff01b commit b5210e637ce32ae99d92c12d683a8288527b53f7
Showing with 34 additions and 1 deletion.
  1. +34 −1 recce.ltx
@@ -2060,7 +2060,7 @@ in the input, at most some finite number
of rightmost derivations.
-\Var{g} is
+\var{g} is
a \dfn{finite derivation grammar} if \var{g}
is an unambiguous grammar,
and if,
@@ -2095,6 +2095,39 @@ where \var{c} is a finite constant,
which may be a function of the grammar.
+A medial YIM must take the form
+ \Veim{medial} = [ \Vdr{medial}, \Vloc{medial-origin} ]
+ \Vdr{medial} = [\Vsym{A} \de \Vsf{pre-dot} \mydot \Vsym{post-dot} ].
+We claim that that number of medial dotted rules
+ \label{e:finite-medial-claim} \var{c} \le \size{\var{dotted-rules}}
+where \size{\var{dotted-rules}} is the cardinality
+of the set of dotted rules in \var{g}.
+To show \eqref{e:finite-medial-claim},
+we assume, for a reductio, that there are
+more medial Earley items at \var{cut} then there are
+medial dotted rules.
+This means that at least two of the medial Earley items,
+call them \Veim{eim1} and \Veim{eim2} must share the
+same dotted rule, call it \Vdr{shared}.
+We can write these two Earley items
+ \Veim{eim1} = [ \Vdr{shared}, \Vloc{origin1} ] \\
+ \text{and} \quad \Veim{eim2} = [ \Vdr{shared}, \Vloc{origin2} ]
+Since by the assumption for the reduction,
+\Veim{eim1} and \Veim{eim2} differ, we have
+$\Vloc{origin1} \neq \Vloc{origin2}$.
Let \var{g} be a finite derivation grammar.
The number of completed YIM's for it in the Earley set \var{cut},

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