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Update for changes to Leo logic

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1 parent f4dcd2b commit 5d2973a37d907d36426a0ea47b49ee1b053d2617 Jeffrey Kegler committed Jun 19, 2013
Showing with 23 additions and 18 deletions.
  1. +23 −18 recce.ltx
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41 recce.ltx
@@ -2046,7 +2046,15 @@ Marpa's Earley sets are correct.
\begin{sloppypar}
The proof
-is by double induction.
+is by triple induction,
+that is, induction with a depth down to 3 levels.
+We number the levels of induction from
+outermost to innermost,
+0, 1 and 2.
+The level 0 induction is usually called the outer induction.
+The level 1 induction is usually called the inner induction.
+The level 2 induction is referred to by its number.
+
The outer induction is on the Earley sets.
The outer induction hypothesis is that all Earley sets
\EVtable{\Marpa}{i},
@@ -2328,7 +2336,7 @@ that
the number of Earley items is \On{},
so a traversal of them must terminate.
-Consider, for the purposes of an inner induction,
+Consider, for the purposes of the level 2 induction,
the reductions of \Leo{} to occur in generations.
Let the scanned Earley items be generation 0.
An EIMT produced by a reduction is generation $\var{n} + 1$
@@ -2337,7 +2345,7 @@ Predicted Earley items do not need to be assigned generations.
In Marpa grammars they can never contain completions,
and therefore can never act as the component of a reduction.
-The induction hypothesis for the inner induction
+The induction hypothesis for the level 2 induction
is that for some \var{n},
the Earley items of \EVtable{\Marpa}{i} for generations 0 through \var{n}
are complete and consistent.
@@ -2347,7 +2355,7 @@ In Section \ref{s:scan-complete},
we showed that generation 0 is complete --
it contains Earley items
corresponding to all of the generation 0 EIMT's of \Leo.
-This is the basis of the inner induction.
+This is the basis of the level 2 induction.
Since we stipulated that \Marpa{} adds Earley items
at the end of each set,
@@ -2356,15 +2364,13 @@ Therefore \Marpa{},
when creating Earley items of generation $\var{n}+1$
while traversing \EVtable{\Marpa}{i},
can rely
-on the inner induction hypothesis for
-the completeness of Earley items in
+on the level 2 induction hypothesis for
+the completeness of Earley items
in generation \var{n}.
Let
-\begin{equation*}
-\Veim{working} = [\Vah{working}, \Vorig{working}]
-\end{equation*}
-be the Earley item in \EVtable{\Marpa}{i}
+$\Veim{working} \in \Ves{i}$
+be the Earley item
currently being considered as a potential component for
an Earley reduction operation.
From the pseudocode, we see
@@ -2397,7 +2403,8 @@ as well as one corresponding
to every prediction that results from
an Earley reduction result
of generation $\var{n}+1$ in \EVtable{\Leo}{i}.
-This shows the case of reduction completeness.
+This shows the level 2 induction
+and the case of reduction completeness.
\subsubsection{Leo reduction is complete}
\label{s:leo-complete}
@@ -3132,20 +3139,18 @@ The size of the input, \size{\Cw},
is the maximum over
\var{tokens} of $\Vloc{start}+\var{length}$.
-\begin{sloppypar}
Multiple tokens can start at a single location.
(This is how \Marpa{} supports ambiguous tokens.)
-Tokens may have multiple lengths.
The variable-length,
ambiguous and overlapping tokens
of \Marpa{}
bend the conceptual framework of ``parse location''
beyond its breaking point,
-and a new term for parse location is introduced,
-the \dfn{earleme}.
-Token length is measured in earlemes,
-and the start and end location of a token is indicated in earlemes.
-\end{sloppypar}
+and a new term for parse location is needed.
+Start and end of tokens are described in terms
+of \dfn{earleme} locations,
+or simply \dfn{earlemes}.
+Token length is also measured in earlemes.
Like standard parse locations, earlemes start at 0,
and run up to \size{\Cw}.

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