Add non-overlap tree metaconstraint

While here, and sepex for necessarily overlapping structures

diff --git a/main.tex b/main.tex
index 85f62267e3e0fd32ae8912b3bb696ed7b956e61d..059107902a2e57e6abb346d2684b0bf0adadc760 100644 (file)
--- a/main.tex
+++ b/main.tex
@@ -43,6 +43,7 @@ or \url{http://www.gnu.org/licenses/agpl-3.0.txt}) or any later version.}}
 \usepackage{tabu}
 \usepackage{natbib}
 \usepackage{hyperref}
+\usepackage{wrapfig}
 
 \usepackage[xindy,toc]{glossaries}     % AFTER: hyperref
 

diff --git a/tree-sepex.tex b/tree-sepex.tex
index 86722102d8d727c45921665f69e531db8fa2ca5c..7c32e1784eaacd423c9ad96db49784e43b1d7532 100644 (file)
--- a/tree-sepex.tex
+++ b/tree-sepex.tex
@@ -50,3 +50,18 @@ able to unfold the description of that state.  In general, that unfolding
 process might not terminate.
 %
 \Note{I would feel a lot better if somebody checked this.}
+
+\subsection{Necessarily Overlapping}
+
+\begin{wrapfigure}{r}{1in}\centering\begin{tikzpicture}
+  \Tree [.f$_{1=22}$ X [.g$_{1=2}$ X X ] ]
+\end{tikzpicture}\end{wrapfigure}
+The tree in the figure to the right is a small example of a structure which
+necessarily involves overlapping constraints, assuming that the nodes
+$X$ are drawn from an infinite set of trees; if there are merely finitely
+many possibilities, then the constraints may be pushed into the state
+label.  Specifically, because we must constrain positions $1$, $21$ and $22$
+to be equal, we are obligated to have at least one of $21$ and $22$ as
+constraint paths in the automata, and then either the other, which overlaps,
+or the constraint $1=2$ at position $2$, which again overlaps.  However,
+this structure is amenable to Opaque description, as shown.

diff --git a/treeautintro.tex b/treeautintro.tex
index 38a0eb8b356ed347d1bace2f0fe9149c6774361f..a9512c9962e7f2d3b0f896097fbc6898d5312c53 100644 (file)
--- a/treeautintro.tex
+++ b/treeautintro.tex
@@ -100,6 +100,13 @@ fragment of the tree being labeled (or generated); that is, constraint paths
 may not involve a $\config$-labeled run node other than at their apex and
 endpoints.
 
+\paragraph{Non-Overlapping} No constraint paths may share an edge; that is,
+if it is possible for a constraint path to traverse $q \in \config$ to reach
+its $i$\textsuperscript{th} child, then no transition into $q$ may constrain
+position $i$.%
+%
+\footnote{See \cite[Exercise 4.8]{tata} for an alternate formalization.}
+
 \paragraph{Opaque} If a position is constrained by a transition, no other
 constraint path may cross this position.