Add chapter for common examples for trees

diff --git a/main.tex b/main.tex
index fc5a4fe7bc8a827d62d65658517facb523ef336c..301e37af84c90197a5f0a92725455998c513f8ff 100644 (file)
--- a/main.tex
+++ b/main.tex
@@ -15,6 +15,7 @@ or \url{http://www.gnu.org/licenses/agpl-3.0.txt}) or any later version.}}
 
 \author{Nathaniel Wesley Filardo}
 
+\usepackage{tikz}
 \usepackage{fullpage}
 \usepackage{xcolor}
 \usepackage{amsmath,amssymb,latexsym}
@@ -28,10 +29,9 @@ or \url{http://www.gnu.org/licenses/agpl-3.0.txt}) or any later version.}}
 \usepackage{multicol}
 \usepackage{multirow}
 \usepackage{proof}
-\usepackage{qtree}
+\usepackage{tikz-qtree}
 \usepackage{listings}
        \lstloadlanguages{Haskell}
-\usepackage{floatrow}
 \usepackage{caption}
 \usepackage{comment}
    % http://stackoverflow.com/questions/2193307/how-to-get-latex-to-hyphenate-a-word-that-contains-a-dash
@@ -253,6 +253,7 @@ or \url{http://www.gnu.org/licenses/agpl-3.0.txt}) or any later version.}}
 \maininclude{Homomorphism-Equality Automata}{zoo-tree/tahom}
 \maininclude{Tree Automata with Global Equality and Disequality}{zoo-tree/taged}
 \maininclude{Rigid Tree Automata}{zoo-tree/rta}
+\maininclude{Simple Separating Examples}{tree-sepex}
 
 %\part{Infinite Trees}
 

diff --git a/tree-sepex.tex b/tree-sepex.tex
new file mode 100644 (file)
index 0000000..8672210
--- /dev/null
+++ b/tree-sepex.tex
@@ -0,0 +1,52 @@
+When thinking about finite tree automata, it is often desirable to have a
+few simple examples in mind to compare classes.  We have collected a few
+such examples here.
+
+\subsection{Equalities Vs. Recursion}
+
+\subsubsection{All-equal lists}
+
+Not all classes of automata permit free interaction of recursion and
+equality constraints; as such, these classes may not be able to describe the
+language of all-equal lists: $\set{ [], [A], [A,A], [A,A,A], \ldots }$ for
+any $A$.  Note, however, that for a fixed, regular tree $A$, the set of
+all-equal lists of $A$ is a regular tree language.  The difficulty emerges
+when we wish to describe all-equal lists over an unbounded number of
+possible elements.
+
+\subsubsection{Markovian-equal lists}
+
+A generalization of the above, consider the language of lists composed of
+regions of size at least two whose elements are equal, for example:
+$[A,A,B,B,B]$ or $[A,A,A,B,B,C,C,D,D,D,D]$.  Rigid tree automata will not be
+able to describe such a class due to the need for unboundedly many
+equivalence classes.
+
+\subsection{Opacity}
+
+Consider the sets of trees described by
+%
+\begin{center}\begin{tabular}{cc}
+    $\set{ f(g(X,Y),g(Z,X)) \middle\vert X,Y,Z \in \mathcal{T}(\Sigma)}$
+  & $\set{ f(W,g(A,B)) \middle\vert A,B,W \in \mathcal{T}(\Sigma) \wedge A\vert_1 = B\vert_2 }$
+  \\
+    \begin{tikzpicture}
+      \Tree [.f [.g X Y ] [.g Z X ] ]
+    \end{tikzpicture}
+  & \begin{tikzpicture}
+      \Tree [.f W [.g$_{11=21}$ A B ] ]
+    \end{tikzpicture}
+\end{tabular}\end{center}
+%
+These descriptions are clearly (given their comprehension form) amenable to
+classification by Opaque constraints.  Their intersection is $\set{
+f(g(A,B),g(C,A)) \middle\vert A,B,C \in \mathcal{T}(\Sigma) \wedge C\vert_1
+= A\vert_2 }$, which can be made amenable to Opaque classification only if
+we are able to unfold $\mathcal{T}(\Sigma)$ for $A$ and $C$.  While we can
+do this for regular trees over finite $\Sigma$, in general what we are
+calling here $A$ will actually be trees accepted by a particular state, so
+we can only describe this intersection with Opaque constraints if we are
+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.}