From b358bccd3f36b7e4a1ddf8c8122755d91d000340 Mon Sep 17 00:00:00 2001
From: Christoph Egger <christoph@christoph-egger.org>
Date: Mon, 25 Apr 2016 16:30:34 +0200
Subject: [PATCH] Misc fixes

---
 chapters/finishingcool.tex | 22 +++++++++-------------
 chapters/optimizations.tex | 14 ++++++--------
 2 files changed, 15 insertions(+), 21 deletions(-)

diff --git a/chapters/finishingcool.tex b/chapters/finishingcool.tex
index 1b747e9..be9565a 100644
--- a/chapters/finishingcool.tex
+++ b/chapters/finishingcool.tex
@@ -3,33 +3,29 @@
 
 \begin{frame}{Where we left (TODO)}
   \begin{itemize}
-  \item actually implementing deferral tracking
-  \item deciding when some deferral can be considered ``finished''
+  \item Actually implementing deferral tracking
+  \item Deciding when some deferral can be considered ``finished''
   \item \dots
-  \item finding ``right'' heuristic for when to try and close the tableaux
-  \item fixing all the bugs
+  \item Finding ``right'' heuristic for when to try and close the tableau
+  \item Fixing all the bugs
   \end{itemize}
 \end{frame}
 
 \begin{frame}{Extra nodes}
   \begin{itemize}
-  \item Different set of deferrals create different Nodes
-  \item Exponential increase in Nodes
+  \item Different set of deferrals create different nodes
+  \item Exponential increase in number of nodes
   \end{itemize}
 \end{frame}
 
-\begin{frame}
-  TODO: graphviz der $p\#q\#$ formel
-\end{frame}
-
 \begin{frame}{Propagation}
   \begin{itemize}
   \item COOL works in rounds
     \begin{itemize}
-    \item One further solution from each Core (propositional)
-    \item All possible modal steps
+    \item One further state for each core (propositional, minisat)
+    \item All possible modal steps (coalgebraic step)
     \end{itemize}
-  \item Check for satisfiability after \#{}OpenStates rounds\pause
+  \item Check for (un)satisfiability after \#{}OpenStates rounds\pause
   \item Ideas
     \begin{itemize}
     \item Creating cycles forces earlier propagation
diff --git a/chapters/optimizations.tex b/chapters/optimizations.tex
index 6517e64..9de0c1e 100644
--- a/chapters/optimizations.tex
+++ b/chapters/optimizations.tex
@@ -1,17 +1,16 @@
 \section{Optimizing}
 \subsection*{Dummy}
 
-\begin{frame}{Semnatic Branching}
+\begin{frame}{Semantic Branching}
   \begin{itemize}
-  \item Idea: Add aditional information when considering second
-    alternative of an or: ${\phi \vee \psi}$ becomes ${\phi \vee
-      (¬\phi \wedge \psi)}$
-  \item Do not reexplore branches already known to be unsat already
+  \item Idea: Add additional information when considering second
+    alternative of a disjunction: ${\phi \vee \psi}$ becomes ${\phi
+      \vee (¬\phi \wedge \psi)}$
+  \item Do not re-explore branches already known to be unsatisfiable already
   \end{itemize}
 \end{frame}
 
 \begin{frame}{Non-atomic Negations}
-  TODO: ?
   \begin{itemize}
   \item Consider $\heartsuit\heartsuit\heartsuit{}p \wedge \overline{\heartsuit}\overline{\heartsuit}\overline{\heartsuit}¬p$
   \item Same as $\heartsuit\heartsuit\heartsuit{}p \wedge
@@ -40,11 +39,10 @@
   \end{itemize}
 \end{frame}
 
-
 \begin{frame}{Removing Edges}
   \begin{block}{Propagation}
     \begin{itemize}
-    \item Runs in $O(edge\cdot nodes)$
+    \item Runs in $\mathcal{O}(\mathit{|edge|}\cdot\mathit{|nodes|})$
     \item Needs to consider all satisfiable nodes
     \item Iterates over all edges
     \end{itemize}
-- 
2.39.2