Be more brief

1 files changed, 7 insertions, 9 deletions

diff --git a/presentation/presentation.md b/presentation/presentation.md
index e2898d4..d3ab7be 100644
--- a/presentation/presentation.md
+++ b/presentation/presentation.md
@@ -69,8 +69,7 @@ singletons.
 
 * I show an $\langle O_{\varphi}(n), O_{\varphi}(m \log{m}) \rangle$ solution to
   MSO query answering.
-* Answering is $O_{\varphi}(1)$ when restricting to only first-order free
-  variables, matching Kazana's result.
+* $O_{\varphi}(1)$ for first-order free variables, matching Kazana's result.
 * Should be understandable by a CS student who has taken undergraduate
   algorithmics and automata theory courses.
 
@@ -117,16 +116,15 @@ Note: computing the partition takes time $O(m \log{m})$ as it uses a sort.
 
 ## Computing Roots of Parts - Subtree
 
-* During preprocessing, we precomputed a run of $A$ over $T$.
-* In a subtree part, no vertices other than the root were relabeled.
-* So we can apply $A$'s transition function to the precomputed states of the
-  root's children, and the root's new label.
+* During preprocessing, we precompute $A$'s run over $T$.
+* In a subtree part, only the root was relabeled.
+* Apply $A$'s transition function to the precomputed states of the root's
+  children and the root's new label.
 
 ## Computing Roots of Parts - Singleton
 
 * Both of the singleton's children are roots of parts.
-* Since we're working bottom-up, we've already computed the states in the roots
-  of those.
+* We're working bottom-up, we've already computed the states in those roots.
 * Apply $A$'s transition function to those and the singleton's new label.
 
 ## Computing Roots of Parts - Subtree With a Hole
@@ -135,7 +133,7 @@ Note: computing the partition takes time $O(m \log{m})$ as it uses a sort.
 * Idea: can be computed by a DFA walking up from hole to root.
 * Chapter 3: Branch Infix Regular Questions
     * Show how to solve this in $\langle O_A(n), O_A(1) \rangle$.
-    * Generalization of a clever algorithm on words.
+    * Generalization of a known algorithm on words.
 
 ## Computing Roots of Parts - Subtree With a Hole