diff --git a/document.pdf b/document.pdf
index 52564cb..5667997 100644
Binary files a/document.pdf and b/document.pdf differ
diff --git a/document.tex b/document.tex
index 6fa722a..a6a87c7 100644
--- a/document.tex
+++ b/document.tex
@@ -156,6 +156,15 @@
         Note: the $\lfp$ is applied to a unary operator, thus it's the least fixpoint of the lattice $\langle \wp(\mathcal{L}), \subseteq \rangle$ whose least element is $\emptyset$.
     \end{definition}
 
+    \subsection{An Approximator for Partial Stable Semantics}
+    
+    \begin{align*}
+        \Gamma(T, P) &\define \{ \head(r) ~|~ r \in \P, T \subseteq \bodyp(r), \bodyn(r) \intersect P = \emptyset \}\\
+        \Gamma(P, T) &\define \{ \head(r) ~|~ r \in \P, P \subseteq \bodyp(r), \bodyn(r) \intersect T = \emptyset \}\\
+        \Phi(T, P) &\define \Bigl( \Gamma(T, P), \Gamma(P, T) \Bigr)
+    \end{align*}
+
+    
     \section{The Polynomial Heirarchy}
     Intuitive definitions of ${\sf NP}$
     \begin{itemize}
@@ -176,4 +185,5 @@
         \item (Alternating Turing machine) A problem that is solved by some path of an algorithm that is allowed to branch in parallel. A branch is allowed to switch to ``ALL'' mode only once and require that all subsequent forks return success
     \end{itemize}
 
+
 \end{document}
\ No newline at end of file