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add a approximator for partial stable semantics

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Spencer Killen 2023-05-14 11:13:05 -06:00
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@ -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 $\emptyset$. 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} \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} \section{The Polynomial Heirarchy}
Intuitive definitions of ${\sf NP}$ Intuitive definitions of ${\sf NP}$
\begin{itemize} \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 \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{itemize}
\end{document} \end{document}