A {\em partial order}$\preceq$ is a relation over a set $S$ such that for every triple of elements $x, y, z \in S$ the following hold
\begin{itemize}
\item (reflexivity) $x \preceq x$
\item (antisymmetry) $(x \preceq y \land y \preceq x)\implies x = y$
\item (transitivity) $(x \preceq y \land y \preceq z)\implies(x \preceq z)$
\end{itemize}
\end{definition}
\begin{definition}
Given a partial order $\preceq$ over a set $S$ and a subset $X \subseteq S$, a {\em lower bound} of $X$ (resp.\ an {\em upper bound} of $X$) is an element $x \in S$ (Note that it may be the case that $x \not\in X$) such that
\begin{itemize}
\item$\forall y \in X, \boxed{x}\preceq y$
\item (resp. $\forall y \in X, \boxed{y}\preceq x$)
\end{itemize}
The {\em greatest lower bound} of a set $X \subseteq S$ (denoted as $\textbf{glb}(X)$) is a unique upperbound of the set of all lowerbounds of $X$.
The {\em least upper bound} of a set $X \subseteq S$ (denoted as $\textbf{lub}(X)$) is a unique lowerbound of the set of all upperbounds of $X$.
In general, $\textbf{lub}(X)$ and $\textbf{glb}(X)$ may not exist.
\end{definition}
\begin{definition}
A (complete) lattice $\langle\mathcal{L}, \preceq\rangle$ is a set of elements $\mathcal{L}$ and a partial order $\preceq$ over $\mathcal{L}$ such that for any set $S \subseteq\mathcal{L}$
\begin{itemize}
\item$\textbf{lub}(X)$ and $\textbf{glb}(X)$ exist and are unique.
\end{itemize}
\end{definition}
\subsection{Fixpoint Operators}
\begin{definition}
Given a complete lattice $\langle\mathcal{L}, \preceq\rangle$, a fixpoint operator over the lattice is a function $\Phi: \mathcal{L}\rightarrow\mathcal{L}$ that is {\em$\preceq$-monotone}
\begin{itemize}
\item$\Phi$ is $\preceq$-monotone if for all $x, y \in\mathcal{L}$
\begin{align*}
x \preceq y \implies\Phi(x) \preceq\Phi(y)
\end{align*}
(Note: this does not mean the function is inflationary, i.e., $x \preceq\Phi(x)$ may not hold)
\end{itemize}
A {\em fixpoint} is an element $x \in\mathcal{L}$ s.t. $\Phi(x)= x$.
\end{definition}
\begin{theorem}[Knaster-Tarski]
The set of all fixpoints of a fixpoint operator $\Phi$ on a complete lattice is itself a complete lattice. The least element of this new lattice exists and is called the least fixpoint (denoted as $\lfp~{\Phi}$)
\end{theorem}
Some intuitions about lattices
\begin{itemize}
\item The entire lattice has a biggest element $\lub{}(\mathcal{L})=\top$ and a smallest element $\glb{}(\mathcal{L})=\bot$
\item When a lattice has a finite height (or finite domain). The least fixed point of a fixpoint operator can be computed by iteratively applying the fixpoint operator to $\bot$
\item An operator may return an element that is not comparable to the input, however, after a comparable element is returned (either greater or less than) that comparability and direction are maintained for all subsequent iterations.
\item Further, because $\bot$ is less than all elements in the lattice, it is always the case that $\bot\preceq\Phi(\bot)$
\end{itemize}
\section{Partial Stable Model Semantics}
\begin{definition}
A (ground and normal) {\em answer set program}$\P$ is a set of rules where each rule $r$ is of the form
\begin{align*}
h \leftarrow a_0,~ a_1,~ \dots,~ a_n,~ \Not b_0,~\Not b_1,~\dots,~\Not b_k
\end{align*}
where we define the following shorthand for a rule $r \in\P$
\begin{align*}
\head(r) &= h\\
\bodyp(r) &= \{ a_0,~ a_1,~ \dots,~ a_n \}\\
\bodyn(r) &= \{ b_0,~ b_1,~ \dots,~ b_k \}
\end{align*}
\end{definition}
\begin{definition}
A two-valued interpretation $I$ of a program $\P$ is a set of atoms that appear in $\P$.
\end{definition}
\begin{definition}
An interpretation $I$ is a \boxedt{model} of a program $\P$ if for each rule $r \in\P$
\begin{itemize}
\item If $\bodyp(r)\subseteq I$ and $\bodyn(r)\intersect I =\emptyset$ then $\head(r)\in I$.
\end{itemize}
\end{definition}
\begin{definition}
An interpretation $I$ is a \boxedt{stable model} of a program $\P$ if $I$ is a model of $\P$\textbf{and}
for every interpretation $I' \subseteq I$ there exists a rule $r \in\P$ such that
\begin{itemize}
\item$\bodyp(r)\subseteq I'$, $\bodyn(r)\intersect I \not=\emptyset$ (Note that this is $I$ and not $I'$) and $\head(r)\not\in I'$
\end{itemize}
\end{definition}
\begin{definition}
A three-valued interpretation $(T, P)$ of a program $\P$ is a pair of sets of atoms such that $T \subseteq P$.
The \boxedt{truth-ordering} respects $\ff < \uu < \tt$ and is defined for two three-valued interpretations $(T, P)$ and $(X, Y)$ as follows.
\begin{align*}
(T, P) \preceq_t (X, Y) \textrm{ iff } T \subseteq X \land P \subseteq Y
\end{align*}
The \boxedt{precision-ordering} respects the partial order $\uu < \tt$, $\uu < \ff$ and is defined for two three-valued interpretations $(T, P)$ and $(X, Y)$ as follows.
\begin{align*}
(T, P) \preceq_p (X, Y) \textrm{ iff } T \subseteq X \land\boxed{Y \subseteq P}
\end{align*}
\end{definition}
\begin{definition}
A three-valued interpretation $(T, P)$ is a {\em model} of a program $\P$ if for each rule $r \in\P$
\begin{itemize}
\item$\body(r)\subseteq P \land\bodyn(r)\intersect T =\emptyset$ implies $\head(r)\in P$, and
\item$\body(r)\subseteq T \land\bodyn(r)\intersect P =\emptyset$ implies $\head(r)\in T$.
\end{itemize}
\end{definition}
\begin{definition}
A three-valued interpretation $(T, P)$ is a {\em stable model} of a program $\P$ if it is a model of $\P$ and if for every three-valued interpretation $(X, Y)$ such that $(X, Y)\preceq_t (T, P)$ there exists a rule $r \in\P$ such that either
\begin{itemize}
\item$\bodyp(r)\subseteq Y \land\bodyn(r)\intersect T =\emptyset$ and $\head(r)\not\in Y$\textbf{OR}
\item$\bodyp(r)\subseteq X \land\bodyn(r)\intersect P =\emptyset$ and $\head(r)\not\in X$
\end{itemize}
\end{definition}
\section{Approximation Fixpoint Theory}
We can think of a three-valued interpretation $(T, P)$ as an approximation on the set of true atoms.
$T$ is a lower bound and $P$ is the upper bound.
\begin{definition}
An {\em approximator} is a fixpoint operator on the complete lattice $\langle\wp(\mathcal{L})^2, \preceq_p \rangle$ (called a bilattice)
\end{definition}
Given a function $f(T, P): S^2\rightarrow S^2$, we define two separate functions
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 $\emptyset$.
\item the class of problems which have algorithms that can verify solutions in polynomial time.
\item A problem that can be solved by reducing it to a SAT expression of the form
\begin{align*}
\exists (a \lor b \lor c) \land (\neg a \lor\neg d \lor c)~\land\cdots
\end{align*}
\item (Alternating turing machine) A problem that is solved by some path of an algorithm that is allowed to branch is parallel
\end{itemize}
Intuitive definitions of ${\sf NP^{NP}}$ a.k.a.\ $\Sigma_2^P$
\begin{itemize}
\item the class of problems which have algorithms that can verify solutions in ${\sf NP}$ time.
\item A problem that can be solved by reducing it to a SAT expression of the form
\begin{align*}
\exists c,~ \forall a,~ b,~ (a \lor b \lor c) \land (\neg a \lor\neg d \lor c)~\land\cdots
\end{align*}
\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
