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clean:
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\usepackage{amsmath}
\usepackage{amsfonts}
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\usepackage{graphicx}
\usepackage{float}
\usepackage{verbatim}
\usepackage{enumerate}
\usepackage{mathtools}
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\usepackage{bm}% boldmath
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\documentclass{article}
\input{common}
\title{Summary of Important Definitions}
\date{May 11th}
\begin{document}
\maketitle
\section{Lattice Theory}
\begin{definition}
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
\begin{align*}
f(\cdot, P)_1:~ &S \rightarrow S\\
f(T, \cdot)_2:~ &S \rightarrow S
\end{align*}
such that
\begin{align*}
f(T, P) = \Big(&(f(\cdot, P)_1)(T), \\&(f(T, \cdot)_2)(P)\Big)
\end{align*}
\begin{definition}
Given an approximator $\Phi(T, P)$ the stable revision operator is defined as follows
\begin{align*}
S(T, P) = (\lfp{(\Phi(\cdot, P)_1)},~ \lfp{(\Phi(T, \cdot)_2)})
\end{align*}
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}
\section{The Polynomial Heirarchy}
Intuitive definitions of ${\sf NP}$
\begin{itemize}
\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
\end{itemize}
\end{document}