422 lines
22 KiB
TeX
422 lines
22 KiB
TeX
\documentclass{article}
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\usepackage{natbib}
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\usepackage{hyperref}
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\input{common}
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\title{AFT and HMKNF}
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\author{}
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\date{May 25th 2023}
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\begin{document}
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\maketitle
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\section{An Approximator Hybrid MKNF knowledge bases}
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A program $\P$ is a set of (normal) rules. An ontology $\OO$ is a first-order formula.
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\begin{definition}
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Given a set of atoms $S$ and an atom $a$. We use $\OBO{S}$ to denote the first-order formula obtained by extending $\OO$ with every atom from $S$ fixed to be true. The relation
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\begin{align*}
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\OBO{S} \models a
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\end{align*}
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Holds if either
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\begin{itemize}
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\item there is no satisfying assignment for $\OBO{S}$ (The principle of explosion)
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\item for every satisfying assignment for $\OBO{S}$, $a$ is true, $\OBO{S}$ is logically equivalent to $\OBO{S \union \{ a\}}$
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\end{itemize}
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Holds if when we fix every atom in $S$ to be true with
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\end{definition}
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The lattice $\langle \L^2, \preceq_p \rangle$
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\begin{align*}
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\Gamma(T, P) &\define \{ \head(r) ~|~ r \in \P,~ T \subseteq \bodyp(r),~ \bodyn(r) \intersect P = \emptyset \}\\
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extract(T) &\define \{ a \in \L ~|~ \OBO{T} \models \neg a \} \\
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\Phi(T, P) &\define \Bigl( \Gamma(T, P), \Gamma(P, T) \setminus extract(T) \Bigr)
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\end{align*}
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\begin{align*}
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S(\Phi)(T, P) \define \Bigl( \lfp~\Gamma(\cdot, P),~ \lfp\,\Bigl( \Gamma(\cdot, T) \setminus extract(T) \Bigr) \Bigr)
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\end{align*}
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\begin{definition}
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A (hybrid MKNF) knowledge base $\KB = (\OO, \P)$ is a program $\P$ and an ontology $\OO$\footnote{See \Section{section-prelim-hmknf} for a formal definition}
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An interpretation $(T, P)$\footnote{See \Section{psms} for definition of interpretations} is an MKNF model of $\KB$ if
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\begin{itemize}
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\item $T \subseteq P$
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\item $\OBO{P}$ is consistent, i.e., $\OBO{P} \not\models \bot$ (It follows that $\OBO{T} \not\models \bot$)
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\item All the rules are satisfied, $\lfp~{\Gamma(\cdot, T)} = P$.
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\end{itemize}
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\end{definition}
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\appendix
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\section{Lattice Theory}
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\begin{definition}
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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
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\begin{itemize}
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\item (reflexivity) $x \preceq x$
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\item (antisymmetry) $(x \preceq y \land y \preceq x) \implies x = y$
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\item (transitivity) $(x \preceq y \land y \preceq z) \implies (x \preceq z)$
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\end{itemize}
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\end{definition}
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\begin{definition}
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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
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\begin{itemize}
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\item $\forall y \in X, \boxed{x} \preceq y$
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\item (resp. $\forall y \in X, \boxed{y} \preceq x$)
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\end{itemize}
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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 $X$.
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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 $X$.
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In general, $\textbf{lub}(X)$ and $\textbf{glb}(X)$ may not exist.
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\end{definition}
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\begin{definition}
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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 $S \subseteq \mathcal{L}$
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\begin{itemize}
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\item $\textbf{lub}(X)$ and $\textbf{glb}(X)$ exist and are unique.
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\end{itemize}
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\end{definition}
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\section{Partial Stable Model Semantics}\label{psms}
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\begin{definition}
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A (ground and normal) {\em answer set program} $\P$ is a set of rules where each rule $r$ is of the form
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\begin{align*}
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h \leftarrow a_0,~ a_1,~ \dots,~ a_n,~ \Not b_0,~\Not b_1,~\dots,~\Not b_k
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\end{align*}
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where we define the following shorthand for a rule $r \in \P$
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\begin{align*}
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\head(r) &= h\\
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\bodyp(r) &= \{ a_0,~ a_1,~ \dots,~ a_n \}\\
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\bodyn(r) &= \{ b_0,~ b_1,~ \dots,~ b_k \}
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\end{align*}
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\end{definition}
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\begin{definition}
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A two-valued interpretation $I$ of a program $\P$ is a set of atoms that appear in $\P$.
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\end{definition}
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\begin{definition}
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An interpretation $I$ is a \boxedt{model} of a program $\P$ if for each rule $r \in \P$
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\begin{itemize}
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\item If $\bodyp(r) \subseteq I$ and $\bodyn(r) \intersect I = \emptyset$ then $\head(r) \in I$.
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\end{itemize}
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\end{definition}
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\begin{definition}
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An interpretation $I$ is a \boxedt{stable model} of a program $\P$ if $I$ is a model of $\P$ \textbf{and}
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for every interpretation $I' \subseteq I$ there exists a rule $r \in \P$ such that
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\begin{itemize}
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\item $\bodyp(r) \subseteq I'$, $\bodyn(r) \intersect I \not= \emptyset$ (Note that this is $I$ and not $I'$) and $\head(r) \not\in I'$
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\end{itemize}
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\end{definition}
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\begin{definition}
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A three-valued interpretation $(T, P)$ of a program $\P$ is a pair of sets of atoms such that $T \subseteq P$.
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The \boxedt{truth-ordering} respects $\ff < \uu < \tt$ and is defined for two three-valued interpretations $(T, P)$ and $(X, Y)$ as follows.
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\begin{align*}
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(T, P) \preceq_t (X, Y) \textrm{ iff } T \subseteq X \land P \subseteq Y
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\end{align*}
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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.
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\begin{align*}
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(T, P) \preceq_p (X, Y) \textrm{ iff } T \subseteq X \land \boxed{Y \subseteq P}
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\end{align*}
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\end{definition}
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\begin{definition}
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A three-valued interpretation $(T, P)$ is a {\em model} of a program $\P$ if for each rule $r \in \P$
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\begin{itemize}
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\item $\body(r) \subseteq P \land \bodyn(r) \intersect T = \emptyset$ implies $\head(r) \in P$, and
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\item $\body(r) \subseteq T \land \bodyn(r) \intersect P = \emptyset$ implies $\head(r) \in T$.
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\end{itemize}
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\end{definition}
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\begin{definition}
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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
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\begin{itemize}
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\item $\bodyp(r) \subseteq Y \land \bodyn(r) \intersect T = \emptyset$ and $\head(r) \not\in Y$ \textbf{OR}
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\item $\bodyp(r) \subseteq X \land \bodyn(r) \intersect P = \emptyset$ and $\head(r) \not\in X$
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\end{itemize}
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\end{definition}
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\subsection{Fixpoint Operators}
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\begin{definition}
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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}
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\begin{itemize}
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\item $\Phi$ is $\preceq$-monotone if for all $x, y \in \mathcal{L}$
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\begin{align*}
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x \preceq y \implies \Phi(x) \preceq \Phi(y)
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\end{align*}
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(Note: this does not mean the function is inflationary, i.e., $x \preceq \Phi(x)$ may not hold)
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\end{itemize}
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A {\em fixpoint} is an element $x \in \mathcal{L}$ s.t. $\Phi(x) = x$.
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\end{definition}
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\begin{theorem}[Knaster-Tarski]
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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 $\lfp~{\Phi}$)
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\end{theorem}
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Some intuitions about lattices
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\begin{itemize}
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\item The entire lattice has a biggest element $\lub{}(\mathcal{L}) = \top$ and a smallest element $\glb{}(\mathcal{L}) = \bot$
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\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 $\bot$
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\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.
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\item Further, because $\bot$ is less than all elements in the lattice, it is always the case that $\bot \preceq \Phi(\bot)$
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\end{itemize}
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\section{Approximation Fixpoint Theory}
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We can think of a three-valued interpretation $(T, P)$ as an approximation on the set of true atoms.
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$T$ is a lower bound and $P$ is the upper bound.
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\begin{definition}
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An {\em approximator} is a fixpoint operator on the complete lattice $\langle \wp(\mathcal{L})^2, \preceq_p \rangle$ (called a bilattice)
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\end{definition}
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Given a function $f(T, P): S^2 \rightarrow S^2$, we define two separate functions
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\begin{align*}
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f(\cdot, P)_1:~ &S \rightarrow S\\
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f(T, \cdot)_2:~ &S \rightarrow S
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\end{align*}
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such that
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\begin{align*}
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f(T, P) = \Big(&(f(\cdot, P)_1)(T), \\&(f(T, \cdot)_2)(P)\Big)
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\end{align*}
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\begin{definition}
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Given an approximator $\Phi(T, P)$ the stable revision operator is defined as follows
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\begin{align*}
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S(T, P) = (\lfp{(\Phi(\cdot, P)_1)},~ \lfp{(\Phi(T, \cdot)_2)})
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\end{align*}
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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 $\emptyset$.
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\end{definition}
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\section{Hybrid MKNF Knowledge Bases}\label{section-prelim-hmknf}
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Motik and Rosati \cite{motikreconciling2010} extend MKNF (Minimal Knowledge and Negation as Failure \cite{lifschitznonmonotonic1991}) to form hybrid MKNF, a nonmonotonic logic that supports classical reasoning with ontologies.
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We use Knorr et al.'s \cite{knorrlocal2011} \mbox{3-valued} semantics for hybrid MKNF;
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Under which there are three truth values: $\ff$ (false), $\uu$ (undefined), and $\tt$ (true) that use the
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ordering $\ff < \uu < \tt$.
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The $min$ and $max$ functions respect this ordering when applied to sets.
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MKNF relies on the standard name assumption under which
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every first-order interpretation in an
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MKNF interpretation is required to be a Herbrand interpretation with a
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countably infinite amount of additional constants.
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We use $\Delta$ to denote the set of all these constants.
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We use $\phi[x \rightarrow \alpha]$ to denote the formula obtained by replacing all free
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occurrences of variable x in $\phi$ with the term $\alpha$.
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A (\mbox{3-valued}) \textit{MKNF structure} is a triple $(I, \M, \N)$ where $I$ is
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a (two-valued first-order) interpretation and $\M = \langle M, M_1 \rangle$ and
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$\N = \langle N, N_1 \rangle$ are pairs of sets of first-order interpretations
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such that $M \supseteq M_1$ and $N \supseteq N_1$.
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Using $\phi$ and $\sigma$ to denote MKNF formulas, the evaluation of an MKNF
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structure is defined as follows:
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\begin{align*}
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&(I, \M, \N)({p(t_1, \dots ,~t_n)}) \define
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\left\{\begin{array}{ll}
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\tt & \textrm{iff } p(t_1, \dots,~t_n) \textrm{ is \boxed{\textrm{true}} in } I
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\\
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\ff & \textrm{iff } p(t_1, \dots,~t_n) \textrm{ is \boxed{\textrm{false}} in } I
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\\
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\end{array}\right.
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\\
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&(I, \M, \N)({\neg \phi}) \define
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\left\{\begin{array}{ll}
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\tt & \textrm{iff } (I, \M, \N)({\phi}) = \ff \\
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\uu & \textrm{iff } (I, \M, \N)({\phi}) = \uu \\
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\ff & \textrm{iff } (I, \M, \N)({\phi}) = \tt \\
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\end{array}\right.
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\\
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&(I, \M, \N)({\exists x, \phi}) \define max\{(I, \M,
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\N)({\phi[\alpha \rightarrow x]}) ~|~\alpha \in \Delta\}
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\\
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&(I, \M, \N)({\forall x, \phi}) \define min\{(I, \M,
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\N)({\phi[\alpha \rightarrow x]}) ~|~\alpha \in \Delta\}
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\\
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&(I, \M, \N)({\phi \land \sigma}) \define min((I, \M, \N)({\phi}), (I,
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\M, \N)({\sigma}))
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\\
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&(I, \M, \N)({\phi \lor \sigma}) \define max((I, \M, \N)({\phi}), (I,
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\M, \N)({\sigma}))
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\\
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&(I, \M, \N)({\phi \subset \sigma}) \define \left\{ \begin{array}{ll}
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\tt & \textit{iff }\vspace{0.3em} (I, \M,
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\N)({\phi}) \geq (I, \M, \N)({\sigma}) \\
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\ff & \textrm{otherwise}
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\end{array}\right.\\
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&(I, \M, \N)({\bfK \phi}) \define
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\left\{\begin{array}{ll}
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\tt & \textrm{iff } (J, \MM, \N)({\phi}) = \tt \textrm{ \boxedt{for all}
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} J
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\in M
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\\
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\ff & \textrm{iff } (J, \MM, \N)({\phi}) = \ff \textrm{ \boxedt{for some} } J
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\in M_1
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\\
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\uu & \textrm{otherwise}
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\end{array}\right.
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\\
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&(I, \M, \N)({\Not \phi}) \define
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\left\{\begin{array}{ll}
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\tt & \textrm{iff } (J, \M, \NN)({\phi}) = \ff \textrm{ \boxedt{for some} } J
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\in N_1
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\\
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\ff & \textrm{iff } (J, \M, \NN)({\phi}) = \tt \textrm{ \boxedt{for all} } J
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\in N
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\\
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\uu & \textrm{otherwise}
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\end{array}\right.
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\end{align*}
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While this logic captures many nonmonotonic semantics, we restrict our focus to hybrid MKNF Knowledge bases,
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a syntactic restriction that captures combining answer set programs with ontologies.
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An (MKNF) program $\P$ is a set of (MKNF) rules. A rule $r$ is written as
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follows:
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\begin{align*}\label{MKNFRules}
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\bfK h \leftarrow \bfK p_0,\dots,~\bfK p_j,~ \Not n_0,\dots,~\Not n_k.
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\end{align*}
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In the above, the atoms $h, p_0, n_0, \dots, p_j, $ and $n_k $ are function-free first-order
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atoms
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of the form $p(t_0, \dots,~t_n )$ where $p$ is a predicate and $t_0,
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\dots,~t_n$ are either constants or variables.
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We call an MKNF formula $\phi$ \textit{ground} if it does not contain variables.
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The corresponding MKNF formula $\pi(r)$ for a rule $r$ is as follows:
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\begin{align*}
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\pi(r) \define \forall \vec{x},~ \bfK h \subset \bfK p_0 \land \dots \land
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\bfK p_j \land \Not n_0 \land \dots \land \Not n_k
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\end{align*}
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where $\vec{x}$ is a vector of all variables appearing in the rule.
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We will use the following abbreviations:
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\begin{align*}
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&\pi(\P) \define \bigwedge\limits_{r \in \P} \pi(r)\\
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&\head(r) = \bfK h \\
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&\bodyp(r) = \{ \bfK p_0, \dots,~ \bfK p_j \} \\
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&\bodyn(r) = \{ \Not n_0, \dots,~ \Not n_k \} \\
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&\bfK(\bodyn(r)) = \{ \bfK a ~|~ \Not a \in \bodyn(r) \}
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\end{align*}
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An \textit{ontology} $\OO$ is a decidable description logic (DL) knowledge base translatable to first-order logic, we denote its translation to first-order logic with $\pi(\OO)$.
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We also assume that the ontology's entailment relation can be checked in polynomial time.
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A {\em normal hybrid MKNF knowledge base} (or knowledge base for short) $\KB
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= (\OO, \P)$ contains a program and an ontology.
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The semantics of a knowledge base corresponds to the MKNF formula $\pi(\KB) = \pi(\P) \land \bfK \pi(\O)$.
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We assume, without loss of generality \cite{knorrlocal2011}, that any given hybrid MKNF knowledge $\KB = (\OO, \P)$ base is ground, that is, $\P$, does not contain any variables\footnote{Not every knowledge base can be grounded. The prevalent class of groundable knowledge bases is the knowledge bases that are DL-safe~\cite{motikreconciling2010}.}
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This ensures that $\KB$ is decidable.
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A {\em (\mbox{3-valued}) MKNF interpretation} $(M, N)$ is a pair of sets of
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first-order interpretations where $\emptyset \subset N \subseteq M$.
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We say an MKNF interpretation $(M, N)$ \textit{satisfies} a knowledge base $\KB
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= (\OO, \P)$ if for each $I \in M$,
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$(I, \MN, \MN) (\pi (\KB)) = \tt$.
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\begin{definition} \label{model}
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A (\mbox{3-valued}) MKNF interpretation $(M, N)$ is a
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\textit{(\mbox{3-valued}) MKNF model} of a normal hybrid MKNF knowledge base
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$\KB$
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if $(M, N)$ satisfies $\pi(\KB)$ and for every \mbox{3-valued} MKNF
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interpretation pair $(M', N')$ where $M \subseteq M'$, $N \subseteq N'$,
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$(M, N) \not= (M', N')$, and we have some $I \in M'$ s.t. $(I, \langle M', N'
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\rangle, \MN)(\pi(\KB)) \not= \tt$.
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\end{definition}
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It is often convenient to only deal with atoms that appear inside $\P$.
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We use $\KAK$ to denote the set of \Katoms{} that appear as either $\bfK a$ or $\Not a$ in the program and we use $\OBO{S}$ to the objective knowledge w.r.t.\ to a set of \Katoms{} $S$.
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\begin{align*}
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\KAK &\define \{ \bfK a ~|~ r \in \P,~ \bfK a \in \Big(\{ \head(r) \} \union
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\bodyp(r) \union \bfK(\bodyn(r)) \Big) \} \\
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\OBO{S} &\define \big\{ \pi(\OO) \big\} \union \big\{ a ~|~ \bfK a \in S
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\}
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\end{align*}
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A \textit{$\bfK$-interpretation} $(T, P) \in \wp(\KAK)^2$ is a pair of sets of \Katoms{}\footnote{We use $\wp(S)$ to denote the powerset of $S$, i.e., $\wp(S) = \{ X ~|~ X \subseteq S \}$}.
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We say that $(T, P)$ is consistent if $T \subseteq P$.
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An MKNF interpretation pair $(M, N)$ uniquely {\em induces} a consistent \bfK-interpretation $(T, P)$ where for
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each $\bfK a \in \KAK$:
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\begin{align*}
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\bfK a \in \boxed{(T \intersect P)} \textrm{ if } \forall I \in M, \mmeval{\bfK a} &= \boxed{\tt}\\
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\bfK a \not\in \boxed{\hspace{0.5cm} P \hspace{0.5cm}} \textrm{ if } \forall I \in M, \mmeval{\bfK a} &= \boxed{\ff}\\
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\bfK a \in \boxed{(P \setminus T)} \textrm{ if } \forall I \in M, \mmeval{\bfK a} &= \boxed{\uu}
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\end{align*}
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We can acquire the set of false \Katoms{} by taking the complement of $P$, e.g., $F = \KAK \setminus P$ and $P = \KAK \setminus F$.
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When we use the letters $T$, $P$, and $F$, we always mean sets of true, possibly true, and false \Katoms{} respectively.
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We say that a \bfK-interpretation $(T, P)$ \textit{extends} to an MKNF interpretation $(M, N)$ if $(T, P)$ is consistent, $\OBO{P}$ is consistent, and
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\begin{align*}
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(M, N) = (\{ I ~|~ \OBO{T} \models I \}, \{ I ~|~ \OBO{P} \models I \})
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\end{align*}
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This operation extends $(T, P)$ to the $\subseteq$-maximal MKNF interpretation that induces $(T, P)$.
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We comment on how the relation \textit{induces} and \textit{extends} are related.
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\begin{remark}
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Let $(M, N)$ be an MKNF {\bf model} of an MKNF knowledge base $\KB$.
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A \bfK-interpretation $(T, P)$ that extends to $(M, N)$ exists, is unique, and is the \bfK-interpretation induced by $(M, N)$.
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\end{remark}
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We say that an MKNF interpretation $(M, N)$ \textit{weakly induces} a {\bfK-interpretation} $(T, P)$ if $(M, N)$ induces a {\bfK-interpretation} $(T^*, P^*)$ where $T \subseteq T^*$ and $P^* \subseteq P$.
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Similarly, a {\bfK-interpretation} \textit{weakly extends} to an MKNF interpretation $(M, N)$ if there exists an interpretation $(T^*, P^*)$ that extends to $(M, N)$ such that $T \subseteq T^*$, $P^* \subseteq P$.
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Intuitively, we are leveraging the knowledge ordering $\uu < \tt$ and $\uu < \ff$. A \bfK-interpretation is weakly induced by an MKNF interpretation if that MKNF interpretation induces a \bfK-interpretation that ``knows more'' than the original \bfK-interpretation.
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There are MKNF interpretations for which no \bfK-interpretation extends to, however, these are of little interest; either $\OBO{P}$ is inconsistent or $(M, N)$ is not maximal w.r.t. atoms that do not appear in $\KAK$. Similarly, there exist \bfK-interpretations that extend to MKNF interpretations that do not induce them. If this is the case, then the \bfK-interpretation is missing some logical consequences of the ontology and this should be corrected.
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We define the class of \bfK-interpretations that excludes the undesirable \bfK-interpretations and MKNF interpretations from focus.
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\begin{definition}\label{saturated}
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A \bfK-interpretation $(T, P)$ of an MKNF knowledge base $\KB$ is {\em saturated} if it can be extended to an MKNF interpretation $(M, N)$ that induces $(T, P)$.
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Equivalently, a \bfK-interpretation is saturated iff $(T, P)$ and $\OBO{P}$ are consistent,
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$ \OBO{P} \not\models a~\textrm{for each } \bfK a \in (\KAK \setminus P)$ and $\OBO{T} \not\models a~\textrm{for each } \bfK a \in (\KAK \setminus T)$.
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\end{definition}
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\subsection{An Example Knowledge Base}
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Hybrid MKNF is equivalent to answer set semantics when the ontology is empty.
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The definition of an MKNF program given above is precisely the formulation of stable model semantics in the logic of MKNF~\cite{lifschitznonmonotonic1991}.
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We give an example of a knowledge base to demonstrate the combined reasoning of an answer set program with an ontology.
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\begin{example}
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Let $\KB = (\OO, \P)$ be the following knowledge base
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\begin{align*}
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\pi(\OO) &= \{ a \lor \neg b \} \\
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(\textrm{The definition of } &\P \textrm{ follows})\\
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\bfK a &\leftarrow \Not b\\
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\bfK b &\leftarrow \Not a
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\end{align*}
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This knowledge base has two MKNF models
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\begin{align*}
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&\Big(\Big\{\{ a, b \}, \{ a, \neg b\}, \{ \neg a, \neg b \}\Big\}, \Big\{\{ a, b \} \Big\}\Big) \\
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&\hspace{1cm}\textrm{ which induces the \bfK-interpretation } (\emptyset, \{ \bfK a, \bfK b \})\\
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&(M, M) \textrm{ where } M = \Big\{\{ a, \neg b\}, \{ a, b \} \Big\}\\
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&\hspace{1cm}\textrm{ which induces the \bfK-interpretation } (\{ \bfK a \}, \{ \bfK a \})
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\end{align*}
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Above we have the answer set program $\P$ that, without $\OO$, admits three partial stable models. The ontology $\OO$ encodes an exclusive choice between the atoms $a$ and $b$. The program allows for both $a$ and $b$ to be undefined.
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The use of the ontology here is effectively a filter that removes all interpretations where $a$ is false but $b$ is true.
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The mixing of negation as failure and classical negation is not always straightforward. For example
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\begin{align*}
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\pi(\OO) &= \{ a \lor b \} \\
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\P &= \emptyset
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\end{align*}
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The above admits just one MKNF model
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\begin{align*}
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&\big(M, M \big) \textrm{ where } M = \Big\{ \{ a, \neg b \}, \{ \neg a, b \}, \{ a, b \} \Big\}\\
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&\hspace{1cm}\textrm{ which induces the \bfK-interpretation } (\emptyset, \emptyset)
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\end{align*}
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From the perspective of $\P$, which is only concerned with \bfK-interpretations, all atoms are false.
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However, the interpretation $\{ \neg a, \neg b \}$ is absent from the model which ensures that $\O$ is still satisfied.
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Recall that $\pi(\KB) = \bfK (a \lor b)$.
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\end{example}
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\bibliographystyle{plain}
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\bibliography{refs}
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\end{document} |