Towards Decision Making via Expressive Probabilistic Ontologies

Abstract

We propose a framework for automated multi-attribute decision making, employing the probabilistic non-monotonic description logics proposed by Lukasiewicz in 2008. Using this framework, we can model artificial agents in decision-making situation, wherein background knowledge, available alternatives and weighted attributes are represented via probabilistic ontologies. It turns out that extending traditional utility theory with such description logics, enables us to model decision-making problems where probabilistic ignorance and default reasoning plays an important role. We provide several decision functions using the notions of expected utility and probability intervals, and study their properties.

Appendix

Consistency, Lexicographic and Logical Consequence. A probabilistic interpretation Pr verifies a conditional constraint $(\psi |\phi )[l, u]$ iff $Pr(\phi ) = 1$ and $Pr(\psi ) \models (\psi |\phi )[l, u]$. Moreover, Pr falsifies $(\psi |\phi )[l,u]$ iff $Pr(\phi ) = 1$ and $Pr(\psi ) \not \models (\psi |\phi )[l, u]$. A set of conditional constraints $\mathcal {F}$ tolerates a conditional constraint $(\psi |\phi )[l,u]$ under a classical knowledge base T, iff there is model Pr of $T \cup \mathcal {F}$ that verifies $(\psi |\phi )[l,u]$ (i.e., $Pr \models T \cup \mathcal {F} \cup \{(\psi | \phi )[l, u], (\phi |\top )[1, 1]\}$). A PTBox $PT = (T,P)$ is consistent iff T is satisfiable, and there exists an ordered partition $(P_0,\dots ,P_k)$ of P such that each $P_i$ (where $i \in \{0, \dots ,k\}$) is the set of all $F \in P \backslash (P_0 \cup \dots \cup P_{i-1})$ that are tolerated under T by $P \backslash (P_0 \cup \dots \cup P_{i-1})$. Following [13], we note that such ordered partition of PT is unique if it exists, and is called z -partition. A probabilistic knowledge base $KB = (T , P , (P_o )_{o\in \mathbf I _P)}$ is consistent iff $PT = (T , P )$ is consistent, and for every probabilistic individuals $o \in \mathbf I _P$, there is a Pr such that $Pr \models T \cup P_o$.

For probabilistic interpretations Pr and $Pr'$, Pr is lexicographically preferable (or lex-preferable) to $Pr'$ iff there exists some $i \in \{ 0, \dots ,k \}$ such that $|\{ F \in P_i \mid Pr \models F\}| > | \{ F \in P_i | Pr' \models F \}|$ and $|\{F \in P_j \mid Pr \models F \}| = | \{ F \in P_j \mid Pr' \models F\}|$ for all $i < j \le k$. A probabilistic interpretation Pr is a lexicographically minimal (or lex-minimal) model of $T \cup \mathcal {F}$ iff $Pr \models T \cup \mathcal {F}$ and there is no $Pr'$ such that $Pr' \models T \cup \mathcal {F}$ and $Pr'$ is lex-preferable to Pr. A conditional constraint $(\psi | \phi )[l,u]$ is a lexicographic consequence (or lex-consequence) of a set of conditional constraints $\mathcal {F}$ under a PTBox PT (or $\mathcal {F} \mid \mid \!\sim ^{lex} (\psi |\phi )[l,u]$) under PT, iff $Pr(\psi ) \in [l,u]$ for every lex-minimal model Pr of $T \cup \mathcal {F} \cup \{(\phi |\top )[1, 1]\}$. Moreover, $PT \mid \mid \!\sim ^{lex} F$, iff $\emptyset \mid \mid \!\sim ^{lex} F$ under PT. Note that the notion of lex-consequence faithfully generalizes the classical class subsumption. That is, given a consistent PTBox $PT = (T, P)$, a set of conditional constraints $\mathcal {F}$, and c-concepts $\phi$ and $\psi$, if $T \models \phi \sqsubseteq \psi$, then $\mathcal {F}\mid \mid \!\sim ^{lex }(\psi |\phi )[1, 1]$ under PT.

Furthermore, we say that $(\psi | \phi )[l, u]$ is a tight lexicographic consequence (or tight lex-consequence) of $\mathcal {F}$ under PT, denoted $F \mid \mid \!\sim ^{lex}_{tight} (\psi |\phi )[l,u]$ under PT, iff $l= \inf \{ Pr(\psi ) \mid Pr \mid \mid \!\sim ^{lex} T \cup \mathcal {F} \cup \{ (\phi | \top ) [1,1]\}$ and $u= \sup \{ Pr(\psi ) \mid Pr \mid \mid \!\sim ^{lex} T \cup \mathcal {F} \cup \{ (\phi | \top ) [1,1]\}$. Moreover, $PT \mid \mid \!\sim ^{lex}_{tight} F$ iff $\emptyset \mid \mid \!\sim ^{lex} F$. Note that $[l,u] = [1, 0]$ (empty interval) when there is no such model. For a probabilistic knowledge base $KB = (T,P, (P_o)_{o \in \mathbf I _P} )$, $KB \mid \mid \!\sim ^{lex} F$ where F is a conditional constraint for $o \in \mathbf I _P$ iff $P_o \mid \mid \!\sim ^{lex} F$ under (T, P). Moreover, $KB \mid \mid \!\sim ^{lex}_{tight} F$ iff $P_o \mid \mid \!\sim ^{lex} _{tight} F$ under (T, P). A conditional constraint $(\psi | \phi )[l,u]$ is a logical consequence of $T \cup \mathcal {F}$ (i.e., $T \cup F\models (\psi |\phi )[l,u]$) iff each model of $T \cup \mathcal {F}$ is also a model of $(\psi |\phi )[l,u]$. Furthermore, $(\psi |\phi )[l,u]$ is a tight logical consequence of $T \cup F$ (i.e., $T \cup \mathcal {F} \models _{tight} (\psi |\phi )[l,u]$, iff $l= \inf \{ Pr(\psi |\phi ) \mid Pr \models T \cup \mathcal {F} \text { and } Pr(\phi ) > 0\}$ and $u= \sup \{ Pr(\psi |\phi ) \mid Pr \models T \cup \mathcal {F} \text { and } Pr(\phi ) > 0\}$. Given a PTBox $PT = (T, P)$, $Q \subseteq P$ is lexicographically preferable (or lex-preferable) to $Q' \subseteq P$ iff there exists some $i \in {0, \dots , k}$ such that $|Q \cap P_i | > |Q' \cap P_i |$ and $|Q \cap P_j | = |Q' \cap P_j|$ for all $i < j\le k$, where $(P_0,\ldots ,P_k)$ is the z-partition of PT. Q is lexicographically minimal (or lex-minimal) in a set S of subsets of P iff $Q \in S$ and no $Q' \in S$ is lex-preferable to Q. Furthermore, let $\mathcal {F}$ be a set of conditional constraints, and $\phi$ and $\psi$ be two concepts, then a set $\mathcal {Q}$ of lexicographically minimal subsets of P exists such that $F \mid \mid \!\sim ^{lex} (\psi |\phi )[l, u]$ under PT iff $T \cup Q \cup \mathcal {F} \cup {(\phi |\top )[1, 1]} \models (\psi |\top )[l, u]$ for all $Q \in \mathcal {Q}$. This is extended to tight case lex-consequence.