## 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.

### Keywords

- Ontological Investigation
- Artificial Agent Models
- Description Logics (DL)
- Expected Utility Interval
- Conditional Constraints

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

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- 1.
- 2.
It is also called

*preference-indifference*relation, since it is the union of strict preference and indifference relation. - 3.
By convention, objects are written with lower case.

- 4.
Note that

*T*is not used to denote a classical TBox anymore but rather the whole classical knowledge base, TBox and ABox. - 5.
See Proposition 4.8 in [13].

- 6.
See Proposition 4.9 in [13].

- 7.
Alternatively, \(\mathcal {U}\) can be studied in two partition, that is, the set of pairs with non-negative (denoted \(\mathcal {U}^+\)) and negative weights (denoted \(\mathcal {U}^-\)). In extreme cases, \(\mathcal {U} = \mathcal {U}^+\) when \(\mathcal {U}^-= \emptyset \) (similarly for \(\mathcal {U} = \mathcal {U}^+\)).

- 8.
Recall that we concern ourselves with desirable attributes, i.e., weights are non-negative.

- 9.
This is done via Lehmann’s lexicographic entailment; in this particular example

*z-partition*is \((P_0, P_1)\) where \(P_0 = \{(\lnot \textit{Desirable} | \exists \textit{hasHotel.FiveStarHotel})[1,1]\)} and \(P_1 = \{(\textit{Desirable} | \exists \textit{hasHotel.FiveStarHotel})[1,1]\}\) that is, \((T, P) \cup \textit{BadFamedFiveStar}\textit{Hotel}(\textit{meridian})\mid \mid \!\sim ^{lex} \lnot \textit{Desirable} (\textit{trip1})\). - 10.
Note that this definition essentially coincides with that choice functions in the imprecise probability literature [8], with the exception that it is allowed to return an empty set.

- 11.

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## Appendix

### 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.

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Acar, E., Thorne, C., Stuckenschmidt, H. (2015). Towards Decision Making via Expressive Probabilistic Ontologies. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_4

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