# Closure-Based Tractable Possibilistic Inference from Partially Ordered DL-Lite Ontologies

Ahmed Laouar, Salem Benferhat

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Ahmed Laouar, Salem Benferhat. Closure-Based Tractable Possibilistic Inference from Partially Ordered DL-Lite Ontologies. European Conference on Logics in Artificial Intelligence (JELIA 2025), Sep 2025, Kutaisi, Georgia, Georgia. pp.240-256, 10.1007/978-3-032-04587-4_15. hal-05385138

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Distributed under a Creative Commons Attribution 4.0 International License
Closure-Based Tractable Possibilistic Inference
from Partially Ordered DL-Lite Ontologies

Ahmed Laouar¹[0009-0028-0028-3234] and Salem Benferhat¹[0000-0002-4853-3637]

CRIL, Univ. Artois & CNRS, UMR 8188, F-62300 Lens, France
{laouar,benferhat}@cril.fr

**Abstract.** Handling partially specified inconsistent information is a major challenge, especially when balancing the richness of query answers with the need to control computational complexity. We address this challenge by proposing an efficient method for computing a consistent and enriched fragment of data, known as a repair, within knowledge bases that rely on a stable terminological component and incorporate partially ordered uncertainty in the data (ABox). Our approach, grounded in possibility theory, avoids exhaustive enumeration of conflicts or justifications by applying a positive deductive closure directly to the partially ordered ABox. This enables repair computation through simple consistency checks over data subsets, ensuring tractability while supporting an extended set of plausible inferences. Beyond this main contribution on efficient repair computation, we briefly introduce a semantic characterization of repairs that generalises the classical notion of models for consistent knowledge bases. Finally, we present an experimental evaluation against existing possibilistic approaches, demonstrating both practical effectiveness and computational benefits.

**Keywords:** Inconsistency · Ontologies · Data Repairs · Partial Orders.

# 1 Introduction

Formal ontologies specify the semantics and domain knowledge of datasets, enabling reliable query answering. A knowledge base consists of conceptual knowledge in the ontology (TBox) and factual knowledge in the dataset (ABox). This article focuses on two key issues: inconsistency, where data violate ontological constraints, and uncertainty, where facts have varying reliability. Many works address inconsistency handling in frameworks like propositional logic [21], first-order logic [5], description logics [22,26,12,3], and databases [29,11]. Our work aligns with research [26,3,13] on tractable inconsistency management, especially for large ABoxes. In such settings, inconsistencies stem from ABox assertions contradicting TBox rules. We assume a reliable TBox and focus on efficient reasoning over conflicting ABox data. A common approach to reasoning under inconsistency is to compute consistent subsets of the ABox, often called repairs, over which queries can be safely evaluated. The ABox Repair (AR) semantics is desirable for its cautious strategy of returning only answers supported by all
repairs [26], but it is generally intractable. To address this, the Intersection of ABox Repairs (IAR) semantics [26] offers a natural and tractable alternative by evaluating queries over the intersection of all repairs. However, IAR sacrifices completeness for tractability, potentially missing answers entailed by all repairs but absent from their intersection. The ongoing challenge is to surpass IAR in inferential or expressive power while preserving computational efficiency.

Several levers have been proposed to overcome the limitations of IAR semantics. In this paper, we rely on two main levers. The first lever involves computing the deductive closure of the ABox, which derives all facts entailed by the positive axioms of the TBox. This allows the recovery of conclusions not explicitly present in the ABox. The closure can be applied either before or after repair computation. Applying it beforehand, as in the Intersection of Closed ABox Repairs (ICAR) semantics [26], preserves tractability but may produce questionable conclusions, whereas applying it afterwards, as in the Intersection of Closed Repairs (ICR) semantics [12], yields richer inferences at the expense of tractability.

The second lever involves the widely used strategy of assigning priorities to handle inconsistency. The idea is to resolve conflicts by preferring certain assertions, typically via a total or partial preorder, enabling a more informed selection of consistent subsets. A substantial body of work has explored the use of priorities to guide conflict resolution across diverse settings. These include total orders [9,14], context-dependent or local orders defined only within specific conflicts [10], interval-based or lattice-based preferences [20], and general partial orders [7,6]. While priority relations provide an intuitive way to direct inconsistency handling, they also significantly expand the range of possible semantics, often as extensions of classical, non-prioritised frameworks.

This paper focuses on tractable semantics based on partial preference relations over the ABox, using possibilistic logic. Partially ordered information is managed by extending the partial order into a family of total orders that refine it, each inducing a prioritised consistent repair. Reasoning then aggregates information from these preferred repairs. Within DL-Lite<sub>R</sub>, two methods illustrate this approach in possibility theory [17]: the π-repair, which directly intersects repairs from the total extensions [6], and the Cπ-repair, which enriches each repair by computing its deductive closure before intersection [24]. Both are tractable and provide meaningful semantics. Moreover, the closed method is more productive, deriving additional consequences from the knowledge base's logical structure.

In this work, we apply deductive closure at an earlier stage—directly on the initial partially ordered ABox. To enable this, we adapt the classical notion of deductive closure to account for the partial order among assertions. This early enrichment helps identify minimal conflicting elements by isolating less certain assertions and allows for a more efficient implementation of the Cπ-repair semantics through a sequence of consistency checks.

The main contribution of this work is an efficient repair process leveraging the deductive closure of the input ABox. We show possibilistic inference can be applied directly to this enriched ABox, yielding the same repaired ABoxes as previous approaches, without explicitly computing assertion supports. This
reduces reasoning to consistency checks over ABox subsets. Additionally, we provide a semantic perspective on possibilistic repairs, outlining a preferential framework with partially ordered interpretations and preferred models. This view supports formal understanding of inference under prioritised inconsistency and will guide future developments. Finally, our experimental evaluation shows computing the deductive closure before generating repairs improves performance over existing methods, validating the practical relevance of our approach.

This paper is organised as follows. Section 2 reviews the DL-Lite<sub>R</sub> language and the C<sub>π</sub>-repair approach. Section 3 introduces the partially preordered closure, characterises c<sub>π</sub>-acceptance, and offers a brief semantic perspective. Section 4 presents experimental results, followed by the conclusion. The implementation code, ontology, and datasets are available at https://github.com/ahmedlaouar/py_reasoner/tree/jelia2025.

## 2 Preliminaries

We briefly recall the basics of the DL-Lite<sub>R</sub> fragment of the DL-Lite description logics [16], and the notions of conflicts and deductive closure. Then, we recall the Closure-Based Partially Ordered Possibilistic Repair (C<sub>π</sub>-repair).

### 2.1 The DL-Lite<sub>R</sub> language

A DL-Lite<sub>R</sub> knowledge base is constructed from three mutually disjoint sets: concept names N<sub>C</sub>, role names N<sub>R</sub> and individual names N<sub>I</sub>, as follows:

$$R := P \mid P^{-} \quad E := R \mid \neg R \quad B := A \mid \exists R \quad C := B \mid \neg B$$

where A is an atomic concept, P is an atomic role and P<sup>-</sup> is the inverse of an atomic role. Concepts B and C are called basic and complex concepts, respectively. Roles R and E are called basic and complex roles, respectively. The negation symbol ¬ designates complement sets (of concepts and roles), and the existential quantifier symbol ∃ denotes existential restriction.

A knowledge base (KB) is a pair K = ⟨T, A⟩ where T is the TBox and A is the ABox. The TBox is a finite set of inclusion axioms between concepts and roles of the form B ⊑ C and R ⊑ E. Axioms with the negation symbol to the right of the inclusion symbol are called negative inclusion axioms, while axioms without the negation symbol are called positive inclusion axioms. The ABox contains a finite set of membership assertions (or ground facts) on atomic concepts and atomic roles of the form A(a) and P(a, b), where a and b are individuals.

The semantics of a DL-Lite<sub>R</sub> KB is given in terms of interpretations. An interpretation I = ⟨Δ<sup>T</sup>, .<sup>T</sup>⟩ consists of a non-empty domain Δ<sup>T</sup> and an interpretation function .<sup>T</sup> assigning meaning to concept names, role names, and individual names. We use the standard definitions of satisfaction of axioms and assertions by an interpretation; see [16] for a comprehensive presentation.

An interpretation I is a model of a KB K = ⟨T, A⟩ if it satisfies every axiom in the TBox T and every assertion in the ABox A. The KB K is consistent if it
admits at least one model; otherwise, it is inconsistent. A TBox $\mathcal{T}$ is incoherent if some concept name is empty in all models of $\mathcal{T}$; otherwise it is coherent. An assertion $\varphi$ is entailed by a knowledge base $\mathcal{K}$, denoted $\mathcal{K} \models \varphi$, if and only if $\varphi$ is satisfied by every model of $\mathcal{K}$.

In a DL-Lite<sub>R</sub> KB, the negative inclusion axioms serve to identify conflicts within the ABox [16]. These are inclusion-minimal inconsistent subsets of the ABox, which contradict the TBox in the case of inconsistency.

**Definition 1.** Let $\mathcal{K} = \langle \mathcal{T}, \mathcal{A} \rangle$ be a DL-Lite<sub>R</sub> KB. A subset of the ABox $\mathcal{C} \subseteq \mathcal{A}$ is a conflict if $\langle \mathcal{T}, \mathcal{C} \rangle$ is inconsistent and $\forall \varphi \in \mathcal{C}, \langle \mathcal{T}, \mathcal{C} \setminus \{\varphi\} \rangle$ is consistent.

The conflict set Cf($\mathcal{A}$) contains all the conflicts in $\mathcal{A}$, each consisting of at most two assertions, since in DL-Lite<sub>R</sub> each negative inclusion axiom in the TBox involves at most two concepts or two roles [16].

We use an example built on the set of concept names $N_C = \{\text{Prof}, \text{FacMember}, \text{FullProf}, \text{Dean}, \text{VisitingProf}, \text{PostDoc}\}$, which represent the basic concepts: professor, faculty member, full professor, dean, visiting professor, and postdoctoral, and the set of individual names $N_I = \{\text{Bob}, \text{Alice}\}$.

*Example 1.* Let $\mathcal{K} = \langle \mathcal{T}, \mathcal{A} \rangle$ be a DL-Lite<sub>R</sub> KB where the TBox $\mathcal{T}$ and the ABox $\mathcal{A}$ are given as follows:

$$ \mathcal{T} = \left\{ \begin{array}{ll} 1. & \text{FullProf} \sqsubseteq \text{Prof} \\ 2. & \text{Prof} \sqsubseteq \text{FacMember} \end{array} \right. \quad \begin{array}{ll} 3. & \text{Dean} \sqsubseteq \text{FacMember} \\ 4. & \text{VisitingProf} \sqsubseteq \neg \text{FullProf} \end{array} \quad \begin{array}{l} 5. \text{Prof} \sqsubseteq \neg \text{PostDoc} \end{array} $$

$$ \mathcal{A} = \left\{ \begin{array}{ll} \varphi_1 = \text{Dean}(\text{Alice}) & \varphi_3 = \text{Prof}(\text{Bob}) \\ \varphi_2 = \text{FullProf}(\text{Bob}) & \varphi_4 = \text{PostDoc}(\text{Bob}) \end{array} \right\} \quad \varphi_5 = \text{VisitingProf}(\text{Bob}) $$

One can check that $\mathcal{K}$ is inconsistent. The conflict set of $\mathcal{A}$ is given by Cf($\mathcal{A}$) = $\{C_1, C_2, C_3\}$, such that $C_1$ contradicts Axiom 4, and $C_2$ and $C_3$ contradict Axiom 5. Namely, $C_1 = \{\text{FullProf}(\text{Bob}), \text{VisitingProf}(\text{Bob})\}$, $C_2 = \{\text{Prof}(\text{Bob}), \text{PostDoc}(\text{Bob})\}$, and $C_3 = \{\text{FullProf}(\text{Bob}), \text{PostDoc}(\text{Bob})\}$.

The positive inclusion axioms are used to derive implied facts from the ABox. The deductive closure applies those positive axioms on ABox assertions to return all the assertions that are implicitly or explicitly stated in the ABox [16,9].

**Definition 2.** Let $\mathcal{T}_p$ denote the set of all positive inclusion axioms of a TBox $\mathcal{T}$. The deductive closure of $\mathcal{A}$ with respect to $\mathcal{T}$ is defined by:

$$ \mathrm{cl}(\mathcal{A}) = \{\varphi \mid \langle \mathcal{T}_p, \mathcal{A} \rangle \models \varphi \text{ s.t. } \varphi \text{ is a concept } B(a) \text{ or a role } R(a, b) \text{ assertion}\} $$

Where B and R are respectively a concept name and a role name in $\mathcal{T}$, a, b are individuals in $\mathcal{A}$, and $\models$ is the standard DL-Lite<sub>R</sub> inference relation.

The deductive closure bears strong similarities with the notion of the *chase*, which originates in the database community and has also been adapted to DL-Lite<sub>R</sub> [16]. A key difference, however, is that while the *chase* may introduce fresh individuals to complete entailment chains by applying positive TBox axioms to
ABox assertions, the deductive closure, as considered here, aims to construct a finite set of entailed assertions restricted to the individuals already present in the ABox. A related notion, referred to as saturation, has been employed in the context of quantified ABoxes [2], where existential variables are used instead of introducing fresh individuals, as in the traditional chase.

In our setting, the deductive closure is obtained by exhaustively applying the positive inclusion axioms from $\mathcal{T}$ (e.g., concept and role inclusions) to the assertions in $\mathcal{A}$ until a fixed point is reached, that is, no further assertions can be derived and added. The following example illustrates this process using the ABox from Example 1.

*Example 2.* Let us continue Example 1 and compute $cl(\mathcal{A})$. The set of positive inclusion axioms of $\mathcal{T}$, denoted by $\mathcal{T}_p$ contains the axioms 1, 2, and 3 of $\mathcal{T}$, namely:

$$ \mathcal{T}_p = \{1. \text{ FullProf} \sqsubseteq \text{Prof} \quad 2. \text{ Prof} \sqsubseteq \text{FacMember} \quad 3. \text{ Dean} \sqsubseteq \text{FacMember}\} $$

The deductive closure of $\mathcal{A}$ is $cl(\mathcal{A}) = \mathcal{A} \cup \{\text{FacMember}(\text{Alice}), \text{FacMember}(\text{Bob})\}$. Such that $\text{FacMember}(\text{Alice})$ is obtained from the axiom 3 applied on the assertion $\text{Dean}(\text{Alice})$, and $\text{FacMember}(\text{Bob})$ is obtained by applying one of the axioms 1 or 2 on the assertions $\text{FullProf}(\text{Bob})$ or $\text{Prof}(\text{Bob})$, respectively.

## 2.2 Partially Ordered Information as a Family of Totally Ordered Possibilistic Repairs

We are interested in partially ordered information, where some data pieces may have incomparable reliability levels. Partial orders are especially useful when it is not feasible to fully compare all information, such as when multiple sources provide data with their own priority criteria [15]. These orders reflect diverse factors like source reliability, specificity, or recency [28,27]. Moreover, combining different total orders often naturally induces a partial order, for example through the Pareto principle [29]. Finally, missing data for some criteria also motivates the use of partial orders, which offer a flexible and realistic way to handle incomparability and uncertainty.

**Possibilistic DL-Lite<sub>R</sub>.** Possibilistic logic is widely used to manage incomplete and uncertain information [18,19,4]. It captures uncertainty by either possibility distributions representing uncertainty in extension, or weighted logics for compact representation. Possibilistic DL-Lite [8] extends standard DL-Lite by assigning weights in the unit interval $[0, 1]$ to TBox axioms and ABox assertions, representing their uncertainty degree. Since TBox axioms are fully reliable, weights apply only to ABox assertions. This assignment yields a totally preordered KB, denoted $\mathcal{K}_> = \langle \mathcal{T}, \mathcal{A}_> \rangle$, where:

$$ \mathcal{A}_> = \{(\varphi_i, w_i) \mid \varphi \text{ is an assertion and } w_i \in [0, 1]\} $$

The possibilistic inference from an inconsistent KB is achieved using an inconsistency degree, denoted $\text{Inc}(\mathcal{A}_>)$. Formally, $\text{Inc}(\mathcal{A}_>) = w_i$ if the sub-base
$\{\varphi | (\varphi, w_j) \in \mathcal{A}_>, w_j > w_i\}$ is consistent and the sub-base $\{\varphi | (\varphi, w_j) \in \mathcal{A}_>, w_j \ge w_i\}$ is inconsistent. The possibilistic repair [9] of $\mathcal{K}_>$ is given by:

$$ \mathcal{R}(\mathcal{A}_>) = \{\varphi | (\varphi, w_i) \in \mathcal{A}_>, w_i > \mathrm{Inc}(\mathcal{A}_>)\} $$

where the inconsistency degree $\mathrm{Inc}(\mathcal{A}_>)$ is the highest weight assigned to an assertion that makes the ABox inconsistent.

**Partially preordered possibilistic DL-Lite<sub>R</sub>.** In partially preordered possibilistic knowledge bases (KBs), symbolic weights are assigned to ABox assertions from a partially ordered set denoted by $U = \emptyset, u_1, \dots, u_n, \mathbb{1}$. This set $U$ is equipped with a strict partial order $>$, forming the uncertainty scale $\mathbb{L} = (U, >)$. By definition, $>^*$ is an irreflexive and transitive relation. Some weights may be incomparable; that is, for any two weights $u_i, u_j \in U$, $u_i \not>^* u_j$ means that neither $u_i >^* u_j$ nor $u_j >^* u_i$ holds. The weight $\mathbb{1}$, representing full certainty, is the greatest element of $U$ in the sense that for every $u_i \in U$, we have $\mathbb{1} >^* u_i$.

A partially preordered DL-Lite<sub>R</sub> KB is a triple $\mathcal{K}_> = (\mathcal{T}, \mathcal{A}_>, \mathbb{L})$, where a single weight from $U$ is assigned to each assertion:

$$ \mathcal{A}_> = \{(\varphi_i, u_i) \mid \varphi \text{ is an assertion and } u_i \in U\} $$

For simplicity, given two assertions $(\varphi_i, u_i)$, $(\varphi_j, u_j) \in \mathcal{A}_>$, we write $\varphi_i >^* \varphi_j$ and $\varphi_i \not>^* \varphi_j$ to denote $u_i >^* u_j$ and $u_i \not>^* u_j$, respectively.

*Example 3.* Assume the uncertainty scale $\mathbb{L} = (U, >)$ given by $U = \{u_1, u_2, u_3, u_4, u_5, \mathbb{1}\}$ and the strict partial order $>^*$ illustrated by Figure 1b (solid arrows represent $>^*$). The weights of $\mathbb{L}$ are assigned to the assertions of $\mathcal{A}_>$, as illustrated by Figure 1a. The obtained partially preordered ABox is denoted by $\mathcal{A}_>$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: The ABox $\mathcal{A}_>$: solid arrows represent strict preference.</figcaption></figure>

A key idea for reasoning with partially ordered information is to extend the partial order into a family of total orders. Each total order preserves the strict relations of the original total order and resolves each incomparability into a strict total preference. Symbolic weights from the uncertainty scale $\mathbb{L}$ are then replaced with numerical values in the interval $[0, 1]$. This family of total orders induces
a corresponding family of totally preordered ABoxes, denoted ($A_{>1},...,A_{>m}$),
said to be compatible with the partially preordered ABox $A_{\succ}$. Formally:

$$
\forall (\varphi_i, w_i), (\varphi_j, w_j) \in A_{>k}, \text{ if } \varphi_i \succ \varphi_j \text{ then } w_i > w_j. \text{ where } w_i, w_j \in [0, 1]
$$

The number of numerical values used to replace the symbolic weights from the interval ]0, 1] is infinite. However, it suffices to consider only a finite number of extensions—namely, those that induce distinct total orderings. Indeed, if two different distributions of weights over formulas induce the same order, they lead to equivalent repairs and thus can be regarded as indistinguishable in this respect.

After generating all compatible totally preordered ABoxes, one can compute
the possibilistic repair $\mathcal{R}(A_{>i})$ for each of them. This process can be further
improved by intersecting the deductive closures of these repairs, rather than the
repairs themselves. This yields a more productive (i.e., larger) repair [23]. The
procedure for obtaining such a productive repair is as follows:

1. Extend the partial order $\succ$ over $U$ to obtain a family of total orders, inducing a set of totally preordered ABoxes $(A_{>1}, \dots, A_{>m})$ compatible with $A_{\succ}$.

2. For each compatible ABox $A_{>i}$, compute its possibilistic repair $\mathcal{R}(A_{>i})$.

3. Compute the deductive closure of each repair $cl(\mathcal{R}(A_{>i}))$.

**Definition 3.** *The closure-based partially preordered possibilistic repair of A$\succ$, denoted c$\pi$(A$\succ$), is obtained as follows:*

$$
c\pi(\mathcal{A}_{\succ}) = \bigcap_{i=1}^{m} \{cl(\mathcal{R}(\mathcal{A}_{>i})) \mid \mathcal{A}_{>i} \text{ is compatible with } \mathcal{A}_{\succ.}\}
$$

*Example 4.* We compute the Cπ-repair of the ABox $\mathcal{A}_{\succ}$ from Example 1:

$$
c\pi(\mathcal{A}_{\succ}) = \{\text{Dean}(\text{Alice}), \text{FacMember}(\text{Alice}), \text{Prof}(\text{Bob}), \text{FacMember}(\text{Bob})\}
$$

Let $>_{1}, \dots, >_{m}$ be the total order extensions of $\succ$. Observe:

* For all $>_{i}$, $u_{1} >_{i}$ $u_{4}$ and $u_{1} >_{i}$ $u_{5}$, so $\text{Dean}(\text{Alice})$ is more certain than the first assertion that makes any $A_{>i}$ inconsistent, thus $\text{Dean}(\text{Alice}) \in \mathcal{R}(A_{>i})$. In addition, $\text{FacMember}(\text{Alice}) \in cl(\mathcal{R}(A_{>i}))$, so both belong to $c\pi(A_{\succ})$.

* For all $>_{i}$, $\text{Prof}(\text{Bob}) \in \mathcal{R}(A_{>i})$ or $\text{FullProf}(\text{Bob}) \in \mathcal{R}(A_{>i})$, this is because at least one of $u_{2} >_{i}$ $u_{5}$ or $u_{3} >_{i}$ $u_{4}$ holds for any $>_{i}$. Hence, for all $>_{i}$, $\text{Prof}(\text{Bob}) \in cl(\mathcal{R}(A_{>i}))$, and $\text{FacMember}(\text{Bob}) \in cl(\mathcal{R}(A_{>i}))$. Therefore, both belong to $c\pi(A_{\succ})$.

* The assertions FullProf(Bob), VisitingProf(Bob), and PostDoc(Bob) are not in-
ferred from $\mathcal{R}(A_{>i})$ for at least one $>_{i}$.

The use of closure in Cπ-repair ensures inferring conclusions implicit in all re-
pairs of compatible ABoxes, making it more productive than the unclosed vari-
ant [24]. However, extending a partial order into total orders is computationally
expensive, as the number of total extensions can grow exponentially.

To address this challenge, we introduce two major improvements. First, we
avoid enumerating every total extension of the partial order. Second, we avoid
computing the deductive closure for each individual repair. Instead, we perform
a single closure operation directly on the initial knowledge base, from which we
compute a single repair that fully characterizes the Definition 3.
# 3 Possibilistic Repairs via Prior Closure of Partial Information: Avoiding Total Enumeration

## 3.1 How to deductively close a partially preordered ABox?

As previously introduced, deductive closure can be applied in two ways when reasoning with flat (unordered), totally ordered, or partially ordered ABoxes. One approach, used in the definition of $C\pi$-repair (Definition 3), applies the standard DL-Lite$_R$ closure to each repair derived from a total order extending the original partial preference. This approach does not require any adaptation of the standard definition of closure, since the repairs are flat (unordered) or totally ordered.

Alternatively, closure can be applied directly to the initial ABox before completing any repairs; this is the approach followed in this article. However, this raises an important question: unlike repairs, the initial ABox is partially ordered, and the standard notion of closure cannot be directly applied. In particular, an assertion may be inferred in multiple ways, each associated with different weights, making it unclear how to assign a single weight to the inferred assertion.

For example, consider the TBox $T_p = \text{Prof} \sqsubseteq \text{FacMember}, \text{Dean} \sqsubseteq \text{FacMember}$ and the ABox $A = (\text{Dean}(jean), u_1), (\text{Prof}(jean), u_2)$. Jean is both a professor and a dean, each with an uncertain weight ($u_1$ or $u_2$). Since both roles imply being a faculty member, we can infer $\text{FacMember}(jean)$ from either assertion. The key question is: what reliability degree should be assigned to this inferred assertion? In standard possibilistic logic under a total order, the natural choice is the maximum of the two degrees, resulting in a unique value. But with only a partial order, selecting a unique degree is not straightforward, complicating reasoning and motivating the need for adapted closure definitions.

In a partially ordered setting, $u_1$ and $u_2$ may be incomparable, so no unique maximum applies. Both weights must be retained, each independently supporting the assertion without preference. Thus, the deductive closure of a partially ordered ABox cannot assign a single weight to each assertion. Instead, it yields an extended ABox where inferred assertions may carry sets of weights, reflecting multiple, possibly incomparable justifications. Assigning subsets of the partially ordered set $U$ to assertions defines a new uncertainty scale comparing them. This scale is given by $\mathbb{L}^U = (2^U, \Delta^{\max})$, where the order $\Delta^{\max}$ is defined as follows:

$$ \forall \mathcal{X}, \mathcal{Y} \subseteq U, \quad \mathcal{X} \Delta^{\max} \mathcal{Y} \text{ iff } \forall u_j \in \mathcal{Y}, \exists u_i \in \mathcal{X} \text{ such that } u_i \Delta u_j $$

Since $\Delta$ is a strict order relation, one can easily check that $\Delta^{\max}$ is also an irreflexive and transitive order relation between subsets of $U$.

The following definition introduces the *partially preordered deductive closure* of a partially preordered ABox $A\Delta$ using the uncertainty scale $\mathbb{L}^U$.

**Definition 4.** Let $T_p$ denote the set of positive inclusion axioms of $T$. The partially preordered deductive closure of $A\Delta$ w.r.t $T$ is defined as:

$$ cl_{\Delta}(A\Delta) = \{(\varphi, U(\varphi)) \mid \varphi \text{ is an assertion and } U(\varphi) \neq \emptyset\} $$
where, for any assertion $\varphi$, $U(\varphi)$ is the set of weights associated with its derivations, defined as:

$$U(\varphi) = \{u_i \in U \mid \exists(\varphi_i, u_i) \in \mathcal{A}_{\triangleright} \text{ such that } \langle \mathcal{T}_p, \{\varphi_i\} \rangle \models \varphi\}$$

If there is no support for $\varphi$ in $\mathcal{A}_{\triangleright}$, then $U(\varphi) = \emptyset$.

Note that, to align with the standard closure of Definition 2, each assertion $\varphi$ is either a concept assertion $B(a)$ or a role assertion $R(a, b)$, where $B$ and $R$ are a concept and a role name in $\mathcal{T}$, and $a, b$ are individuals mentioned in $\mathcal{A}_{\triangleright}$. Furthermore, since DL-Lite$_R$ does not allow for conjunction in TBox axioms, the minimal set to derive any assertion contains one assertion at most [24]. Note also that if two assertions $\varphi_i$ and $\varphi_j$ are such that $\langle \mathcal{T}_p, \{\varphi_i\} \rangle \models \varphi_j$, then all the weights supporting $\varphi_i$ also support $\varphi_j$; that is, $U(\varphi_i) \subseteq U(\varphi_j)$.

Example 5. Let us apply the partially preordered closure on $\mathcal{A}_{\triangleright}$ from Example 3. The TBox’s positive inclusion axioms $\mathcal{T}_p$ are:

$$\mathcal{T}_p = \{ \text{1. FullProf} \sqsubseteq \text{Prof} \quad \text{2. Prof} \sqsubseteq \text{FacMember} \quad \text{3. Dean} \sqsubseteq \text{FacMember} \}$$

The partially preordered closure of $\mathcal{A}_{\triangleright}$ is given by:

$$cl_{\triangleright}(\mathcal{A}_{\triangleright}) = \left\{ \begin{array}{ll} (\text{Dean}(\text{Alice}), \{u_1\}), & (\text{FullProf}(\text{Bob}), \{u_2\}), \\ (\text{Prof}(\text{Bob}), \{u_2, u_3\}), & (\text{PostDoc}(\text{Bob}), \{u_4\}), \\ (\text{VisitingProf}(\text{Bob}), \{u_5\}), & (\text{FacMember}(\text{Alice}), \{u_1\}), \\ (\text{FacMember}(\text{Bob}), \{u_2, u_3\}) & \end{array} \right\}$$

## 3.2 The cπ-acceptance method

To fully leverage the structure of partially ordered knowledge bases, we propose a new characterisation of Cπ-repair that starts with the partially preordered deductive closure defined in Definition 4. The key idea is the following: an assertion should only be accepted if it is consistent with all the information that is at least as reliable as itself, meaning assertions that are strictly more certain, equally certain, or incomparable. In this framework, each inferred assertion $\varphi$ is annotated with the set of weights $U(\varphi)$ it receives through the closure $cl_{\triangleright}(\mathcal{A}_{\triangleright})$.

Instead of verifying consistency with the entire ABox, we narrow the scope to a relevant subset: those assertions whose weights are not strictly less reliable than any weight in $U(\varphi)$. In other words, we test whether $\varphi$ conflicts with what is already known and sufficiently trusted. If such a conflict arises, the assertion must be discarded from the Cπ-repair. To make this selection possible, we refine the $\triangleright^{\text{max}}$ relation to compare a set of weights $U(\varphi_i) \subseteq U$ with a singleton $\{u_j\}$, where $u_j \in U$. The comparison is defined as follows:

$$U(\varphi_i) \triangleright^{\text{max}} \{u_j\} \iff \exists u_i \in U(\varphi_i) \text{ such that } u_i \triangleright u_j$$

This helps us identify assertions that are strictly less certain than a given assertion $(\varphi_i, U(\varphi_i)) \in cl_{\triangleright}(\mathcal{A}_{\triangleright})$ and exclude them from the consistency checking.
**Definition 5.** Let $(\varphi_i, U(\varphi_i))$ be an assertion from $cl_{\triangle}(A_{\triangle})$. The set of asser-
tions that are strictly less certain than $\varphi_i$ is defined as:

$$
\mathrm{SLT}(\varphi_i) = \{ \varphi_j \mid (\varphi_j, u_j) \in A_{\triangle} \text{ and } U(\varphi_i) \triangleright^{\max} \{u_j\} \}
$$

The complementary set containing assertions not strictly less certain than $\varphi_i$ is:

$$
\Delta(\varphi_i) = A_{\triangle} \setminus \mathrm{SLT}(\varphi_i)
$$

We now introduce the crucial notion of cπ-acceptance, which plays a key role in
characterizing whether an inferred assertion belongs to the cπ(A<sub>▷</sub>), the closed
possibilistic repair given by Definition 3.:

**Definition 6.** An assertion $(\varphi_i, U(\varphi_i)) \in cl_{\triangle}(A_{\triangle})$ is said to be cπ-accepted if and only if the pair $\langle T, \Delta(\varphi_i) \rangle$ is consistent.

This definition is fundamental, as it reduces the problem of checking whether
an inferred assertion belongs to the intersection of the closures of all repairs
to a simple consistency check. Hence, the Cπ-repair of a partially preordered
ABox A<sub>▷</sub> can be computed by applying Definition 6 to all assertions in cl<sub>▷</sub>(A<sub>▷</sub>)
and collecting those that are cπ-accepted.

*Example 6.* Let us now apply the *cπ-acceptance characterisation of Definition 6* on the partially preordered closed ABox of Example 5. For each assertion $\varphi_i$, we use its set of weights $U(\varphi_i)$ to exhibit $\Delta(\varphi_i)$, on which a consistency test is performed. The test result is indicated using (✓) if successful and (✗) otherwise.

$$
\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l}
\varphi_1 = \text{Dean}(\text{Alice}), & U(\varphi_1) = \{u_1\}, & \Delta(\varphi_1) = \{\varphi_1, \varphi_2, \varphi_3\}, & \checkmark \\
\varphi_2 = \text{FullProf}(\text{Bob}), & U(\varphi_2) = \{u_2\}, & \Delta(\varphi_2) = \{\varphi_1, \varphi_2, \varphi_3, \varphi_5\}, & \times \\
\varphi_3 = \text{Prof}(\text{Bob}), & U(\varphi_3) = \{u_2, u_3\}, & \Delta(\varphi_3) = \{\varphi_1, \varphi_2, \varphi_3\}, & \checkmark \\
\varphi_4 = \text{PostDoc}(\text{Bob}), & U(\varphi_4) = \{u_4\}, & \Delta(\varphi_4) = \{\varphi_1, \varphi_2, \varphi_3, \varphi_4, \varphi_5\}, & \times \\
\varphi_5 = \text{VisitingProf}(\text{Bob}), & U(\varphi_5) = \{u_5\}, & \Delta(\varphi_5) = \{\varphi_1, \varphi_2, \varphi_3, \varphi_4, \varphi_5\}, & \times \\
\varphi_6 = \text{FacMember}(\text{Alice}), & U(\varphi_6) = \{u_1\}, & \Delta(\varphi_6) = \{\varphi_1, \varphi_2, \varphi_3\}, & \checkmark \\
\varphi_7 = \text{FacMember}(\text{Bob}), & U(\varphi_7) = \{u_2, u_3\}, & \Delta(\varphi_7) = \{\varphi_1, \varphi_2, \varphi_3\}, & \checkmark
\end{array}
$$

Successful consistency checks mean cπ-acceptance, one can see that the set of
the cπ-accepted assertions is equivalent to cπ(A<sub>▷</sub>) computed in Example 4.

The following proposition shows that, indeed, the repair obtained by checking
$c\pi$-acceptance of every assertion in the partially preordered closure of a partially
preordered knowledge base is equivalent to the $C\pi$-repair given in Definition 3.

**Proposition 1.** An assertion $\varphi$ is accepted using the $c\pi$-acceptance approach, as per Definition 6, if and only if $\varphi \in c\pi(A_{\triangle})$, as per Definition 3.

This above proposition offers two major advantages. First, it avoids the need
to enumerate all possible extensions of the partially preordered ABox. Second,
it eliminates the necessity of computing the closure for each individual repair
<figure><img src="image_3.png" /><figcaption>Fig. 2: $c\pi$-acceptance vs. the process of computing the $C\pi$-repair</figcaption></figure>

associated with those extensions. Instead, the closure is computed once, globally, over the initial structure, relying only on the weighted subsets associated with each inferred assertion. Figure 2 illustrates the process of computing the $C\pi$-repair alongside the equivalence with the new characterisation.

In DL-Lite<sub>R</sub>, consistency checking is performed in AC<sup>0</sup> with respect to data complexity [16,1]. Consistency testing is a crucial task in our approach, as each assertion must be checked for consistency against relevant subsets of the knowledge base. Each individual consistency test is in AC<sup>0</sup>. The overall repair computation process, which involves iterating over all assertions and verifying their consistency, remains polynomial and tractable. The following proposition formalises the computation of the complete repair in polynomial time.

**Proposition 2.** *The computation of the set of assertions $(\varphi, U(\varphi)) \in cl_{\Delta}(A_{\Delta})$ that are $c\pi$-accepted is achieved in polynomial time, in accordance with the data complexity of DL-Lite<sub>R</sub>.*

Before turning to the experimental evaluation, we briefly outline a possible semantic characterisation of inference under inconsistency and partial priority in ABoxes. This is intended as a preliminary sketch rather than a complete treatment; a more detailed exploration of such semantic foundations is left for future work. Here, we simply introduce the key ingredients that could underlie the semantics of possibilistic repairs.

In the standard setting, where the ABox is consistent and no priority is specified, DL-Lite semantics relies on the classical model/countermodel distinction: an interpretation is a model if it satisfies both the TBox *T* and the ABox *A*, and a countermodel otherwise. This dichotomy can be captured using a binary plausibility scale, where models are assigned value 1 and countermodels 0.

However, when the ABox is inconsistent and equipped with a partial preorder $\le_A$ (or $\Delta_A$) over its assertions, representing priority or certainty, this binary view
becomes inadequate. To account for varying degrees of plausibility, we use again the graded uncertainty scale $\mathcal{U} = \{\emptyset, u_1, \dots, u_n, \mathbb{1}\}$, where $\emptyset$ represents complete impossibility and $\mathbb{1}$ represents full plausibility. This approach echoes preferential semantics commonly used in non-monotonic reasoning.

Each interpretation $\mathcal{I}$ is then evaluated based on how it satisfies the TBox and (partially ordered) ABox. If $\mathcal{I} \not= \mathcal{T}$, or it violates an assertion $\varphi$ such that weight($\varphi$) = $\mathbb{1}$, it receives $\emptyset$. If it satisfies both $\mathcal{T}$ and all assertions in $\mathcal{A}$, it receives $\mathbb{1}$. Otherwise, it receives a set of values $\{u_i \in \mathcal{U} \mid \mathcal{I} \not= \varphi \text{ for some } (\varphi, u_i) \in \mathcal{A}\}$, reflecting the uncertainty degrees of the violated assertions.

This gives rise to a partial possibility distribution $\sigma : \Omega \to 2^{\mathcal{U}}$, where $\Omega$ is the set of interpretations. Interpretations are then compared based on the plausibility sets assigned by $\sigma$, using a minimality relation (such as the $\succmax$ ordering introduced earlier). The preferred interpretations $\mathcal{P}(\mathcal{T}, \mathcal{A})$ are those for which no strictly more plausible interpretation exists, namely interpretations $\mathcal{I}$ such that there is no $\mathcal{I}'$ with $\sigma(\mathcal{I}') > \sigma(\mathcal{I})$. These preferred interpretations, which comply with the TBox and are as consistent as possible with the partially ordered ABox, are then used as the semantic basis for answering queries: a query answer is considered valid if it is satisfied by all preferred interpretations.

# 4 Experimental evaluation

We implemented our method in python, using the `py_reasoner` tool. This tool integrates a PostgreSQL back-end for ABox storage, the `RDFLib` library to enable efficient access to OWL ontologies in memory, and the `Rapid` query rewriting engine to rewrite queries related to conflicts and assertion supports.

## 4.1 Experimental setting

**TBox.** We used the latest version of the DBpedia ontology [25], which is populated via mappings from Wikipedia infoboxes. To preserve its original structure, we employed interfacing methods for axiom retrieval and query rewriting restricted to $DL-Lite_R$ axioms. The ontology includes negative inclusion axioms over classes and object properties, consistent with the $DL-Lite_R$ syntax. The TBox contains 34,653 axioms, including 27 explicit negative inclusions, resulting in a total of 614,670 implicit and explicit negative inclusion axioms.

**Datasets.** We built ABoxes from DBpedia RDF type statements and object relations, supplemented with LHD inferred types from an earlier version of the ontology [25]. The latter includes unclean inferred types, making it well suited for inconsistency experiments. We extracted subsets with progressively increasing ABox and conflict set sizes. Table 1 details the ABoxes used in our evaluation.

We defined a partial preorder $\succ$ over ABox assertions using three criteria. The first two rely on timestamps: *derivationTimestamp* from DBpedia and
<table><caption>Table 1: ABoxes in terms of: size, percentage of assertions in conflicts, number of conflicts, and time to compute the conflict set, size of the deductive closure, time to compute the standard closure, and the partially ordered closure.</caption><thead><tr><th>ABox (A)</th><th>#A</th><th>%Cf(A)</th><th>#Cf(A)</th><th>Cf(A)(s)</th><th>#cl(A)</th><th>cl(A)(s)</th><th>cl<sub>▷</sub>(A)(s)</th></tr></thead><tbody><tr><td>dbr_n1e03_p5e-02</td><td>1k</td><td>8.3</td><td>72</td><td>59.29</td><td>3833</td><td>3.0</td><td>2.98</td></tr><tr><td>dbr_n1e03_p3e-01</td><td>1k</td><td>32.7</td><td>794</td><td>60.05</td><td>4716</td><td>2.87</td><td>3.05</td></tr><tr><td>dbr_n1e03_p5e-01</td><td>1k</td><td>52.4</td><td>1643</td><td>60.34</td><td>5248</td><td>3.0</td><td>2.94</td></tr><tr><td>dbr_n1e04_p5e-02</td><td>10k</td><td>7.03</td><td>1782</td><td>62.15</td><td>34968</td><td>2.73</td><td>3.58</td></tr><tr><td>dbr_n1e04_p3e-01</td><td>10k</td><td>32.36</td><td>28098</td><td>64.95</td><td>42691</td><td>3.01</td><td>3.34</td></tr><tr><td>dbr_n1e04_p5e-01</td><td>10k</td><td>51.56</td><td>51495</td><td>63.06</td><td>49535</td><td>3.31</td><td>3.3</td></tr><tr><td>dbr_n5e04_p5e-02</td><td>50k</td><td>10.68</td><td>25643</td><td>69.6</td><td>175376</td><td>2.88</td><td>4.38</td></tr><tr><td>dbr_n5e04_p3e-01</td><td>50k</td><td>33.02</td><td>230548</td><td>74.6</td><td>210992</td><td>3.38</td><td>6.76</td></tr><tr><td>dbr_n5e04_p5e-01</td><td>50k</td><td>51.79</td><td>397370</td><td>67.36</td><td>238008</td><td>3.63</td><td>7.07</td></tr></tbody></table>

wikiTimestamp from Wikidata (via the Wikipedia API), both ordered by re-cency. The third labels assertions by their source RDF (preferred) or LHD (less preferred). These yield orders >_dts, >_wts, and >_src, such that $\varphi \succ \varphi'$ if:

$$ \varphi >_{\text{dts}} \varphi' \text{ or } \varphi >_{\text{wts}} \varphi' \text{ or } \varphi >_{\text{src}} \varphi' \wedge \neg(\varphi' >_{\text{dts}} \varphi \text{ or } \varphi' >_{\text{wts}} \varphi \text{ or } \varphi' >_{\text{src}} \varphi) $$

The evaluation compares the performance of the new characterisation from Definition 6 with the earlier “dominant supports” method introduced in [24].

## 4.2 Experimental results

We implemented the dominant supports characterisation as follows. We first compute the conflict set of the ABox. Then, for each assertion in $cl(A \succ)$ (Definition 2), we compute its support sets, each linear in ABox size. Finally, we check that every conflict has at least one dominant support per assertion.

The new characterisation starts by computing the partially preordered closure, then checks consistency for each assertion with those not strictly less certain. This check can be done in two ways. The first uses a SQL query that detects violations of TBox negative inclusion axioms, ignoring those involving assertions less certain than the one checked. Since SQL involves I/O operations, the second runs in memory using the conflict set, identifying violations where no assertion is strictly less certain. We used the latter for weakly conflicting ABoxes, where the conflict set fits in memory.

Our experiments demonstrate the stable performance of the “$c\pi$-acceptance” method compared to the so-called “dominant supports” method in computing the $C\pi$-repair. Figure 3 [left, middle] plots running times across ABox sizes, grouped by percentage of conflicting assertions. The new method is at least 2× faster on small ABoxes (1k), 10× on medium (10k), and over 40× on large (50k). As both methods process each assertion in the deductive closure, we also plotted repair time against closure size in Figure 3 [right]. While “dominant supports” slows down with larger closures, the new method remains stable.

*Discussion.* The experimental evaluation highlights the efficiency and scalability of the proposed method. In particular, computing the $C\pi$-repair via the partially
<figure><img src="image_4.png" /><figcaption>Fig. 3: Comparison of Cπ-repair Computation Time using (2) Characterisations.</figcaption></figure>

preordered closure avoids the exhaustive search over support sets required by the “dominant supports” method. While applying the deductive closure is inherently costly, especially under partial orders, our method mitigates this by limiting consistency checks to relevant assertion subsets. The results show that it scales gracefully with both the ABox size and the closure size, delivering substantial time savings even in large and highly conflicting datasets. This demonstrates the practical advantage of relying on a more targeted and order-aware closure.

## 5 Conclusion

A natural semantics for handling inconsistent and partially ordered ontologies is to consider the family of total order extensions of the partial order. However, this approach is computationally expensive, as the number of such extensions grows exponentially. In this paper, we tackled the issue of efficiently computing a productive repair for partially preordered DL-Lite<sub>R</sub> ABoxes by introducing a new partially preordered closure operator and by pushing the repair computation to the most basic reasoning task: consistency checking. One of the main strengths of our approach lies in applying deductive closure directly to the initial partially preordered ABox, before any repair computation. Unlike previous approaches, this early application does not introduce unwanted consequences nor degrade the quality of the repairs. On the contrary, it allows us to preserve tractability by reducing the problem to consistency checks over subsets of the closed ABox. This closure-based process yields the same repairs as those obtained by computing and then closing all compatible total order extensions of the ABox, but in a much more efficient manner. We show the efficiency of this method particularly through comparisons with previous approaches that rely on exhaustive searches over the conflict sets of the ABox and the support sets of assertions.

Future work includes extending the closure-based approach and its associated tractability guarantees to more expressive logical languages.

**Acknowledgments.** This research was supported by the European Union's Horizon research and innovation programme under the MSCA-SE (Marie Skłodowska-Curie Ac-
tions Staff Exchange); Call: HORIZON-MSCA-2021-SE-01; Project title: STARWARS (STormwAteR and WastewAteR networkS heterogeneous data AI-driven management) [grant agreement 101086252]. A. Laouar's PhD is supported by the ANR project Vivah (Vers une intelligence artificielle à visage humain) [grant number ANR-20-THIA-0004]. This research has also received support from the ANR project EXPIDA (EXplainable and parsimonious Preference models to get the most out of Inconsistent DAtabases), [grant number ANR-22-CE23-0017]. The second author has also received support from the French national project ANR (Agence Nationale de la Recherche) CROQUIS (Collecte, représentation, complétion, fusion et interrogation de données hétérogènes et incertaines de réseaux d'eaux urbains) [grant number ANR-21-CE23-0004]. Finally, we thank the anonymous referees for their valuable feedback on this paper.

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