The More the Merrier: Combining Properties for ABox Abduction under Repair Semantics in ELbot

Abstract: Abduction is a central approach to explain missing entailments from a knowledge base by providing a hypothesis, that would, if added to the knowledge base, make the missing entailment become true. Abduction under repair semantics has recently been investigated in detail, where several desirable properties and optimality criteria were considered, such as signature-restrictions and minimality in size and of introduced conflicts. Naturally, hypotheses that satisfy more than one of these properties or combine a property with an optimality criterion would be even more desirable for applications. So far, such hypotheses have not been investigated in the literature. In the present paper, we consider the ABox abduction problem for hypotheses satisfying more than one property or additional optimality criteria, for EL_bot under brave and AR semantics. Our main observation is that often requiring additional properties for hypotheses does not lead to an increase of complexity.

• The paper presents an analysis of how combining structural properties, such as signature restriction and conflict-confinement, influences hypothesis existence and verification.
• It employs reductions from quantified Boolean formulas to rigorously evaluate the computational impact of minimality and conflict-based constraints.
• The study reveals that, with few exceptions, the overall complexity is governed by the hardest individual property, informing scalable ontology repair methods.

Combining Properties for ABox Abduction Under Repair Semantics in $\mathcal{EL}_\bot$

Introduction and Motivation

The paper systematically investigates ABox abduction under repair semantics in the description logic $\mathcal{EL}_\bot$, focusing on the computational effects of combining desirable structural properties for abductive hypotheses. Abduction, as employed here, is used to explain missing entailments from a knowledge base by identifying hypotheses whose integration would allow these entailments. In the context of inconsistency-tolerant reasoning—especially under repair semantics (brave, AR)—multiple properties have emerged as significant for practical and theoretical considerations: signature restriction, non-triviality, minimality (subset and cardinality), and conflict-confinement. The primary question addressed is whether requiring these properties simultaneously induces additional computational complexity for the existence and verification of hypotheses.

Formal Framework and Problem Formulation

The authors precisely define ABox abduction for inconsistent KBs under repair semantics, restricting hypotheses to ABox assertions only (no fresh individuals, only flat/simple assertions). Given an inconsistent knowledge base $(\mathcal{T}, \mathcal{A})$ and an observation $\alpha$, a hypothesis $\mathcal{H}$ is sought such that $$(\mathcal{T}, \mathcal{A} \cup \mathcal{H}) \models_S \alpha$$, where $S$ denotes either brave or AR semantics. Multiple properties are considered for $\mathcal{H}$:

• Signature restriction: $\mathcal{H}$ contains only symbols from a predefined signature.
• Non-triviality: $\alpha \notin \mathcal{H}$.
• Minimality: $\mathcal{EL}_\bot$0 is minimal in subset or cardinality among all hypotheses.
• Conflict-confinement: Hypothesis does not introduce new minimal conflicts in the KB.

The central reasoning tasks are hypothesis existence (is there any $\mathcal{EL}_\bot$1 satisfying given constraints?) and verification (does a given $\mathcal{EL}_\bot$2 satisfy the constraints?).

Complexity Analysis for Combined Properties

Combining Properties Without Minimality

The paper provides detailed complexity characterizations for combinations of non-minimal properties:

• Signature restriction + non-triviality: Complexity for existence (NP-complete for brave, $\mathcal{EL}_\bot$3-complete for AR) and verification (\textbf{NP}-complete for brave, \textbf{coNP}-complete for AR) remains unchanged compared to isolated properties.
• Conflict-confinement + non-triviality or signature restriction: Both existence and verification problems rise to $\mathcal{EL}_\bot$4-complete for both brave and AR semantics, reflecting an increase due to the conflict property.

The authors formally establish, via polynomial reductions and combinatorial arguments, that most combinations do not compound hardness beyond the hardest individual property involved, except for certain interactions involving conflict-confinement.

Pairing Properties With Minimality Constraints

Analysis extends to minimality (subset/minimality, cardinality-minimality, conflict-minimality):

• Subset-minimality or cardinality-minimality with signature restriction or non-triviality: The verification problem is $\mathcal{EL}_\bot$5-complete for brave semantics and $\mathcal{EL}_\bot$6-complete for AR.
• Conflict-confinement with minimality: The complexity for verification is $\mathcal{EL}_\bot$7-complete for brave and coNP-complete for AR. Notably, the trivial singleton observation may not suffice under these combined criteria, and smart constructions are used to demonstrate the complexity bounds.

For conflict-minimality (minimizing the number or subset of new conflicts), the authors prove that verifying $\mathcal{EL}_\bot$8-minimal non-trivial or signature-restricted hypotheses is $\mathcal{EL}_\bot$9-complete for both brave and AR semantics. However, the complexity for cardinality-based conflict-minimality remains open, and counting introduced conflicts may be $(\mathcal{T}, \mathcal{A})$0-hard.

Reduction Techniques and Proof Strategies

Hardness proofs utilize reductions from variants of quantified Boolean formulas (QBF), leveraging the correspondence of variable assignments to assertions in the ABox. The technical appendix provides detailed construction for these reductions, notably demonstrating that signature constraints can be enforced via conflict-confinement, and minimality checks can encode gaps between assignments and coverage of all possible repairs.

Implications and Theoretical Significance

The results yield a comprehensive complexity landscape for repair-based abduction in $(\mathcal{T}, \mathcal{A})$1:

• Practical implication: For ontology engineering and diagnosis where both signature restriction and conflict-confinement are desired, algorithms can refer to the established complexity class without expecting an exponential leap. This enables better optimization and resource allocation for tools that require explanations compliant with multiple properties.
• Theoretical implication: The closure properties of the complexity classes under intersection reinforce that the hardest individual property governs computational feasibility for combined constraints, except for certain conflict-minimality interactions.
• Generalization: Many results apply to propositional Horn logic, promising transferability beyond description logics.

The paper also discusses how the behavior changes for more expressive DLs (e.g., $(\mathcal{T}, \mathcal{A})$2) where entailment is already ExpTime-complete, making abduction complexity less interesting as a distinguishing factor. Some properties, such as conflict-confinement in AR semantics, display simplified structure in fragments like $(\mathcal{T}, \mathcal{A})$3-Lite.

Open Questions and Future Directions

Several aspects remain open or partially characterized:

• Counting conflicts for cardinality-based conflict-minimality: Likely $(\mathcal{T}, \mathcal{A})$4-hardness, which would pose significant practical barriers.
• IAR semantics: The landscape is largely unexplored; prior results are limited to practical cases and upper bounds.
• Combinations in $(\mathcal{T}, \mathcal{A})$5-Lite: Additional analysis is required for conflict-confinement, as singleton hypotheses may always suffice, simplifying complexity in some settings.
• Role for expressive DLs: The paper suggests further work to fully characterize abduction complexity beyond basic fragments.

Conclusion

This study establishes that combining structural properties for ABox abduction hypotheses under repair semantics in $(\mathcal{T}, \mathcal{A})$6 generally does not elevate problem complexity beyond the hardest single property, except for special cases involving conflict-minimality. The results provide a foundation for designing explainable reasoning systems with fine-grained control over hypothesis properties, informing both complexity-theoretic understanding and practical tool development. The complexity landscape is well-developed for basic and subset-minimal properties but leaves open questions for conflict-minimality based cardinality and extension to other semantics. Future work should target these gaps and explore broader DL fragments, other repair semantics, and algorithmic strategies for scalable abduction in knowledge-based AI systems.

Reference: "The More the Merrier: Combining Properties for ABox Abduction under Repair Semantics in ELbot" (2606.19197)