A Distance-based Paraconsistent Semantics for DL-Lite

DL-Lite is an important family of description logics. Recently, there is an increasing interest in handling inconsistency in DL-Lite as the constraint imposed by a TBox can be easily violated by assertions in ABox in DL-Lite. In this paper, we present a distance-based paraconsistent semantics based on the notion of feature in DL-Lite, which provides a novel way to rationally draw meaningful conclusions even from an inconsistent knowledge base. Finally, we investigate several important logical properties of this entailment relation based on the new semantics and show its promising advantages in non-monotonic reasoning for DL-Lite.

💡 Research Summary

DL‑Lite is a lightweight family of description logics that underpins the OWL 2 QL profile. Because its TBox constraints are often violated by ABox assertions, DL‑Lite knowledge bases (KBs) can easily become inconsistent. Under classical DL semantics an inconsistent KB entails every possible axiom (the triviality problem), which makes reasoning useless in practice. Existing work on inconsistent DL‑Lite KBs falls into two main streams. The first stream treats inconsistencies as data errors and attempts to “repair” the KB by extracting a maximal consistent subset (MCS). This approach suffers from the multi‑extension problem (there may be many different MCSs) and discards potentially useful information. The second stream adopts paraconsistent or four‑valued semantics (e.g., Belnap’s four‑valued logic, quasi‑classical semantics) to tolerate contradictions. While these approaches avoid triviality, they can be overly skeptical: a conclusion that would follow from a consistent KB under classical semantics may be lost under the paraconsistent semantics, even when the inconsistency is unrelated to the query.

The paper proposes a novel distance‑based paraconsistent semantics that leverages the notion of features (also called Herbrand‑style interpretations with limited structure) introduced for DL‑Lite. A feature consists of a finite set of types (sets of basic concepts) together with a Herbrand set of ground assertions. Features guarantee role coherence: for every role R, the presence of ∃R in the union of the types forces the presence of ∃R⁻ in the same union, and vice‑versa. Because each KB has only finitely many features and each feature is finite, the approach sidesteps the infinite‑model problem that hampers model‑based semantics for expressive DLs.

The core technical contribution is the definition of a distance function between two types. Two concrete distance measures are considered: (i) the Hamming distance, i.e., the size of the symmetric difference of the two type sets, and (ii) a “drastic” distance that is 0 for identical types and 1 otherwise. A distance function d must be symmetric and return 0 only for equal types. For a type τ and a set of types Ξ, the distance d(τ, Ξ) is defined as the minimum distance between τ and any member of Ξ; if Ξ is empty a large default value d is used to penalise contradictions.

Since a TBox may contain several inclusions, the paper introduces a type group Π = {Ξ₁,…,Ξₙ}, where each Ξᵢ is the set of types satisfying a single inclusion. To aggregate the multiple distances d(τ, Ξᵢ) into a single score, an aggregation function f is employed. f must be monotone, return 0 only when all arguments are 0, and be identity on singletons. Typical choices include the sum, the maximum, or weighted averages. The combined score λ_{d,f}(τ, Π) = f({d(τ, Ξ₁),…,d(τ, Ξₙ)}) quantifies how “close” τ is to satisfying the whole TBox.

A type τ is called d_f‑minimal with respect to Π if its λ‑score is minimal among all types. The set Λ_{d,f}(Π, Ξ) of all d_f‑minimal types inside a candidate set Ξ is always non‑empty (Proposition 1) and, when the TBox is consistent, coincides with the intersection of the Ξᵢ’s (i.e., the exact model types). However, minimal types alone may violate role coherence, as illustrated by an example where ∃P appears without the required ∃P⁻.

To restore role coherence, the authors define an inflationary operator µ_{d,f}. Given a type set Ξ, µ_{d,f}(Ξ) adds to Ξ any type that (a) introduces the missing counterpart of a role appearing in Ξ, and (b) is at least as close (according to λ) as any other type that could provide that counterpart. Iterating µ_{d,f} until a fixed point yields Ξ⁺, the coherent closure of Ξ. The minimal model type set of a TBox T is then defined as Λ⁺{d,f}(Π(T), T_Σ) = Λ{d,f}(Π(T), T_Σ) ∪ … obtained after closure. Proposition 2 shows that this set is always non‑empty, respects role coherence, and collapses to the exact model types when the TBox is coherent (i.e., when ∃R and ∃R⁻ appear together).

With the minimal model type set in hand, the paper defines a distance‑based entailment relation: a KB K distance‑entails an axiom φ (written K ⊨_{d,f} φ) iff every minimal model feature of K (i.e., a feature whose type component belongs to the minimal model type set) satisfies φ. This entailment enjoys several desirable logical properties:

1. Paraconsistency – In the presence of inconsistency, not every axiom follows; only those supported by the closest coherent features are derived.
2. Non‑monotonicity – Adding new assertions can invalidate previously derived conclusions, reflecting the intuition that new contradictory information should cause a re‑evaluation.
3. Cautiousness – Compared with four‑valued semantics, the distance‑based approach is more conservative, avoiding derivations that are not supported by any minimal coherent feature.
4. Consistency preservation – If K is classically consistent, distance‑based entailment coincides exactly with classical entailment, a property many paraconsistent approaches lack.

The authors prove these properties formally (proofs are omitted from the main text due to space constraints but are available in an extended technical report). They also argue that although the development is carried out for DL‑Liteⁿᵇᵒᵒˡ (the most expressive member of the DL‑Lite family), the methodology extends straightforwardly to other DL‑Lite variants because the underlying notions of types, features, and distance functions are independent of the specific constructors.

In summary, the paper introduces a principled, distance‑driven semantics for inconsistent DL‑Lite KBs that bridges the gap between model‑based reasoning (which suffers from infinite models) and paraconsistent reasoning (which can be overly skeptical). By grounding the semantics in finite features and a well‑behaved distance measure, the approach retains as much useful information as possible, respects role coherence, and provides a robust, non‑monotonic entailment relation that aligns with classical reasoning when the KB is consistent. This contribution advances the state of the art in inconsistency‑tolerant reasoning for lightweight description logics and opens avenues for further research on distance‑based semantics in richer DL families.