Data Complexity of Querying Description Logic Knowledge Bases under Cost-Based Semantics

TL;DR

The paper advances the data-complexity analysis of querying inconsistent weighted description logic knowledge bases under cost-based semantics, extending to DL-Lite fragments with inverse roles and role inclusions. It introduces a small-interpretation quotient/interlacing framework and a partial normal form to obtain tight upper bounds (Δ^p_2) for optimal-cost semantics and tractable FO rewritings in a key DL-Lite case (DL-Lite_bool^H) under fixed cost, while establishing strong lower bounds (NP/CO-NP/Δ^p_2) across several DL fragments. The work delivers a practical pathway to FO rewritings and AC^0 data complexity for certain fixed-cost query answering, supported by a robust theoretical apparatus including a very small interpretation property and quotient constructions. Overall, the results significantly sharpen the understanding of cost-based semantics in dynamic DL contexts and point toward feasible implementations for tractable OMQA in the DL-Lite milieu.

Abstract

In this paper, we study the data complexity of querying inconsistent weighted description logic (DL) knowledge bases under recently-introduced cost-based semantics. In a nutshell, the idea is to assign each interpretation a cost based upon the weights of the violated axioms and assertions, and certain and possible query answers are determined by considering all (resp. some) interpretations having optimal or bounded cost. Whereas the initial study of cost-based semantics focused on DLs between $\mathcal{EL}_\bot$ and $\mathcal{ALCO}$, we consider DLs that may contain inverse roles and role inclusions, thus covering prominent DL-Lite dialects. Our data complexity analysis goes significantly beyond existing results by sharpening several lower bounds and pinpointing the precise complexity of optimal-cost certain answer semantics (no non-trivial upper bound was known). Moreover, while all existing results show the intractability of cost-based semantics, our most challenging and surprising result establishes that if we consider $\text{DL-Lite}^\mathcal{H}_\mathsf{bool}$ ontologies and a fixed cost bound, certain answers for instance queries and possible answers for conjunctive queries can be computed using first-order rewriting and thus enjoy the lowest possible data complexity ($\mathsf{TC}_0$).

Figures (1)

• Figure 1: Two interpretations $\mathcal{I}$ and $\mathcal{J}$ of the same WKB. Symbols $\heartsuit, \diamondsuit$ indicate $1$-types without unary violations; $\clubsuit, \spadesuit$ indicate $1$-types with unary violations. An arrow indicates the role $\mathsf{r}$; a bold arrow indicates a satisfied $\mathsf{r}$-role assertion; a dashed bold arrow indicates the violation of an $\mathsf{r}$-role assertion. Superscripts indicate the ABox type of an element; note that $\mathcal{I}$ features one anonymous element. We omit $k$ and pretend $t_0$ is rare, while $t_1$ and $t_2$ are not (alternatively, one could set $k \geq 6$ and consider sufficiently many other individuals with ABox types $t_1$ and $t_2$). Shades of gray, from darker to lighter, highlight the pre-core, the critical elements, the core, and the individuals. $\mathcal{J}$ is obtained by applying Lemma \ref{['lemma:main-lemma-for-dllite']} on $\mathcal{I}$, using the 'perfect' individual with $1$-type $\heartsuit$ (resp. $\diamondsuit$) for ABox type $t_1$ (resp. $t_2$). Note that the witness$w_\spadesuit$ is given $1$-type $\spadesuit$ and thus carries some unary violations.

Theorems & Definitions (36)

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