Approximate minimization of interpretations in fuzzy description logics under the Gödel semantics
Linh Anh Nguyen
TL;DR
The paper addresses the problem of minimizing finite fuzzy interpretations in fuzzy description logics under Gödel semantics, specifically without the Baaz projection operator and without the universal role. It introduces the first algorithm that computes a minimal interpretation although it preserves fuzzy concept assertions up to a degree \(\gamma\in(0,1]\), for a broad class of FDLs ranging from a sublogic of \(f\!\mathcal{ALC}_{reg}\) to extensions with inverse roles and nominals. The method hinges on constructing the greatest fuzzy \(\Phi\)-auto-bisimulation to obtain a compact fuzzy partition and then building a quotient interpretation \(\mathcal{I}'\) that preserves essential truth degrees, with a proven time complexity of \(O((m\log l + n)\log n)\). An extended formulation supports approximate preservation, and the authors provide both theoretical guarantees and an implementation. The work demonstrates that, in many cases, minimizing without Baaz/U can yield smaller domain reductions than approaches that rely on those constructors, offering practical improvements for reasoning tasks in fuzzy ontologies and fuzzy Kripke-like models.
Abstract
The problem of minimizing fuzzy interpretations in fuzzy description logics (FDLs) is important both theoretically and practically. For instance, fuzzy or weighted social networks can be modeled as fuzzy interpretations, where individuals represent actors and roles capture interactions. Minimizing such interpretations yields more compact representations, which can significantly improve the efficiency of reasoning and analysis tasks in knowledge-based systems. We present the first algorithm that minimizes a finite fuzzy interpretation while preserving fuzzy concept assertions in FDLs without the Baaz projection operator and the universal role, under the Gödel semantics. The considered class of FDLs ranges from the sublogic of $f\!\mathcal{ALC}$ without the union operator and universal restriction to the FDL that extends $f\!\mathcal{ALC}_{reg}$ with inverse roles and nominals. Our algorithm is given in an extended form that supports approximate preservation: it minimizes a finite fuzzy interpretation $\mathcal{I}$ while preserving fuzzy concept assertions up to a degree $γ\in (0,1]$. Its time complexity is $O((m\log{l} + n)\log{n})$, where $n$ is the size of the domain of $\mathcal{I}$, $m$ is the number of nonzero instances of atomic roles in $\mathcal{I}$, and $l$ is the number of distinct fuzzy values used in such instances plus 2.