Approximate minimization of interpretations in fuzzy description logics under the G\"odel semantics

TL;DR

This paper introduces the first algorithm for approximate minimization of finite fuzzy interpretations in certain fuzzy description logics under G"odel semantics, enhancing reasoning efficiency in knowledge systems.

Contribution

It presents a novel algorithm that minimizes fuzzy interpretations without quotient constructions, supporting approximate preservation and applicable to a broad class of FDLs.

Findings

Algorithm preserves fuzzy concept assertions up to degree γ.

Time complexity is O((m log l + n) log n).

Supports approximate minimization in diverse FDLs.

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 G\"odel 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 $\gamma \in (0,1]$. 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. Methodologically, our approach fundamentally differs from existing ones, as it avoids quotient constructions traditionally employed for minimizing fuzzy interpretations and fuzzy automata.