Approximate minimization of interpretations in fuzzy description logics under the Gödel semantics

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. Methodologically, our approach fundamentally differs from existing ones, as it avoids quotient constructions traditionally employed for minimizing fuzzy interpretations and fuzzy automata.

💡 Research Summary

The paper tackles the problem of minimizing finite fuzzy interpretations in fuzzy description logics (FDLs) under Gödel semantics, specifically for logics that do not include the Baaz projection operator (△) or the universal role (U). While previous work on minimization relied on quotient constructions and was limited to languages that contain either △ or U, the authors present the first algorithm that works for the broader class ranging from the sub‑logic of f‑ALC without union and universal restriction up to f‑ALC_reg extended with inverse roles and nominals.

The core idea is to avoid the traditional quotient approach. Instead, the algorithm first computes the greatest fuzzy Φ‑auto‑bisimulation of the given interpretation I, where Φ specifies which constructors (inverse roles, nominals) are available. From this bisimulation a compact fuzzy partition is derived: the partition recursively groups domain elements whose fuzzy equivalence degree exceeds the infimum d of the bisimulation relation. Each block of the partition is either a crisp block (degree = 1) or a fuzzy block (degree = d < 1) that is further subdivided.

Once the partition is built, the algorithm selects a representative element from each block to form a reduced domain Δ′. The selection respects an approximation parameter γ ∈ (0,1]: for any concept assertion C(a) in the original interpretation, the reduced interpretation I′ must satisfy C_I′(rep(a)) ≥ γ·C_I(a) (or ≤ γ·C_I(a) for lower bounds). This γ‑approximation allows the method to achieve substantial compression even when exact preservation would be impossible.

Complexity analysis shows that constructing the partition costs O(m log l) time, where m is the number of non‑zero atomic role instances and l is the number of distinct fuzzy values used (plus two). Selecting representatives and building the final interpretation costs O(n log n), with n = |Δ|. Hence the total running time is O((m log l + n) log n) and memory usage is linear in n + m.

The authors prove two main theorems: (1) Correctness – the reduced interpretation preserves all fuzzy concept assertions up to the factor γ; (2) Minimality – no interpretation with a strictly smaller domain can satisfy the same γ‑preservation condition. Thus the algorithm yields a truly minimal reduction under the given approximation tolerance.

Experimental evaluation on synthetic benchmarks and a real‑world fuzzy social network (researchers, topics, and weighted collaborations) demonstrates the practical impact. With γ = 1 (exact preservation) the algorithm still reduces domain size by about 35 % on average; with γ = 0.8 the reduction exceeds 50 %; and with γ = 0.6 it reaches over 60 %. Runtime remains modest (seconds for domains of size 10⁴), outperforming the quotient‑based methods of prior work by a factor of 3–5.

The paper situates its contribution within related work, highlighting that unlike