Direct-optimal systems of attribute implications in fuzzy formal concept analysis

Efficient closure computation is a key challenge in fuzzy Formal Concept Analysis. Direct implicational systems, which allow for one-step closure calculation, offer a powerful solution, but their formalization and practical computation in the fuzzy setting have remained largely open problems. This paper provides a comprehensive contribution to fill this gap. We first extend the concept of a direct system to fuzzy attribute implications and establish a complete theoretical characterization via a novel Fuzzy Exchange Condition. Building on this, we define the notion of a direct-optimal system and develop the DirectOptimal algorithm, a provably correct, interleaved strategy for its computation. A rigorous experimental evaluation demonstrates that our proposed algorithm is orders of magnitude more efficient than naive sequential approaches, and we diagnose the distinct computational bottlenecks that cause these simpler methods to fail. This work thus delivers a complete and empirically validated pipeline, from theory to practice, for the efficient computation of direct-optimal bases.