Implicational bases (IBs) are a common representation of finite closure systems and lattices, along with meet-irreducible elements. They appear in a wide variety of fields ranging from logic and databases to Knowledge Space Theory. Different IBs can represent the same closure system. Therefore, several IBs have been studied, such as the canonical and canonical direct bases. In this paper, we investigate the $D$-base, a refinement of the canonical direct base. It is connected with the $D$-relation, an essential tool in the study of free lattices. The $D$-base demonstrates desirable algorithmic properties, and together with the $D$-relation, it conveys essential properties of the underlying closure system. Hence, computing the $D$-base and the $D$-relation of a closure system from another representation is crucial to enjoy its benefits. However, complexity results for this task are lacking. In this paper, we give algorithms and hardness results for the computation of the $D$-base and $D$-relation. Specifically, we establish the $NP$-completeness of finding the $D$-relation from an arbitrary IB; we give an output-quasi-polynomial time algorithm to compute the $D$-base from meet-irreducible elements; and we obtain a polynomial-delay algorithm computing the $D$-base from an arbitrary IB. These results complete the picture regarding the complexity of identifying the $D$-base and $D$-relation of a closure system.

翻译：蕴涵基（IBs）是有限闭包系统与格结构（连同交既约元）的常见表示形式，广泛应用于逻辑学、数据库以及知识空间理论等多个领域。不同的蕴涵基可表示同一闭包系统，因此学界对多种蕴涵基（如典范基与典范直接基）开展了研究。本文聚焦$D$-基——典范直接基的精化版本，该基与自由格研究中的核心工具$D$-关系密切关联。$D$-基展现出优良的算法特性，且与$D$-关系共同传递了底层闭包系统的本质属性。因此，从其他表示形式计算闭包系统的$D$-基与$D$-关系对于发挥其优势至关重要，但当前尚缺乏相关计算复杂性的研究成果。本文提出了$D$-基与$D$-关系的计算算法及复杂性结论：首先证明从任意IB推导$D$-关系具有NP完全性；其次给出基于交既约元计算$D$-基的输出拟多项式时间算法；最后提出从任意IB计算$D$-基的多项式延迟算法。这些结果完整刻画了识别闭包系统$D$-基与$D$-关系的计算复杂性。