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：蕴含基（IBs）是有限闭包系统与格（连同交不可约元素）的一种常见表示形式。它们广泛应用于从逻辑学、数据库到知识空间理论等众多领域。不同的蕴含基可以表示同一个闭包系统，因此研究者提出了多种蕴含基，例如典范基与典范直基。本文研究了 $D$-基——典范直基的一种精炼形式。它与自由格研究中的核心工具 $D$-关系相关联。$D$-基展现出良好的算法性质，且与 $D$-关系共同传递出底层闭包系统的本质特性。因此，从其他表示形式计算闭包系统的 $D$-基与 $D$-关系对于利用其优势至关重要。然而，该任务的计算复杂度研究目前尚存空白。本文给出了计算 $D$-基与 $D$-关系的算法与复杂性结果。具体而言：我们证明了从任意蕴含基中寻找 $D$-关系是 $NP$-完全的；给出了一个从交不可约元素计算 $D$-基的输出拟多项式时间算法；并提出了一个从任意蕴含基计算 $D$-基的多项式延迟算法。这些结果完善了闭包系统 $D$-基与 $D$-关系识别复杂性的理论图景。