The paper explores the correspondence between balanced incomplete block designs (BIBD) and certain linear CNF formulas by identifying the points of a block design with the clauses of the Boolean formula and blocks with Boolean variables. Parallel classes in BIBDs correspond to XSAT solutions in the corresponding formula. This correspondence allows for transfers of results from one field to the other. As a new result we deduce from known satisfiability theorems that the problem of finding a parallel class in a partially balanced incomplete block design is NP-complete.

中文翻译：

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平衡不完全块设计和精确可满足性

本文通过使用布尔公式的子句和带有布尔变量的块来识别块设计的点，探讨了平衡不完全块设计 (BIBD) 和某些线性 CNF 公式之间的对应关系。BIBD 中的并行类对应于相应公式中的 XSAT 解决方案。这种对应允许将结果从一个字段传输到另一个字段。作为一个新结果，我们从已知的可满足性定理推导出在部分平衡的不完全块设计中找到并行类的问题是 NP 完全的。