We introduce the completeness problem for Modal Logic (possibly with fixpoint operators) and examine its complexity. A formula is called complete, if any two satisfying processes are bisimilar. The completeness problem is closely connected to the characterization problem, which asks whether a given formula characterizes a given process up to bisimulation equivalence. We discover that completeness, characterization, and validity have the same complexity — with some exceptions for which there are, in general, no complete formulae. To prove our upper bounds, we present a non-deterministic procedure with an oracle for validity that combines tableaux and a test for bisimilarity, and determines whether a formula is complete.

我们介绍了模态逻辑的完整性问题（可能有定点运算符），并研究了其复杂性。如果有两个令人满意的过程是双相似的，则公式称为完成。完整性问题与表征问题密切相关，后者要求一个给定的公式是否可以表征一个给定的过程，直到双仿真对等。我们发现完整性，特征和有效性具有相同的复杂性-除了某些例外，通常没有完整的公式。为了证明我们的上限，我们提出了一个非确定性的过程，其中使用了一个预言性的oracle，它结合了表格和双相似性检验，并确定公式是否完整。