We analyze the computational complexity of admissibility and unifiability with parameters in transitive modal logics. The class of cluster-extensible (clx) logics was introduced in the first part of this series of papers [8]. We completely classify the complexity of unifiability or inadmissibility in any clx logic as being complete for one of ${\mathrm{\Sigma }}_2^{\exp }$, NEXP, coNEXP, PSPACE, or ${\mathrm{\Pi }}_2^p$. In addition to the main case where arbitrary parameters are allowed, we consider restricted problems with the number of parameters bounded by a constant, and the parameter-free case.

Our upper bounds are specific to clx logics, but we also include similar results for logics of bounded depth and width. In contrast, our lower bounds are very general: they apply each to a class of all transitive logics whose frames allow occurrence of certain finite subframes.

We also discuss the baseline problem of complexity of derivability: it is coNP-complete or PSPACE-complete for each clx logic. In particular, we prove PSPACE-hardness of derivability for a broad class of transitive logics that includes all logics with the disjunction property.

我们分析了传递模态逻辑中参数的可容许性和统一性的计算复杂性。本系列论文的第一部分介绍了类可扩展（clx）逻辑[8]。我们将任何clx逻辑中不可统一性或不可接纳性的复杂性完全归类为对于以下其中一项是完整的${\mathrm{\Sigma }}_2^{\mathrm{经验值}}$，NEXP，coNEXP，PSPACE或 ${\mathrm{\Pi }}_2^p$。除了允许使用任意参数的主要情况之外，我们还考虑了以常量为边界的参数数量和无参数情况的受限问题。

我们的上限是特定于clx逻辑的，但是对于有界深度和宽度的逻辑，我们也包括类似的结果。相反，我们的下限非常笼统：它们适用于所有传递逻辑的一类，这些逻辑的框架允许出现某些有限子帧。

我们还讨论了可导性复杂度的基线问题：每个clx逻辑都是coNP完全或PSPACE完全。特别是，我们证明了广泛的传递逻辑包括所有具有析取属性的逻辑的可导性的PSPACE硬度。