Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of tQCTL as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that tQCTL restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When tQCTL restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExpPol-complete; AExpPol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of tQCTL restricted to EF or to EXEF and of the well-known modal logics such as K, KD, GL, K4 and S4 with propositional quantification under a semantics based on classes of trees.

翻译：在模型逻辑K、T或S4中添加参数量化已知会导致不可降解性,但在树文语义(tQCTL)下以标定性量化的CTL承认了一个非元素性塔完全可测量性问题。我们调查了tQCTL严格碎片的复杂性,以及在树文语义下以标定性量化的模型逻辑K的复杂性。更具体地说,我们证明,TQCTL限于时间操作器EX已经是塔硬的,这是意料之外的,因为EX只能强制执行本地特性。当限制为EX的tQCTL在N ⁇ 2 的树上被解释为N ⁇ 2 时,我们证明可覆盖的 tQCTL 是AExpPol-完整; ExpPol-硬性是通过减少最近引入的标注问题而确定的,这有助于研究模型核对时间逻辑的间隔问题。我们证明, tQCTL 的塔硬性是塔硬性,因为EXEF 或ExEF 以及众所周知的模型逻辑以K4树S4 和Smanticle为基础,以K4树的S4 和SDrmatical 的等级为基础。