Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

Abstract

Adding propositional quantification to the modal logics $\mathsf{K, T}$ or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (QCTLt) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of $\mathsf{QCTL}^{t}$ as well as of the modal logic $\mathsf{K}$ with propositional quantification under the tree semantics. More specifically, we show that $\mathsf{QCTL}^{t}$ restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When QCTLt restricted to EX is interpreted on N-bounded trees for some $N\geq 2$ , we prove that the satisfiability problem is $\mathbf{AExp}_{\mathrm{pol}^{-}}$ complete; $\mathbf{AExp}_{\mathrm{pol}}$ -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 QCTLtrestricted to EF or to EXEF and of the well-known modal logics $\mathsf{K}$ , KD, GL, S4, K4 and D4, with propositional quantification under a semantics based on classes of trees.