Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

摘要

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 (QCTL^t) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of QCTL^t as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that QCTL^t restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When QCTL^t 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 QCTL^trestricted to EF or to EXEF and of the well-known modal logics K, KD, GL, S4, K4 and D4, with propositional quantification under a semantics based on classes of trees.