Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

Abstract : Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecid-ability 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 AExp pol-complete; AExp 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 QCTL t restricted 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.
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Contributor : Stéphane Demri <>
Submitted on : Wednesday, November 6, 2019 - 6:28:26 AM
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Bartosz Bednarczyk, Stéphane Demri. Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?. 34th Annual ACM/IEEE Symposium on Logic In Computer Science (LICS'19), Jun 2019, Vancouver, France. ⟨hal-02350328⟩



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