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 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 AExp polcomplete; 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.

An Auxiliary Logic on Trees: on the Tower-Hardness of Logics Featuring Reachability and Submodel Reasoning

“…For instance, T ( ) denotes the set of formulae restricted to the operator , which can be nested at most times. This fragment of T is shown to be -NEXPTIME-hard in [5], which directly leads to the TOWER-hardness of T ( ) and T ( ). By analysing our translation it is easy to show that T ( 0 ), i.e.…”

“…Here, we reprove the result for by first tackling the TOWER-hardness of the logic with the exists-until ( ), and then show that this operator can be defined using . Differently from [5] and thanks to the properties of ALT, our reduction does not imbricate until operators, showing that this extension of CTL remains TOWER-hard even when ( ) is restricted so that and are Boolean combinations of propositional symbols.…”

“…We now consider Computation Tree Logic (CTL), a well-known logic for branching time model checking [14,13]. Among its extensions, in [5,22,28] the addition of propo-sitional quantification is considered. The satisfiability problem of the resulting logic is undecidable on Kripke structures, and TOWER-complete on trees [28].…”

“…The satisfiability problem of the resulting logic is undecidable on Kripke structures, and TOWER-complete on trees [28]. In [5], the authors show that the problem is TOWER-hard even when considering just one operator among exists-next or exists-finally (the definitions are below). Here, we reprove the result for by first tackling the TOWER-hardness of the logic with the exists-until ( ), and then show that this operator can be defined using .…”

“…T . A good amount of research has been done independently on these logics [5,9,17,28], culminating with the TOWER-hardness of ( * ) with two quantified variables [17] and the TOWER-hardness of T with just one temporal operator between exists-finally and exists-next [5] (see Section 4 for the definitions). Connections between these two formalisms have not been explicitly developed so far, perhaps because of the quite different logics:…”