Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?

TL;DR

This paper explores the computational complexity of propositional quantification in modal and temporal logics on trees, revealing unexpected hardness results even for restricted fragments like EX, and establishing new complexity bounds.

Contribution

It demonstrates that even simple fragments of tQCTL, such as EX, are Tower-hard, and provides precise complexity classifications for various logic fragments on trees.

Findings

tQCTL with EX operator is Tower-hard.

Satisfiability for tQCTL restricted to EX on N-bounded trees is AExpPol-complete.

Several modal logics with propositional quantification are Tower-hard on classes of trees.

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 (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.