Modal Logics and Local Quantifiers: A Zoo in the Elementary Hierarchy

Abstract

AbstractWe study a family of modal logics interpreted on tree-like structures, and featuring local quantifiers $$\exists ^{k}p$$ ∃ k p that bind the proposition p to worlds that are accessible from the current one in at most k steps. We consider a first-order and a second-order semantics for the quantifiers, which enables us to relate several well-known formalisms, such as hybrid logics, $$\textsf {S5Q}$$ S 5 Q and graded modal logic. To better stress these connections, we explore fragments of our logics, called herein round-bounded fragments. Depending on whether first or second-order semantics is considered, these fragments populate the hierarchy $${2\textsc {NExp} \subset 3\textsc {NExp} \subset \cdots }$$ 2 NE X P ⊂ 3 NE X P ⊂ ⋯ or the hierarchy $${2\textsc {AExp}_{pol} \subset 3\textsc {AExp}_{pol} \subset \cdots }$$ 2 AE X P pol ⊂ 3 AE X P pol ⊂ ⋯ , respectively. For formulae up-to modal depth k, the complexity improves by one exponential.