Alternating Set Quantifiers in Modal Logic

TL;DR

This paper proves the strictness of set quantifier alternation hierarchies in modal logic over finite graphs, extending known results from monadic second-order logic and resolving longstanding open questions in the field.

Contribution

It establishes the strictness of modal logic hierarchies based on set quantifier alternations, answering a question posed over four decades ago.

Findings

Hierarchy strictness is confirmed for various classes of finite graphs.

Extends the known hierarchy results from monadic second-order logic to modal logic.

Resolves a long-standing open problem in modal logic hierarchy structure.

Abstract

We establish the strictness of several set quantifier alternation hierarchies that are based on modal logic, evaluated on various classes of finite graphs. This extends to the modal setting a celebrated result of Matz, Schweikardt and Thomas (2002), which states that the analogous hierarchy of monadic second-order logic is strict. Thereby, the present paper settles a question raised by van Benthem (1983), revived by ten Cate (2006), and partially answered by Kuusisto (2008, 2015).