Diamonds and Dominoes: Impossibility Results for Associative Modal Logics

TL;DR

This paper proves that any associative modal logic valid over a specific frame involving Boolean algebras with an operator is undecidable, resolving a long-standing open problem in the field.

Contribution

It establishes the undecidability of a broad class of associative modal logics over certain frames, settling a major open question in the area.

Findings

Undecidability of the equational theory for Boolean algebras with associative operators.

Resolution of the open problem on the decidability of hyperboolean modal logic.

Implications for the limits of algorithmic reasoning in associative modal systems.

Abstract

We show that for any class of Boolean algebras with an associative operator, if it contains the complex algebra of (P(N), U), its equational theory is undecidable. Equivalently, any associative normal modal logic valid over the frame (P(N), U) is undecidable. This settles a long-open question on the decidability of hyperboolean modal logic (Goranko and Vakarelov, 1999), and addresses several related problems.