Modal operators and toric ideals

TL;DR

This paper explores the algebraic structure of modal propositional logic, translating modal constraints into polynomial ideals and establishing conditions under which toric ideals fully characterize these constraints.

Contribution

It introduces a novel algebraic framework linking modal logic constraints to toric ideals, expanding the application of algebraic statistics to modal logic.

Findings

Constraints on modal propositions are represented by binomial ideals.

Toric ideals can fully describe modal constraints under certain conditions.

The approach bridges modal logic and algebraic geometry, enabling new analytical tools.

Abstract

In the present paper we consider modal propositional logic and look for the constraints that are imposed to the propositions of the special type $\Box a$ by the structure of the relevant finite Kripke frame. We translate the usual language of modal propositional logic in terms of notions of commutative algebra, namely polynomial rings, ideals, and bases of ideals. We use extensively the perspective obtained in previous works in Algebraic Statistics. We prove that the constraints on $\Box a$ can be derived through a binomial ideal containing a toric ideal and we give sufficient conditions under which the toric ideal fully describes the constraints.