A Function-Set Framework: General Properties and Applications to Modal Logic
Luke Bayzid, Alexandre Madeira, Manuel A. Martins
TL;DR
The paper addresses the challenge of comparing heterogeneous representations across domains by proposing a unified, set-theoretic, time-indexed framework built from a state set $S$, an entity set $E$, and a totally ordered time $T$, with contexts $Ω ⊆ S^{E×T}$. It demonstrates that modal logics can be represented within this framework via modal contexts that define operators $oxempty$ and $ abla$, and shows a constructive correspondence between Kripke frames and modal contexts, including the use of equivalence classes and derived relations to preserve standard semantics. Key contributions include formalizing determinability (and its countable-time iterator equivalence), introducing the notion of a modal context, and providing explicit mappings that realize modal logic as a specialization of the framework. The approach offers a unifying foundation for comparing representations and sets the stage for extensions to epistemic, temporal, and dynamic logics, with avenues for investigating decidability, bisimulations, and finite-model properties.
Abstract
Representations are essential to mathematically model phenomena, but there are many options available. While each of those options provides useful properties with which to solve problems related to the phenomena in study, comparing results between these representations can be non-trivial, as different frameworks are used for different contexts. We present a general structure based on set-theoretic concepts that accommodates many situations related to logical and semantic frameworks. We show the versatility of this approach by presenting alternative constructions of modal logic; in particular, all modal logics can be represented within the framework.