When do modal definability and preservation theorems transfer to the finite?

We study which classic modal definability and preservation results survive when attention is restricted to finite structures, where many first-order transfer theorems are known to break down. Several semantic characterizations for modal formula classes survive the passage to the finite, while a number of first-order preservation theorems for basic frame operations fail. Our main positive result is that the Bisimulation Safety Theorem does transfer to finite structures. We also discuss computability aspects, and analogues in the finite for the Goldblatt-Thomason theorem and for modal correspondence theory.

💡 Research Summary

The paper investigates the extent to which classic results concerning modal definability and preservation theorems survive when the underlying structures are restricted to finite Kripke models. The authors begin by recalling that many fundamental metatheorems of first‑order logic—compactness, the Löwenheim‑Skolem theorem, the Los‑Tarski preservation theorem, Craig interpolation—break down in the finite. Nevertheless, modal logic enjoys a “finite model property” (every formula that is satisfiable is satisfiable in a finite model) and a bisimulation‑invariant fragment characterization, both of which suggest that some modal results may transfer to the finite setting.

The first technical part (Section 2) revisits four well‑known preservation theorems for modal formulas interpreted over all Kripke frames and shows that each of them lifts to the finite by exploiting the finite model property.

1. Monotonicity: A formula ϕ is monotone in a propositional variable p iff it is equivalent to a p‑positive formula. The authors restate this as the validity of ϕ → ϕ