Modal logics of almost sure validities in some classes of euclidean and transitive frames
Vladislav Sliusarev
TL;DR
The paper studies almost-sure validity of modal formulas on random finite Kripke frames drawn from a frame class \mathcal{C}. It defines the almost-sure logic Log^{\mathbf{as}}(\mathcal{C}) and shows it forms a normal modal logic extending Log(\mathcal{C}); a key reduction shows a.a.s. validity in random frames is governed by connected frames (Con Fr). Using this framework, the authors obtain finite axiomatizations for the almost-sure logics of KD5, KD45, K5B, S5, Grz.3, and GL.3, and prove that for these classes the a.s. logics coincide with the base logics (e.g., KD5^{as} = KD5, GL.3^{as} = GL.3). They further analyze Euclidean and transitive frames, including random inverse trees, to substantiate these results and discuss implications for zero-one laws and broader classifications. Overall, the work demonstrates that several natural modal logics retain their expressive strength under asymptotic random-frame semantics and provides a methodological path toward finite axiomatizations for almost-sure validities in finite frame classes.
Abstract
Given a class C of finite Kripke frames, we consider the uniform distribution on the frames from C with n states. A formula is almost surely valid in C if the probability that it is valid in a random C-frame with n states tends to 1 as n tends to infinity. The formulas that are almost surely valid in C form a normal modal logic. We find complete and sound axiomatizations for the logics of almost sure validities in the classes of finite frames defined by the logics K5, KD5, K45, KD45, K5B, S5, Grz.3, and GL.3.