Modal logics of almost sure validities in some classes of euclidean and transitive frames

TL;DR

This paper investigates the modal logics of formulas that become almost surely valid in large finite Kripke frames within certain classes, providing axiomatizations for these logics.

Contribution

It introduces the concept of almost sure validity in finite frames and offers complete axiomatizations for these modal logics across various frame classes.

Findings

Axiomatizations for almost sure validities in multiple frame classes

Identification of the limit behavior of validity in large frames

Formalization of almost sure validity as a modal logic concept

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.