Modal Logic over Finite Structures, by Eric Rosen
In this paper, we develop various aspects of the finite model theory of
propositional modal logic. In particular, we show that certain results
about the expressive power of modal logic over the class of all structures,
due to van Benthem and his collaborators, remain true over the class of
finite structures. We establish that a first-order definable class of
finite models is closed under bisimulations iff it is definable by a
`modal first-order sentence'. We show that a class of finite models that
is defined by a modal sentence is closed under extensions iff it is
defined by a diamond-modal sentence.
In sharp contrast, it is well known that many classical
results for first-order logic, including various preservation theorems,
fail for the class of finite models.