Modal Logic over Finite Structures
Eric Rosen

Abstract:
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.