Characterization and definability in modal first-order fragments

Characterization results for modal logics below first order identifies them as a fragment of first order logic. These results are very important to analyze the expressive power of a given logic. The first work in this direction was done by van Benthem [12] who used bisimulations to characterize the basic modal logic as the bisimulation invariant fragment of first order logic. There exists a huge number of modal logics, to enumerate some of them: temporal logic [7], propositional dynamic logic [8], description logic [3], sub-boolean modal logics [11], etc. Each of these logics was specially designed to fit a particular purpose, that is one of the jobs modal logics do best. There is no single definition of bisimulation, each of the aforementioned logics has its own definition to match its particular semantics. It is interesting to analyze to which fragment of first order logic each of those logics correspond. Analogue results to van Benthem’s characterization hold for many other logics besides basic modal logic but, as the basic elements have changed (the bisimulation definition, the logic, etc.), there comes the need to re-prove the result. It is well known that the proofs of those results for several modal logics have, somehow, the same ‘taste’. Definability results identify the properties that a class of models should satisfy in order to have a formula or a set of formulas whose models are exactly the class we are trying to define. This question had previously been stated and answered for classical first order logic [6], basic modal logic [4] and many others [11, 10]. As with characterization results, each logic has its own different proof although they share most of the key ideas.