Lindström theorems for fragments of first-order logic

Lindström theorems characterize logics in terms of model-theoretic conditions such as Compactness and the Löwenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindström theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindström theorems for several fragments of first-order logic, including the k-variable fragments for k> 2, Tarski’s relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindström proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems. Characterizing the 2-variable fragment or the full guarded fragment remain open problems.