Modal and guarded characterisation theorems over finite transition systems

Characterisation theorems for modal and guarded fragments of first-order logic are explored over finite transition systems. We show that the classical characterisations in terms of semantic invariance under the appropriate forms of bisimulation equivalence can be recovered at the level of finite model theory. The new, more constructive proofs naturally extend to alternative proofs of the classical variants. The finite model theory version of van Benthem's characterisation of basic modal logic is due to E. Rosen. That proof is simplified and the result slightly strengthened in terms of quantitative bounds. The main theme, however is a uniform treatment that extends to incorporate universal and inverse modalities and guarded quantification over transition systems. Technically, the present treatment exploits first-order locality in the context of a new finitary construction of locally acyclic bisimilar covers. These serve as graded finite analogues of tree unravellings, giving local control over first-order logic infinite bisimilar companion structures.