Decidable fragments of many-sorted logic

Many natural specifications use types. We investigate the decidability of fragments of many-sorted first-order logic. We identified some decidable fragments and illustrated their usefulness by formalizing specifications considered in the literature. Often the intended interpretations of specifications are finite. We prove that the formulas in these fragments are valid iff they are valid over the finite structures. We extend these results to logics that allow a restricted form of transitive closure. We tried to extend the classical classification of the quantifier prefixes into decidable/undecidable classes to the many-sorted logic. However, our results indicate that a naive extension fails and more subtle classification is needed.