Unification in Order-Sorted Logic with Term Declarations

This paper provides two results concerning Order-Sorted Logic with Term Declarations. First, we show that linear term declarations can be transformed conservatively into function declarations, thus yielding elementary signatures. This provides a simple proof of the well known fact that unification in linear signatures is decidable. A similar transformation exists for semi-linear term declarations, resulting in shallow term declarations. Secondly, we provide an inference system transforming sort constraints over an arbitrary signature into almost solved form. The step from almost solved forms to solved forms requires a procedure to decide emptiness of sort intersections, which is not possible in general. This shows that it is the sort intersection problem that accounts for the undecidability of unification in signatures with term declarations