Specifying Type Systems with Multi-Level Order-Sorted Algebra

We propose to use order-sorted algebras (OSA) on multiple levels to describe languages together with their type systems. It is demonstrated that even advanced aspects can be modeled, including, parametric polymorphism, complex relationships between diierent sorts of an operation's rank, the speciication of a variable number of parameters for operations, and type constructors using values (and not only types) as arguments. The basic idea is to use a signature to describe a type system where sorts denote sets of type names and operations denote type constructors. The values of an algebra for such a signature are then used as sorts of another signature now describing a language having the previously deened type system. This way of modeling is not restricted to two levels, and we will show useful applications of three-level algebras.