Characterization of Type Hierarchies with Open Specification

Type hierarchies which arise in applications are often described incompletely; this missing information may be handled in a variety of ways. In this work, such incomplete hierarchies are viewed as open specifications; that is, descriptions which are sets of constraints. The actual hierarchy is then any structure satisfying these constraints. For such specifications, two forms of characterization are provided. The first is algebraic and utilizes a generalization of weak partial lattices; it provides a structure-based characterization in which optimality is characterized via an initial construction. The second is logical, an inference-based representation, in which models are characterized as products of models of propositional-based specifications.