On sound and complete axiomatization of coinductive subtyping for object-oriented languages

Coinductive abstract compilation is a novel technique, which has been recently introduced for defining precise type systems for objectoriented languages. In this approach, type inference consists in translating the program to be analyzed into a Horn formula f , and in resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of f . Type systems defined in this way are idealized, since types and, consequently, goal derivations, are not finitely representable. Hence, sound implementable approximations have to rely on the notions of regular types and derivations, and of subtyping and subsumption between types and atoms, respectively. In this paper we address the problem of defining a sound and complete axiomatization of a subtyping relation between coinductive object and union types, defined as set inclusion between type interpretations. Besides being an important theoretical result, completeness is useful for reasoning about possible implementations of the subtyping relation, when restricted to regular types.