Coinductive subtyping for abstract compilation of object-oriented languages into Horn formulas

TL;DR

This paper develops a formal subtyping relation for coinductive types used in abstract compilation of object-oriented languages into Horn formulas, ensuring soundness and handling empty types.

Contribution

It introduces a subtyping relation for coinductive terms with union and object types, and proves its soundness, improving type approximation in abstract compilation.

Findings

Subtyping relation is sound with respect to type interpretation.

Simplified the notion of contractive derivation.

Identified and addressed gaps in previous empty type representations.

Abstract

In recent work we have shown how it is possible to define very precise type systems for object-oriented languages by abstractly compiling a program into a Horn formula f. Then type inference amounts to 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 in the most interesting instantiations both the terms of the coinductive Herbrand universe and goal derivations cannot be finitely represented. However, sound and quite expressive approximations can be implemented by considering only regular terms and derivations. In doing so, it is essential to introduce a proper subtyping relation formalizing the notion of approximation between types. In this paper we study a subtyping relation on coinductive terms built on union and object type constructors. We define an interpretation of types as set of values induced by a quite intuitive relation of membership of values to types, and prove that the definition of subtyping is sound w.r.t. subset inclusion between type interpretations. The proof of soundness has allowed us to simplify the notion of contractive derivation and to discover that the previously given definition of subtyping did not cover all possible representations of the empty type.