Coinductive Type Systems for Object-Oriented Languages

Abstract

We propose a novel approach based on coinductive logic to specify type systems of programming languages. The approach consists in encoding programs in Horn formulas which are interpreted w.r.t. their coinductive Herbrand model. We illustrate the approach by first specifying a standard type system for a small object-oriented language similar to Featherweight Java. Then we define an idealized type system for a variant of the language where type annotations can be omitted. The type system involves infinite terms and proof trees not representable in a finite way, thus providing a theoretical limit to type inference of object-oriented programs, since only sound approximations of the system can be implemented. Approximation is naturally captured by the notions of subtyping and subsumption; indeed, rather than increasing the expressive power of the system, as it usually happens, here subtyping is needed for approximating infinite non regular types and proof trees with regular ones.

Semantic Subtyping for Objects and Classes

“…The restriction to well-formed types is required to avoid meaningless types; the same choice is used in [13]. A different restriction, called contractiveness, is used for instance in [4], where non-regular types are also allowed.…”

Union and intersection types to support both dynamic and static typing

“…Union types were already included in object-oriented languages, in type systems where they were explicitly declared [12] or inferred from implicitly typed references [3]. We have taken the subtyping rules defined by other authors such as [19] and [8] (S-SUnionL and S-SUnionR), adding the new dynamic typing rule S-DUnionL.…”

Coinductive big-step operational semantics for type soundness of Java-like languages

“…We recall that interpreting recursive definitions or rules coinductively is equivalent to considering also infinite terms and proof trees [7,4,11,15]. The CBS judgment uses value environments (see below), just for uniformity with the type judgment defined in Section 4.…”

“…This work is mainly inspired on previous work by Ancona et al on abstract compilation for object-oriented languages [7,3,6,5,4], a novel approach which aims at reconciling type analysis and symbolic execution, where programs are compiled into a constraint logic program, and type analysis corresponds to solving a certain goal w.r.t. the coinductive semantics of CLP.…”