Decidability of higher-order subtyping with intersection types

Abstract

The combination of higher-order subtyping with intersection types yields a typed model of object-oriented programming with multiple inheritance [11]. The target calculus, F ^ω_⋀ , a natural generalization of Girard's system F ^ω with intersection types and bounded polymorphism, is of independent interest, and is our subject of study.

Our main contribution is the proof that subtyping in F ^ω_⋀ is decidable. This yields as a corollary the decidability of subtyping in F ^ω_≤ , its intersection free fragment, because the F ^ω_⋀ subtyping system is a conservative extension of that of F ^ω_≤ .

The calculus presented in [8] has no reductions on types. In the F ^ω_⋀ subtyping system the presence of β∧-conversion — an extension of β-conversion with distributivity laws — drastically increases the complexity of proving the decidability of the subtyping relation. Our proof consists of, firstly, defining an algorithmic presentation of the subtyping system of F ^ω_⋀ , secondly, proving that this new presentation is sound and complete with respect to the original one, and finally, proving that the algorithm always terminates.

Moreover, we establish basic structural properties of the language of types of F ^ω_⋀ such as strong normalization and Church-Rosser.

Among the novel aspects of the present solution is the use of term rewriting techniques to present intersection types, which clearly splits the computational semantics (reduction rules) from the syntax (inference rules) of the system. Another original feature is the use of a choice operator to model the behavior of variables during subtype checking.

This research was supported by the Dutch organization for scientific research, NWO-SION project Typed lambda calculus, 612-316-030, and by EPSRC GRANT, GR/G 55792 Constructive logic as a basis for program development.