Semantic subtyping

Abstract

Usually subtyping relations are defined either syntactically by a formal system or semantically by an interpretation of types in an untyped denotational model. In this work we show how to define a subtyping relation semantically, for a language whose operational semantics is driven by types; we consider a rich type algebra, with product, arrow, recursive, intersection, union and complement types. Our approach is to "bootstrap" the subtyping relation through a notion of set-theoretic model of the type algebra.The advantages of the semantic approach are manifold. Foremost we get "for free" many properties (e.g., the transitivity of subtyping) that, with axiomatized subtyping, would require tedious and error prone proofs. Equally important is that the semantic approach allows one to derive complete algorithms for the subtyping relation or the propagation of types through patterns. As the subtyping relation has a natural (inasmuch as semantic) interpretation, the type system can give informative error messages when static type-checking fails. Last but not least the approach has an immediate impact in the definition and the implementation of languages manipulating XML documents, as this was our original motivation.

Semantic subtyping

“…The aim of Frisch et al [2002] and Frisch [2004] was to provide a theoretical foundation on the top of which to build the language Duce [Benzaken et al 2003], an XML-oriented transformation language. The key theoretical contribution of the work is a new approach to define semantic subtyping when straightforward set-theoretic interpretation does not work, in particular for arrow types.…”

Subtyping Supports Safe Session Substitution

“…3.3 Channel-oriented subtyping by semantic subtyping Castagna et al (2009) use the idea of semantic subtyping (Frisch et al, 2002) to develop a set-theoretic model of session types in which subtyping is simply set inclusion. The foundation for their model is a duality relation on session types, which characterises safe communication.…”

A Sound Type System for Typing Runtime Errors

“…Frisch et al [11], [12] introduced semantic subtyping. They presented a set-theoretic model of types independent from the semantics of expressions, and defined the subtyping relation by the set inclusion relation on this model.…”

“…* 3 These algorithms are complete w.r.t. semantic subtyping, but not for the axiomatized intersection types [11], [15]. check for dynamically typed languages.…”

“…To bring out the full benefit of E, our system has a certain expressive power including intersection type [7], [20] σ ∩ τ, settheoretic union type [3], [17] σ∪τ, complement type [11], [24] τ C , universal quantification [13], [14], [18] ∀α. τ, existential quantification * 1 ∃α.…”