Pure Subtype Systems, by DeLesley S. Hutchins:

This paper introduces a new approach to type theory called pure subtype systems. Pure subtype systems differ from traditional approaches to type theory (such as pure type systems) because the theory is based on subtyping, rather than typing. Proper types and typing are completely absent from the theory; the subtype relation is defined directly over objects. The traditional typing relation is shown to be a special case of subtyping, so the loss of types comes without any loss of generality.

Pure subtype systems provide a uniform framework which seamlessly integrates subtyping with dependent and singleton types. The framework was designed as a theoretical foundation for several problems of practical interest, including mixin modules, virtual classes, and feature-oriented programming.

The cost of using pure subtype systems is the complexity of the meta-theory. We formulate the subtype relation as an abstract reduction system, and show that the theory is sound if the underlying reductions commute. We are able to show that the reductions commute locally, but have thus far been unable to show that they commute globally. Although the proof is incomplete, it is "close enough" to rule out obvious counter-examples. We present it as an open problem in type theory.

A thought-provoking take on type theory using subtyping as the foundation for all relations. He collapses the type hierarchy and unifies types and terms via the subtyping relation. This also has the side-effect of combining type checking and partial evaluation. Functions can accept "types" and can also return "types".

Of course, it's not all sunshine and roses. As the abstract explains, the metatheory is quite complicated and soundness is still an open question. Not too surprising considering type checking Type:Type is undecidable.

Hutchins' thesis is also available for a more thorough treatment. This work is all in pursuit of Hitchens' goal of feature-oriented programming.

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