Typed self-representation

Abstract

Self-representation -the ability to represent programs in their own language -has important applications in reflective languages and many other domains of programming language design. Although approaches to designing typed program representations for sublanguages of some base language have become quite popular recently, the question whether a fully metacircular typed selfrepresentation is possible is still open. This paper makes a big step towards this aim by defining the F * ω calculus, an extension of the higher-order polymorphic lambda calculus Fω that allows typed self-representations. While the usability of these representations for metaprogramming is still limited, we believe that our approach makes a significant step towards a new generation of reflective languages that are both safe and efficient.

Self-Representation in Girard's System U

“…The idea of a selfrecognizer was first studied by Kleene [18] in 1936 for an untyped λ-calculus. There are several examples of typed recognizers and self-recognizers [26,27] that are implemented by iteration. Iteration is desirable because it can be supported by languages that don't include recursion.…”

“…They did not study representation of F + ω . In 2009 Rendel, Ostermann and Hofer [27] studied the representation of kind polymorphism, and conjectured that it would require "another, higher form of polymorphism". Their solution was to combine the categories of types and kinds, so that kind polymorphism is represented in the same way as type polymorphism.…”

“…They implemented a self-recognizer and the first typed self-enactor, a self-interpreter that implements multi-step reduction on typed representations. Like [27], type checking is undecidable in their calculus. Their representation technique was designed for combinators, and does not appear to be easily translated to a λ-calculus.…”

“…We represent types using higher order abstract syntax (HOAS), inspired by [27] and [30]. Type representations are types of kind U, which is defined inductively from four constructors.…”

Towards typing for small-step direct reflection

“…Closer still to achieving part of what we want is the work of Rendel, Ostermann and Hofer [10], who define a typed selfrepresentation of the (pure) λ-calculus. To achieve this, they first leverage a technique from [2] whereby they abstract over a type constructor, and then repeat this at the type level (to introduce kind-polymorphism).…”