Abstract. We have built the first family of tagless interpretations for a higher-order typed object language in a typed metalanguage (Haskell or ML) that require no dependent types, generalized algebraic data types, or postprocessing to eliminate tags. The statically type-preserving interpretations include an evaluator, a compiler (or staged evaluator), a partial evaluator, and call-by-name and call-by-value CPS transformers. Our main idea is to encode HOAS using cogen functions rather than data constructors. In other words, we represent object terms not in an initial algebra but using the coalgebraic structure of the λ-calculus. Our representation also simulates inductive maps from types to types, which are required for typed partial evaluation and CPS transformations. Our encoding of an object term abstracts over the various ways to interpret it, yet statically assures that the interpreters never get stuck. To achieve self-interpretation and show Jones-optimality, we relate this exemplar of higher-rank and higher-kind polymorphism to plugging a term into a context of let-polymorphic bindings. It should also be possible to define languages with a highly refined syntactic type structure. Ideally, such a treatment should be metacircular, in the sense that the type structure used in the defined language should be adequate for the defining language. John Reynolds [28] 1

We show that, in a parametric model of polymorphism, the type ∀α.((α → α) → α) → (α → α → α) → α is isomorphic to closed de Bruijn terms. That is, the type of closed higher-order abstract syntax terms is isomorphic to a concrete representation. To demonstrate the proof we have constructed a model of parametric polymorphism inside the Coq proof assistant. The proof of the theorem requires parametricity over Kripke relations. We also investigate some variants of this representation.

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-interpreters can be roughly divided into two sorts: self-recognisers that recover the input program from a canonical representation, and self-enactors that execute the input program. Major progress for statically-typed languages was achieved in 2009 by Rendel, Ostermann, and Hofer who presented the first typed selfrecogniser that allows representations of different terms to have different types. A key feature of their type system is a type:type rule that renders the kind system of their language inconsistent. In this paper we present the first statically-typed language that not only allows representations of different terms to have different types, and supports a self-recogniser, but also supports a selfenactor. Our language is a factorisation calculus in the style of Jay and Given-Wilson, a combinatory calculus with a factorisation operator that is powerful enough to support the pattern-matching functions necessary for a self-interpreter. This allows us to avoid a type:type rule. Indeed, the types of System F are sufficient. We have implemented our approach and our experiments support the theory.