A universe of binding and computation

Abstract

We construct a logical framework supporting datatypes that mix binding and computation, implemented as a universe in the dependently typed programming language Agda 2. We represent binding pronominally, using well-scoped de Bruijn indices, so that types can be used to reason about the scoping of variables. We equip our universe with datatype-generic implementations of weakening, substitution, exchange, contraction, and subordination-based strengthening, so that programmers need not reimplement these operations for each individual language they define. In our mixed, pronominal setting, weakening and substitution hold only under some conditions on types, but we show that these conditions can be discharged automatically in many cases. Finally, we program a variety of standard difficult test cases from the literature, such as normalization-by-evaluation for the untyped λ-calculus, demonstrating that we can express detailed invariants about variable usage in a program's type while still writing clean and clear code.

GMeta: A Generic Formal Metatheory Framework for First-Order Representations

“…Chlipala (2007) used an approach inspired by proof by reflection techniques (Boutin 1997) to provide several generic operations on well-typed terms represented by well-scoped de Bruijn indices. Licata and Harper (2009) proposed a universe in Agda that permits definitions that mix binding and computation. The obvious difference is that GMeta works with traditional (non-well-scoped) first-order representations instead of well-scoped de Bruijn indices.…”

“…This difference of representation means that the universes and generic functions have to deal with significantly different issues and that they are quite different in nature. More fundamentally, Chlipala's (2007) and Licata and Harper's (2009) work can be viewed as trying to develop new ways to formalize metatheory in which many of the invariants hold by construction, that would have to be proved otherwise. This is different from our goal: we are not proposing new ways to formalize metatheory, rather we wish to make well-established ways to formalize metatheory with first-order representations less painful to use.…”

A unified treatment of syntax with binders

“…This algorithm has been previously used as a benchmark by several researchers Pitts, 2006;Licata & Harper, 2009;Cave & Pientka, 2012). The challenge lies in the way in which the algorithm mixes computational functions, name abstractions, and fresh name generation.…”

“…If world polymorphism is appropriately exploited, the system can guarantee not only that every index is within range, but also that free variables are treated abstractly (section 11.2), which implies that certain programming errors, such as "forgetting to shift" a free variable, can be detected by the type system. Licata and Harper (2009) aim to provide substitution for free when possible, whereas we do not explicitly explore this aspect. They impose the use of well-scoped de Bruijn indices to the programmer, whereas our "names" are more flexible and can be used as atoms or as de Bruijn indices.…”

Generic conversions of abstract syntax representations

“…Datatype-generic treatment of abstract syntax has already been addressed in the literature [4,7,8,12] using different approaches to abstract syntax with different trade-offs in terms of elegance, usability and expressivity.…”