An effectful way to eliminate addiction to dependence

Abstract

We define a monadic translation of type theory, called weaning translation, that allows for a large range of effects in dependent type theory-such as exceptions, non-termination, non-determinism or writing operation. Through the light of a call-by-push-value decomposition, we explain why the traditional approach fails with type dependency and justify our approach. Crucially, the construction requires that the universe of algebras of the monad forms itself an algebra. The weaning translation applies to a version of the Calculus of Inductive Constructions (CIC) with a restricted version of dependent elimination. Finally, we show how to recover a translation of full CIC by mixing parametricity techniques with the weaning translation. This provides the first effectful version of CIC.Proposition 10. If I is a first-order type, then there is a term eval I :By Proposition 8, full dependent elimination is interpreted, so that by putting all these properties together we recover this interesting extraction result by taking E • := I.Theorem 2. If I is a first-order type and there is a proof of ¬¬I in CIC, then there is a proof of I in CIC. C. Non-determinismWe sketch a model of BTT with non-determinism here.Definition 11. The non-empty list monad is defined as T A := A × list A together with El (X, (nil _)) := X El (X, [X 1 ; . . . ; X n ]) := (X.π 1 × X 1 .π 1 . . . × X n .π 1 , µ)

Gradual type theory

“…Additionally, the combination of dependent typing with gradual typing is worth exploring, especially because while dependent contract checking been used for some time and was explicitly inspired in part by dependent typing, no semantic connection has been established between dependent contracts and category-theoretic models of dependent typing. The main difficulty will be the combination of dependent typing with effects, but there has been much recent work in this area [1,41,27].…”

A Classical Sequent Calculus with Dependent Types

“…[1], by Vákár [35,36] or by Pédrot and Tabareau who proposed a systematical way to add effects to type theory [33]. Side effects-that are impure computations in functional programming-are often interpreted by means of monads.…”

“…Since the same problem arises with a wider class of effects, it seems that we are facing the following dilemma: either we choose an effectful language (allowing us to write more programs) while accepting the lack of dependent types, or we choose a dependently typed language (allowing us to write finer specifications) and give up effects.Many works have tried to fill the gap between effectful programming languages and logic, by accommodating weaker forms of dependent types with computational effects (e.g., divergence, I/O, local references, exceptions). Amongst other works, we can cite the recent works by Ahman et al[1], by Vákár [35,36] or by Pédrot and Tabareau who proposed a systematical way to add effects to type theory [33]. Side effects-that are impure computations in functional programming-are often interpreted by means of monads.…”

Towards Certified Meta-Programming with Typed Template-Coq

“…Our long term goal is to leverage this technique to extend the logical and computational power of Coq using, for instance, the forcing translation [14] or the weaning translation [20].…”