Degrees of Relatedness

Abstract

Dependent type theory allows us to write programs and to prove properties about those programs in the same language. However, some properties do not require much proof, as they are evident from a program's implementation, e.g. if a polymorphic program is not ad hoc but relationally parametric, then we get parametricity theorems for free. If we want to safely shortcut proofs by relying on the evident good behaviour of a program, then we need a type-level guarantee that the program is indeed well-behaved. This can be achieved by annotating function types with a modality describing the behaviour of functions.We consider a dependent type system with modalities for relational parametricity, irrelevance (i.e. type-checking time erasability of an argument) and ad hoc polymorphism. The interplay of three modalities and dependent types raises a number of questions. For example: If a term depends on a variable with a given modality, then how should its type depend on it? Are all modalities always applicable, e.g. should we consider parametric functions from the booleans to the naturals? Do we need any further modalities in order to properly reason about these three?We develop a type system, based on a denotational presheaf model, that answers these questions. The core idea is to equip every type with a number of relations -just equality for small types, but more for larger types -and to describe function behaviour by saying how functions act on those relations. The system has modality-aware equality judgements (ignoring irrelevant parts) and modality-aware proving operators (for proving free theorems) which are even self-applicable. It also supports sized types, some form of intersection and union types, and parametric quantification over algebraic structures. We prove soundness in a denotational presheaf model. CCS Concepts • Theory of computation → Type theory;

A role for dependent types in Haskell

“…More recently, modal type theories have refined our understanding of parametric, erasable and irrelevant arguments in dependent type theories [Abel and Scherer 2012;Abel et al 2017;Mishra-Linger and Sheard 2008;Nuyts et al 2017]. 32 In particular, Nuyts and Devriese [2018] presents a unified framework that includes these and other modalities. It is possible that roles could also be understood as a new modality in this sense, and, once this is accomplished, enjoy a unified treatment with irrelevant quantification.…”

“…The uniform system of Nuyts and Devriese [2018] could form the basis of the second task. More specifically, this system includes structures that can resolve the differences in the treatment of co-domain of Π types between irrelevant and roled arguments.…”

“…In contrast, roled arguments must appear according to their indicated role everywhere. Furthermore, Nuyts and Devriese [2018] includes a subsumption relationship between modalities that could generalize sub-roleing.…”

“…However, even with the addition of role-annotated Π-types, there would be differences in the treatment of roles in System DR and in the general treatment of modalities in Nuyts and Devriese [2018] that would need to be resolved. First, the general system would need to be extended by a role-indexed operational semantics; roles control the unfolding of axioms and must be considered in any reduction relation.…”

Multimodal Dependent Type Theory

“…However, the consequent gain in expressivity is substantial, and so it is well worth the effort. For example, modalities have been used to express guarded recursive definitions [10,15,16,33], parametric quantification [54,55], proof irrelevance [3,54,57], and to define global operations which cannot be localized to an arbitrary context [42]. There has also been concerted effort towards the development of a dependent type theory corresponding to Lawvere's axiomatic cohesion [41], which has many interesting applications [32,40,64,65,67].…”