Integrating Linear and Dependent Types

Krishnaswami, Neelakantan R.; Pradic, Pierre; Benton, Nick

doi:10.1145/2676726.2676969

Cited by 27 publications

(7 citation statements)

References 46 publications

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“…Combining linear and dependent types with full generality has been a long-standing challenge. Various attempts settle on the compromise that types can depend only on non-linear values [Barber and Plotkin 1996;Cervesato and Pfenning 2002;Krishnaswami et al 2015]. Recent work by McBride [2016], refined by Atkey [2018], however resolves the interaction of linear and dependent types by augmenting a linear system with usage annotations capturing the number of times a variable is used computationally, akin to grades but in an implicit style.…”

Section: Further Work and Conclusionmentioning

confidence: 99%

Quantitative program reasoning with graded modal types

Orchard

Liepelt

Eades

2019

Proc. ACM Program. Lang.

110

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In programming, some data acts as a resource (e.g., file handles, channels) subject to usage constraints. This poses a challenge to software correctness as most languages are agnostic to constraints on data. The approach of linear types provides a partial remedy, delineating data into resources to be used but never copied or discarded, and unconstrained values. Bounded Linear Logic provides a more fine-grained approach, quantifying non-linear use via an indexed-family of modalities. Recent work on coeffect types generalises this idea to graded comonads, providing type systems which can capture various program properties. Here, we propose the umbrella notion of graded modal types, encompassing coeffect types and dual notions of type-based effect reasoning via graded monads. In combination with linear and indexed types, we show that graded modal types provide an expressive type theory for quantitative program reasoning, advancing the reach of type systems to capture and verify a broader set of program properties. We demonstrate this approach via a type system embodied in a fully-fledged functional language called Granule, exploring various examples.Quantitative Program Reasoning with Graded Modal Types 110:3[2016]. Our graded modalities can thus be considered to be computationally trivial, i.e., not requiring additional underlying semantics, though we provide a graded possibility encapsulating I/O effects.At the moment, Granule is not designed as a general-use surface-level language. Rather, our aim is to demonstrate the reasoning power provided by combining linear, graded, and indexed types in the context of standard language features like data types, pattern matching, and recursion. A TASTE OF GRANULEWe begin with various example programs in Granule, building from the established concept of linear types up to the graded modalities of this paper. We show how linear and graded modal types allow us to document, discover, and enforce program properties, complementing and extending the reasoning provided by parametric polymorphism and indexed types.Granule syntactically resembles Haskell. Programs comprise mutually recursive definitions, with functions given by sequences of equations, using pattern matching to distinguish their cases. Toplevel definitions must have a type signature (inference and principal types is further work, ğ10). The T E X source of this section is a literate Granule file; everything here is real code. Ill-typed definitions are marked by ✗. We invite the reader to run the type checker and interpreter themselves. 1 LinearityTo ease into the syntax, the following are two well-typed polymorphic functions in Granule:Polymorphic type variables are explicit, given with their kind. These functions are both linear: they use their inputs exactly once. The id function is the linear function par excellence and flip switches around the order in which a function takes its arguments. From flip we can deduce that the structural exchange rule is allowed. However, the other two structural rules, weakening and con...

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Section: Further Work and Conclusionmentioning

confidence: 99%

Quantitative program reasoning with graded modal types

Orchard

Liepelt

Eades

2019

Proc. ACM Program. Lang.

110

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“…Thus, the judgement tracks information about the two different kinds of usage. Cervesato and Pfenning [9], and later Krishnaswami et al [18] and Vákár [35] adapted this to dependent types. Exploiting the fact that variables in the unrestricted context act as they do in normal Type Theory, there is no problem in allowing types to depend on the variables x i .…”

Section: Dependency and Accountancymentioning

confidence: 99%

“…In our introductory section we have already described the main predecessors of this work. In the split context tradition, where intuitionistic and linear types are kept separate, the original work is Cervesato and Pfenning's [9] Linear Logical Framework, which Vákár [35] provided a categorical semantics for; Krishnaswami et al [18] also investigated a variant of the split context system, also based on Benton's Linear Non-Linear Logic [6].…”

Section: Related Workmentioning

confidence: 99%

Syntax and Semantics of Quantitative Type Theory

Atkey

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories.

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“…(1) but are not all (co)monads, the more extensively studied construction in the context of dependent type theory (Krishnaswami et al 2015; de Paiva and Ritter 2016; Shulman 2018; Vákár 2017. Motivated in part by these examples, in this paper, we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent generalisation of) the K axiom 2 of modal logic.…”

Section: Introductionmentioning

confidence: 99%

Modal dependent type theory and dependent right adjoints

Birkedal

Clouston

Mannaa³

et al. 2019

Math. Struct. Comp. Sci.

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In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.

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Integrating Linear and Dependent Types

Cited by 27 publications

References 46 publications

Quantitative program reasoning with graded modal types

Quantitative program reasoning with graded modal types

Syntax and Semantics of Quantitative Type Theory

Modal dependent type theory and dependent right adjoints

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