Syntax and Semantics of Quantitative Type Theory

Atkey, Robert

doi:10.1145/3209108.3209189

Cited by 68 publications

(64 citation statements)

References 32 publications

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“…Our framework is a good starting point for further investigation towards modeling multiparty session types. It will be interesting to see if recent developments, e.g., in quantative type theory [Atkey 2018], can be used to simplify the handling of linearity. It would also be fruitful to further explore the connection of splitting with separation logic with the goal of obtaining cleaner, more modular proofs.…”

Section: Discussionmentioning

confidence: 99%

Intrinsically-Typed Mechanized Semantics for Session Types

Thiemann

2019

Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming

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Session types have emerged as a powerful paradigm for structuring communication-based programs. They guarantee type soundness and session fidelity for concurrent programs with sophisticated communication protocols. As type soundness proofs for languages with session types are tedious and technically involved, it is rare to see mechanized soundness proofs for these systems.We present an executable intrinsically typed small-step semantics for a realistic functional session type calculus. The calculus includes linearity, recursion, and recursive sessions with subtyping. Asynchronous communication is modeled with an encoding.The semantics is implemented in Agda as an intrinsically typed, interruptible CEK machine. This implementation proves type preservation and a particular notion of progress by construction. CCS CONCEPTS• Software and its engineering → Concurrent programming structures; Semantics; • Theory of computation → Process calculi.

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Section: Discussionmentioning

confidence: 99%

Intrinsically-Typed Mechanized Semantics for Session Types

Thiemann

2019

Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming

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“…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. Usage at the type-level is accounted for by 0 of a semiring and term-level use is tracked similarly to coeffect types.…”

Section: Further Work and Conclusionmentioning

confidence: 99%

Quantitative program reasoning with graded modal types

Orchard

Liepelt

Eades

2019

Proc. ACM Program. Lang.

103

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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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“…Quantitative analysis. For quantitative analyses [Atkey 2018;Brunel et al 2014;Ghica and Smith 2014], a world w : W in a R w b denotes the resources needed to construct a or b. Insufficient resources w prevent a R w b from holding, and excessive resources may have the same effect if we model strict linearity rather than just affinity.…”

Section: Relational Model For Parametricity and Usage-tracking: Framementioning

confidence: 99%

A unified view of modalities in type systems

Abel

Bernardy

2020

Proc. ACM Program. Lang.

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We propose to unify the treatment of a broad range of modalities in typed lambda calculi. We do so by defining a generic structure of modalities, and show that this structure arises naturally from the structure of intuitionistic logic, and as such finds instances in a wide range of type systems previously described in literature. Despite this generality, this structure has a rich metatheory, which we expose.

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Syntax and Semantics of Quantitative Type Theory

Cited by 68 publications

References 32 publications

Intrinsically-Typed Mechanized Semantics for Session Types

Intrinsically-Typed Mechanized Semantics for Session Types

Quantitative program reasoning with graded modal types

A unified view of modalities in type systems

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