Graded Modal Dependent Type Theory

Moon, Benjamin; Eades, Harley; Orchard, Dominic

doi:10.1007/978-3-030-72019-3_17

Cited by 15 publications

(7 citation statements)

References 55 publications

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“…Another possibility is to bring uniqueness into the realm of dependent types. Recent work on graded modal dependent type theory (GrTT) [31] allows for capturing requirements on variable usage at both the type and computation levels; grades come in pairs, where the first component is the computation-level grading and the second component is the type-level grading. Strictly linear usage in types is rare -is there value in being able to represent uniqueness here?…”

Section: Future Workmentioning

confidence: 99%

Linearity and Uniqueness: An Entente Cordiale

Marshall

Vollmer

Orchard

2022

Lecture Notes in Computer Science

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Substructural type systems are growing in popularity because they allow for a resourceful interpretation of data which can be used to rule out various software bugs. Indeed, substructurality is finally taking hold in modern programming; Haskell now has linear types roughly based on Girard’s linear logic but integrated via graded function arrows, Clean has uniqueness types designed to ensure that values have at most a single reference to them, and Rust has an intricate ownership system for guaranteeing memory safety. But despite this broad range of resourceful type systems, there is comparatively little understanding of their relative strengths and weaknesses or whether their underlying frameworks can be unified. There is often confusion about whether linearity and uniqueness are essentially the same, or are instead ‘dual’ to one another, or somewhere in between. This paper formalises the relationship between these two well-studied but rarely contrasted ideas, building on two distinct bodies of literature, showing that it is possible and advantageous to have both linear and unique types in the same type system. We study the guarantees of the resulting system and provide a practical implementation in the graded modal setting of the Granule language, adding a third kind of modality alongside coeffect and effect modalities. We then demonstrate via a benchmark that our implementation benefits from expected efficiency gains enabled by adding uniqueness to a language that already has a linear basis.

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Section: Future Workmentioning

confidence: 99%

Linearity and Uniqueness: An Entente Cordiale

Marshall

Vollmer

Orchard

2022

Lecture Notes in Computer Science

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“…However, we can complete this example with a dual process (recvVec) that is then connected to sendVec via forkNonLinear: Note that the receiver needs to know how many elements to receive, so this information has to be passed separately (via an indexed natural number N n). A system with dependent session types [20,21] could avoid this by first sending the length, but this is not (yet) possible in Granule; the Gerty prototype language provides full dependent types and graded modal types which would be a good starting point [15].…”

Section: Singleaction Endmentioning

confidence: 99%

Replicate, Reuse, Repeat: Capturing Non-Linear Communication via Session Types and Graded Modal Types

Marshall

Orchard

2022

Electron. Proc. Theor. Comput. Sci.

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Session types provide guarantees about concurrent behaviour and can be understood through their correspondence with linear logic, with propositions as sessions and proofs as processes. However, a strictly linear setting is somewhat limiting, since there exist various useful patterns of communication that rely on non-linear behaviours. For example, shared channels provide a way to repeatedly spawn a process with binary communication along a fresh linear session-typed channel. Non-linearity can be introduced in a controlled way in programming through the general concept of graded modal types, which are a framework encompassing various kinds of coeffect typing (describing how computations make demands on their context). This paper shows how graded modal types can be leveraged alongside session types to enable various non-linear concurrency behaviours to be re-introduced in a precise manner in a type system with a linear basis. The ideas here are demonstrated using Granule, a functional programming language with linear, indexed, and graded modal types.

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“…Graded modalities indexed by a semiring R are used in many linear calculi and type theories to enable resource sensitive reasoning [40,9,20,37,1,14,36]. In this section, we introduce graded modalities in the framework of primary linear doctrines and use them to formulate a quantitative variant of elementary primary linear doctrines.…”

Section: Quantitative Equality Via Graded Modalitiesmentioning

confidence: 99%

“…Studies of Id-Types in linear type theories are available in the literature, but to the best of our knowledge, they treat Id-Types as intuitionistic types (see Krishnaswami et al [27] and Vákár [42]). There are also dependent type theories using grading, e.g., by Atkey [4] and Moon et al [36], but they do not consider Id-Types. Therefore, a direction that we will explore is to use the machinery introduced in this paper, possibly extended to indexed categories and fibrations, to study quantitative Id-Types in linear type theories.…”

Section: Related and Future Workmentioning

confidence: 99%

Quantitative Equality in Substructural Logic via Lipschitz Doctrines

Dagnino¹,

Pasquali²

2021

Preprint

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Quantitative reasoning provides a flexible approach capable to deal with the uncertainty and imprecision that affects modern software systems due to their complexity. Its distinguishing feature is the use of distances, instead of equivalence relations, so that one can measure how much two objects are similar, rather than just saying whether they are equivalent or not. In this paper we aim at providing a solid logical ground to quantitative reasoning, using the categorical language of Lawvere's hyperdoctrines. The key idea is to see distances as equality predicates in Linear Logic. Adding equality to Linear Logic, however, is not as trivial as it might appear. The linear version of the usual rules for equality, asserting that it is reflexive and substitutive, has the consequence that equality collapses to an equivalence relation, as it can be used an arbitrary number of times. To overcome this issue, we consider the extension of Linear Logic with graded modalities and use them to write a resource sensitive substitution rule that keeps equality quantitative. We introduce a deductive calculus for (Graded) Linear Logic with quantitative equality and the notion of Lipschitz doctrine to give semantics to it. Main examples are based on metric spaces and Lipschitz maps and on quantitative realisability.

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Graded Modal Dependent Type Theory

Cited by 15 publications

References 55 publications

Linearity and Uniqueness: An Entente Cordiale

Linearity and Uniqueness: An Entente Cordiale

Replicate, Reuse, Repeat: Capturing Non-Linear Communication via Session Types and Graded Modal Types

Quantitative Equality in Substructural Logic via Lipschitz Doctrines

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