Adjoint Logic with a 2-Category of Modes

Licata, Daniel R.; Shulman, Michael

doi:10.1007/978-3-319-27683-0_16

Cited by 16 publications

(22 citation statements)

References 10 publications

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“…Many classical results about adjunctions can be replayed inside MTT. For instance, by carrying out a proof that left adjoints preserve colimits we recover modal or crisp induction principles for the left adjoint [43,67]. We can then show e.g.…”

Section: Internal Adjunctionsmentioning

confidence: 96%

“…We will now show concretely how MTT can be used in specific modal situations by varying the mode theory. We will focus on two different examples: guarded recursion [16,21,51], which captures productive recursive definitions through a combination of modalities, and adjoint modalities [43,44,61,67,71], where two modalities form an adjunction internal to the type theory. In both cases we will show how to reconstruct examples from op.…”

Section: Applying Mttmentioning

confidence: 99%

“…At a high level, MTT can be thought of as a machine that converts a concrete description of modes and modalities into a type theory. This description, which is often called a mode theory, is given in the form of a small strict 2-category [43,44,61]. A mode theory gives rise to the following correspondence: object ∼ mode morphism ∼ modality…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multimodal Dependent Type Theory

Gratzer

Kavvos

Nuyts

et al. 2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion-demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

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Section: Internal Adjunctionsmentioning

confidence: 96%

Section: Applying Mttmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multimodal Dependent Type Theory

Gratzer

Kavvos

Nuyts

et al. 2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Now we have to choose how to represent S. One possibility would be to add further new judgement forms so that it could be characterized similarly to and . The general theory of Licata and Shulman (2016) immediately suggests how to do this, but the judgmental structure of the resulting theory would be substantially more complicated. Moreover, semantically it seems likely to correspond to indexing the base topos over the cohesive topos by means of p !…”

Section: The Problem Of Comonadic Modalitiesmentioning

confidence: 99%

Brouwer's fixed-point theorem in real-cohesive homotopy type theory

Shulman¹

2017

Math. Struct. Comp. Sci.

Self Cite

149

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We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp variables". This yields type theories that we call "spatial" and "cohesive", in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the "fundamental ∞-groupoid" or "shape"), disentangling the "identifications" of Homotopy Type Theory from the "continuous paths" of topology. In a further refinement called "real-cohesion", the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.

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“…A type system that uniformly integrates all of these patterns is not obvious if we want to preserve the desirable properties of session fidelity and deadlock freedom that we obtain from binary session types. Underlying our approach is adjoint logic [24,15,23], which generalizes intuitionistic linear logic [12,11] and LNL [2] by decomposing the usual exponential modality !A into two adjoint modal operators and also affords individual control over the structural rules of weakening and contraction. We provide a formulation of adjoint logic in which cut reduction corresponds to asynchronous communication, and from which session fidelity and deadlock freedom derive.…”

Section: Introductionmentioning

confidence: 99%

A Message-Passing Interpretation of Adjoint Logic

Pruiksma

Pfenning

2019

Electron. Proc. Theor. Comput. Sci.

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We present a system of session types based on adjoint logic which generalize standard binary session types [14]. Our system allows us to uniformly capture several new behaviors in the space of asynchronous message-passing communication, including multicast, where a process sends a single message to multiple clients, replicable services, which have multiple clients and replicate themselves on-demand to handle requests from those clients, and cancellation, where a process discards a channel without communicating along it. We provide session fidelity and deadlock-freedom results for this system, from which we then derive a logically justified form of garbage collection.

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Adjoint Logic with a 2-Category of Modes

Cited by 16 publications

References 10 publications

Multimodal Dependent Type Theory

Multimodal Dependent Type Theory

Brouwer's fixed-point theorem in real-cohesive homotopy type theory

A Message-Passing Interpretation of Adjoint Logic

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