
Modality via Iterated Enrichment
This paper investigates modal type theories by using a new categorical s...
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Guarded Computational Type Theory
Nakano's later modality can be used to specify and define recursive func...
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Truly Visual Polymorphic Algebraic Data Structures through Maramafication
This paper presents a socalled maramafication of an essential part of f...
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CHAD for Expressive Total Languages
We show how to apply forward and reverse mode Combinatory Homomorphic Au...
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A simpler encoding of indexed types
In functional programming languages, generalized algebraic data types (G...
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A Light Modality for Recursion
We investigate the interplay between a modality for controlling the beha...
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Topos and Stacks of Deep Neural Networks
Every known artificial deep neural network (DNN) corresponds to an objec...
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A Generalized Modality for Recursion
Nakano's later modality allows types to express that the output of a function does not immediately depend on its input, and thus that computing its fixpoint is safe. This idea, guarded recursion, has proved useful in various contexts, from functional programming with infinite data structures to formulations of stepindexing internal to type theory. Categorical models have revealed that the later modality corresponds in essence to a simple reindexing of the discrete time scale. Unfortunately, existing guarded type theories suffer from significant limitations for programming purposes. These limitations stem from the fact that the later modality is not expressive enough to capture precise inputoutput dependencies of functions. As a consequence, guarded type theories reject many productive definitions. Combining insights from guarded type theories and synchronous programming languages, we propose a new modality for guarded recursion. This modality can apply any wellbehaved reindexing of the time scale to a type. We call such reindexings time warps. Several modalities from the literature, including later, correspond to fixed time warps, and thus arise as special cases of ours.
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