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.

中文翻译：

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多峰依存类型理论

我们介绍了MTT，这是一种支持多种模式的从属类型理论。MTT由模式理论参数化，该模式理论指定了模式，模态及其之间的转换的集合。我们证明，模式理论的不同选择使我们可以在许多模态情况下使用相同的类型理论进行计算和推理，包括保护递归，公理内聚和参数量化。我们重现了先前在保护递归和公理内聚方面的工作中的示例-证明MTT构成了一种简单且可用的语法，其实例直观地对应于以前的手工模态类型理论。在某些情况下，将MTT实例化为特定情况会发掘以前未知的类型理论，该理论将对现有系统进行改进。最后，我们研究了MTT的元理论。我们证明了MTT的一致性，并通过扩展最新的类型理论胶合技术来建立规范性。这些结果与模式理论的选择无关，因此适用于多种模式情况。