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.

翻译：我们引入了MTT, 这是一种支持多种模式的依附型理论。 MTT 被一种模式理论的配对, 这是一种模式理论, 它规定了模式、 模式和变异的集合。 我们显示, 不同的模式理论选择允许我们使用同一类型的理论来计算和解释许多模式情况, 包括有节制的循环、 轴心凝固和参数量化。 我们复制了以前在有节制的循环和轴心凝固中工作的例子, 表明 MTT 是一个简单和可用的语法, 其即即时和先前手工制作的模式类型理论直接对应的。 在某些情况下, 将MTT 快速化到一个特定情况, 一种在先前系统上得到改进的未知类型理论。 最后, 我们调查MTTT的元论。 我们证明MTT的连贯性, 并通过推广最近的类型神学凝固技术来建立可控性。 这些结果与模式理论的选择无关, 并因此适用于多种模式。