Layered Modal Type Theory

Abstract

We introduce layering to modal type theory to combine type theory with intensional analysis. In particular, we demonstrate this idea by developing a 2-layered modal type theory. At the core of this type theory (layer 0) is a simply typed $$\lambda $$ λ -calculus with no modality. Layer 1 is obtained by extending the core language with one layer of contextual $$\square $$ □ types to support pattern matching on potentially open code from layer 0 while retaining normalization. Although both layers fundamentally share the same language and the same typing judgment, we only allow computation at layer 1. As a consequence, layer 0 accurately captures the syntactic representation of code in contrast to the computational behaviors at layer 1. The system is justified by normalization by evaluation (NbE) using a presheaf model. The normalization algorithm extracted from the model is sound and complete and is implemented in Agda.Layered modal type theory provides a uniform foundation for meta-programming with intensional analysis. We see this work as an important step towards a foundational way to support meta-programming in proof assistants.