Computational higher-dimensional type theory

Abstract

Formal constructive type theory has proved to be an effective language for mechanized proof. By avoiding non-constructive principles, such as the law of the excluded middle, type theory admits sharper proofs and broader interpretations of results. From a computer science perspective, interest in type theory arises from its applications to programming languages. Standard constructive type theories used in mechanization admit computational interpretations based on meta-mathematical normalization theorems. These proofs are notoriously brittle; any change to the theory potentially invalidates its computational meaning. As a case in point, Voevodsky's univalence axiom raises questions about the computational meaning of proofs. We consider the question: Can higher-dimensional type theory be construed as a programming language? We answer this question affirmatively by providing a direct, deterministic operational interpretation for a representative higher-dimensional dependent type theory with higher inductive types and an instance of univalence. Rather than being a formal type theory defined by rules, it is instead a computational type theory in the sense of Martin-Löf's meaning explanations and of the NuPRL semantics. The definition of the type theory starts with programs; types are specifications of program behavior. The main result is a canonicity theorem stating that closed programs of boolean type evaluate to true or false.

Free Higher Groups in Homotopy Type Theory

“…Theories such as Voevodsky's homotopy type system (HTS) [25], two-level type theory (2LTT) [3,7] or computational higher type theory [6] are variations of standard HoTT in which such infinite structures can be constructed. We believe it would be worth investigating whether the "suspension of a set" problem can be resolved in such systems.…”

On Higher Inductive Types in Cubical Type Theory

“…The papers [2,4,3,7] present cubical type theories inspired by an alternative cubical set category with different fibrancy structure, but with the same decomposition of the composition operation in a homogeneous composition and a transport operation. This decomposition was introduced in an early version of [8] precisely to solve the problem of the interpretation of higher inductive types with parameters.…”

“…The suspensions are covered in [2], and [7] defines a schema for higher inductive types formulated in this setting. The papers [4,3,7] describe computational type theories in the style of Nuprl with a semantics where types are interpreted as partial equivalence relations which gives canonicity for booleans. The schema presented in [7] covers all of the examples of higher inductive types considered in this paper.…”

Code Generation for Higher Inductive Types

“…Voevodsky added univalence to type theories as an axiom, asserting new identifications without providing a means to compute with them. While more recent work arranges the computational mechanisms of the type theory such that univalence can be derived, as is done in cubical type theories [26] [25], we are concerned with modeling concepts from homotopy type theory in existing, mature implementations of type theory, so we follow Univalent Foundations Program [1] in modeling paths using Martin-Löf's identity type. Higher-dimensional structure can arise from univalence, but it can also be introduced by defining new type formers that introduce not only introduction and elimination principles, but also new non-trivial identifications.…”

“…While work proceeds on prototype implementations of higher-dimensional type theories [26] [25], much work remains before they will be as convenient for experimentation with new ideas as Coq, Agda, or Idris is today. In the meantime, it is useful to be able to experiment with ideas from higher-dimensional type theory in our existing systems.…”