Computational Higher-Dimensional Type Theory

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-Lof'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.