Canonicity for 2-dimensional type theory

Abstract

Higher-dimensional dependent type theory enriches conventional one-dimensional dependent type theory with additional structure expressing equivalence of elements of a type. This structure may be employed in a variety of ways to capture rather coarse identifications of elements, such as a universe of sets considered modulo isomorphism. Equivalence must be respected by all families of types and terms, as witnessed computationally by a type-generic program. Higher-dimensional type theory has applications to code reuse for dependently typed programming, and to the formalization of mathematics. In this paper, we develop a novel judgemental formulation of a two-dimensional type theory, which enjoys a canonicity property: a closed term of boolean type is definitionally equal to true or false. Canonicity is a necessary condition for a computational interpretation of type theory as a programming language, and does not hold for existing axiomatic presentations of higherdimensional type theory. The method of proof is a generalization of the NuPRL semantics, interpreting types as syntactic groupoids rather than equivalence relations.

Canonicity for 2-dimensional type theory

“…To show that the judgemental operations are reducible, we will observe that syntactic presentation of [[−]] ensures that all closed instances of these operations exist, and that, using associativity laws, the definition of reducibility reduces to checking closed instances. We show some representative cases for δ, M , and α; the other cases, including those for θ, are available in the extended version [13]. …”

“…We make two simplifications: first, the universe contains a code for exactly one set, booleans; second, equivalences are given explicitly by the two automorphisms on 2, refl (the identity function) and not. The approach readily scales to a richer universe closed under Π and Σ, following our previous work [14], and to programmer-defined isomorphisms given by iso(f, g, α, β), as we show in the extended version of this article [13]. The first four rules define the type set and the family El(−), and give reflection and uip for El(−), expressing discreteness of sets in the universe.…”

“…The interchange law relates 2-resp and transitivity: transitivity followed by 2-resp is the same as 2-resp followed by transitivity. This is a coherence requirement between the two forms of composition in a 2-category; we discuss some special cases in the extended version of this article [13].…”

“…The rules for dependent pairs, which are analogous to the rules for Γ , x :A, are presented in the extended version [13].…”

Homotopical patch theory

“…The computational interpretation of homotopy type theory as a programming language is a subject of active research, though some special cases have been solved, and work in progress is promising [4,5,21,31]. The main lesson of this work is that, in homotopy type theory, proofs of equality have computational content, and can influence how a program runs.…”