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.

Homotopical patch theory

“…Our representation of patch theories requires points, paths, and homotopies; reasoning about these patch theories can require paths between homotopies (e.g., the commuting tetrahedron in Section 7). Because we only use three dimensions of structure, it might be advantageous to work inside a dimensionally truncated homotopy type theory (Licata & Harper, 2012), or explicitly truncate all types (as discussed in Section 5.3).…”

“…It is conjectured that one can restore canonicity and the computational interpretation by adding more definitional equalities; doing so is an active area of research (Licata & Harper, 2012;Altenkirch & Kaposi, 2015;Cohen et al, 2016). (Current attempts change how the identity type is axiomatized, in order to simplify its definitional equalities.)…”

“…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 (Licata & Harper, 2012;Bezem et al, 2014;Shulman, 2015;Altenkirch & Kaposi, 2015;Barras et al, 2015;Polonsky, 2015;Cohen et al, 2016). 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.…”

“…Because a computational interpretation of homotopy type theory is work in progress, there is no complete operational semantics that can evaluate the programs in this paper. However, we will use a notion of computation-up-to-paths-based on existing work on this topic (Licata & Harper, 2012;Shulman, 2015;Altenkirch & Kaposi, 2015;Cohen et al, 2016)-in order to compute with the programs we define in this paper.…”

A generalization of the Takeuti–Gandy interpretation

“…this shows that any proof of a property on a structure on A (to be a fixpoint) can be transported to a proof of the corresponding property on the isomorphic structure B. We can cover in this way examples similar to the ones in [8] but also with computations on open terms.…”