Computational Higher Type Theory I: Abstract Cubical Realizability

The goal of this work is to develop a computation-based account of higher-dimensional type theory for which canonicity at observable types is true by construction. Types are considered as descriptions of the computational behavior of terms, rather than as formal syntax to which meaning is attached separately. Types are structured as collections of terms of each finite dimension. At dimension zero the terms of a type are its ordinary members; at higher dimension terms are lines between terms of the next lower dimension. The terms of each dimension satisfy coherence conditions ensuring that the terms may be seen as abstract cubes. Each line is to be interpreted as an identification of two cubes in that it provides evidence for their exchangeability in all contexts. It is required that there be sufficiently many lines that this interpretation is tenable. For example, lines must be reversible and closed under concatenation, so that the identifications present the structure of a pre-groupoid. Moreover, there must be further lines witnessing the unit, inverse, and associativity laws of concatention, the structure of an ∞-groupoid. In this paper we give a “meaning explanation” of a computational higher type theory in the style of Martin-Löf and of Constable and Allen, et al. [Martin-Löf, 1984; Martin-Löf, 1984; Constable, et al., 1985; Allen et al., 2006]. Such an explanation starts with a dimension-stratified collection of terms endowed with a deterministic operational semantics defining what it means to evaluate closed terms of any dimension to canonical form. The dimension of a term is the finite set of dimension names it contains; these dimension names may be thought of as variables ranging over an abstract interval, in which case terms may be thought of as tracing out lines in a type. The end points, 0 and 1, of the interval may be substituted to obtain the end points of such lines. Dimension names may be substituted for one another without restriction, allowing dimensions to be renamed, identified, or duplicated. The semantics of types is given by specifying, at each dimension, when canonical elements are equal, when general elements are equal, and when these