Homotopical patch theory

Abstract

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches-syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

Code Generation for Higher Inductive Types

“…Section 4 then generalizes this feature to automate the production of eliminators for higher inductive types using the rewrite mechanism. Section 5 revisits Angiuli et al's encoding of Darcs's patch theory [2] and demonstrates that the higher inductive types employed in that paper can be generated succinctly using our library 1 .…”

“…We reimplemented Angiuli et al's patch theory [2] using our code generator in Agda. We implemented basic patches such as the insertion of a string as line l 1 in a file and deletion of a line l 2 from a file.…”

“…Nevertheless, there are a few applications of homotopy type theory to programming. Applications such as homotopical patch theory [2] discuss a model of the core of the of Darcs [7] version control system using patch theory [17] encoded as a HIT. Containers in homotopy type theory [12,16] implement data structures such as multisets and cycles.…”

“…Using Agda's elaborator reflection, we automatically generate the support code for many useful higher inductive types, specifically those that include additional paths between constructors, but not paths between paths, which is sufficient for treating various interesting examples on the programming side [2][21] [32]. We automate the production of the recursion principles, induction principles, and their computational behavior.…”

“…We automate the production of the recursion principles, induction principles, and their computational behavior. Angiuli et al's encoding of patch theory as a higher inductive type [2] requires approximately 1500 lines of code when represented using rewrite mechanism. Using our library, the encoding can be expressed in just 70 lines.…”