Internalizing representation independence with univalence

Abstract

In their usual form, representation independence metatheorems provide an external guarantee that two implementations of an abstract interface are interchangeable when they are related by an operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations by transferring them from simpler, related implementations. Recent work in proof assistants has shown that Voevodsky's univalence principle allows transferring theorems between isomorphic types, but many instances of representation independence in programming involve non-isomorphic representations. In this paper, we develop techniques for establishing internal relational representation independence results in dependent type theory, by using higher inductive types to simultaneously quotient two related implementation types by a heterogeneous correspondence between them. The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence. We illustrate our techniques by considering applications to matrices, queues, and finite multisets. Our results are all formalized in Cubical Agda, a recent extension of Agda which supports univalence and higher inductive types in a computationally well-behaved way.

Symmetries in reversible programming: from symmetric rig groupoids to reversible programming languages

“…Thus, our results suggest a new way of thinking about computing using the univalence principle, which provides an intuition on why certain constructions are hard (or impossible in the general case). There are other, different approaches to computing with univalence, in [Angiuli et al 2021;Tabareau et al 2021], and in Cubical Type Theory [Angiuli 2019;Vezzosi et al 2019].…”

Cubical methods in homotopy type theory and univalent foundations

“…This way we achieve a clear separation of concerns (Dijkstra, 1974) where proofs can be done using unary numbers and then transported over to binary numbers which are better suited for doing computations than proving properties. A cubical version of the SIP and many of its consequences for formalizing mathematics and computer science can be found in the paper of Angiuli et al (2021b).…”

“…A tutorial to Cubical Agda with many exercises can be found at https://github.com/HoTT/EPIT-2020/tree/main/ 04-cubical-type-theory. There are also quite a few papers reporting on formalization projects using Cubical Agda, including a cubical version of the SIP (Angiuli et al, 2021b), synthetic homotopy theory (Mörtberg and Pujet, 2020), proof theory and ordinal notations (Forsberg et al, 2020), and a formalization of π-calculus (Veltri and Vezzosi, 2020).…”

Free Commutative Monoids in Homotopy Type Theory

“…Various schemas for defining HITs have been proposed [35,14,11,22,23]. HITs have also been used in other computer science applications [5,24,6,4].…”

“…Other definiitons of finite multisets in type theory have been considered before, for instance using setoids [15,26,4]. None of these encodings prove the universal property of free commutative monoids and most results use the assumption of decidable equality.…”