Homotopical patch theory

Angiuli, Carlo; Morehouse, Edward; Licata, Daniel R.; Harper, Robert

doi:10.1017/s0956796816000198

Cited by 14 publications

(25 citation statements)

References 24 publications

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“…Examples of ideas from HoTT/UF in computer science include [Angiuli et al 2016] where the authors use univalence and HITs to model Darcs style patch theory. This work envisioned what could be done if these notions were computing, but at the time it was unknown how to make this happen.…”

Section: Related Workmentioning

confidence: 99%

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Vezzosi

Mörtberg

Abel

2019

Proc. ACM Program. Lang.

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Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types. This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of higher inductive types. These new primitives make function and propositional extensionality as well as quotient types directly definable with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. This extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.or invariance up to equivalence, in the sense that equivalent types will share the same structures and properties. The fact that equality is proof relevant in dependent type theory is the key to enabling this; the data of an equality proof can store the equivalence and transporting along this equality should then apply the function underlying the equivalence. In particular, this allows programs and properties to be transported between equivalent types, hereby increasing modularity and decreasing code duplication. A concrete example are the equivalent representations of natural numbers in unary and binary format. In a univalent system it is possible develop theory about natural numbers using the unary representation, but compute using the binary representation, and as the two representations are equivalent they share the same properties.The principle of univalence is the major new addition in Homotopy Type Theory and Univalent Foundations (HoTT/UF) [Univalent Foundations Program 2013]. However, these new type theoretic foundations add univalence as an axiom which disrupts the good constructive properties of type theory. In particular, if we transport addition on binary numbers to the unary representation we will not be able to compute with it as the system would not know how to reduce the univalence axiom. Cubical type theory [Cohen et al. 2018] addresses this problem by introducing a novel representation of equality proofs and thereby providing computational content to univalence. This makes it possible to constructively transport programs and properties between equivalent types. This representation of equality proofs has many other useful consequences, in particular function and propositional extensionality (pointwise equal functions and logically equivalent propositions are equal), and the equivalence between bisi...

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Section: Related Workmentioning

confidence: 99%

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Vezzosi

Mörtberg

Abel

2019

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

show abstract

“…In the above code, (OTP-equiv key) is of the type given by equation (2). equiv 1 forms a proof element of the type given by equation 3.…”

Section: Implementation Of One-time Pad In the Universementioning

confidence: 99%

“…Containers [28] [29] in homotopy type theory can be used to implement data structures such as multisets and cycles. Patch theory [2] shows modeling of Darcs [32] version control system using the concepts of homotopy type theory.…”

Section: Introductionmentioning

confidence: 99%

“…We use the singleton framework [2] to implement the cryptDB protocol model discussed in this paper. This framework can be generalized to support the class of protocols implemented with encryption schemes that can be expressed using a contractible type.…”

Section: Introductionmentioning

confidence: 99%

“…For examples described in this paper, we will use mkqinv to obtain a proof of equivalence from quasi-inverse. For a path p : A = U B, we have a function coe [2] that coerces along p. The following equation gives the type of coe.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

HoTT-Crypt : A Study in Homotopy Type Theory based on Cryptography

Vivekanandan¹

Kalpa Publications in Computing

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This paper investigates a preliminary application of homotopy type theory in cryptography. It discusses specifying a cryptographic protocol using homotopy type theory which adds the notion of higher inductive type and univalence to Martin-Lo ̈f’s intensional type theory. A higher inductive type specification can act as a front-end mapped to a concrete cryptographic implementation in the universe. By having a higher inductive type front-end, we can encode domain-specific laws of the cryptographic implementation as higher-dimensional paths. The higher inductive type gives us a graphical computational model and can be used to extract functions from underlying concrete implementation. Us- ing this model we can extend types to act as formal certificates guaranteeing on correctness properties of a cryptographic implementation.

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Code Generation for Higher Inductive Types

Vivekanandan

2019

Functional and Constraint Logic Programming

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Higher inductive types are inductive types that include nontrivial higher-dimensional structure, represented as identifications that are not reflexivity. While work proceeds on type theories with a computational interpretation of univalence and higher inductive types, it is convenient to encode these structures in more traditional type theories with mature implementations. However, these encodings involve a great deal of error-prone additional syntax. We present a library that uses Agda's metaprogramming facilities to automate this process, allowing higher inductive types to be specified with minimal additional syntax.

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Homotopical patch theory

Cited by 14 publications

References 24 publications

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Cubical agda: a dependently typed programming language with univalence and higher inductive types

HoTT-Crypt : A Study in Homotopy Type Theory based on Cryptography

Code Generation for Higher Inductive Types

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