scholarly journals Foundations of dependent interoperability

2018 ◽  
Vol 28 ◽  
Author(s):  
PIERRE-ÉVARISTE DAGAND ◽  
NICOLAS TABAREAU ◽  
ÉRIC TANTER
Keyword(s):  
Type Theory ◽  
Dependent Types ◽  
Full Spectrum ◽  
Galois Connections ◽  
Domain Specific ◽  
Semantic Account ◽  
System Calls ◽  
Homotopy Type Theory ◽  
Library Users ◽  
Certified Software

AbstractFull-spectrum dependent types promise to enable the development of correct-by-construction software. However, even certified software needs to interact with simply-typed or untyped programs, be it to perform system calls, or to use legacy libraries. Trading static guarantees for runtime checks, thedependent interoperabilityframework provides a mechanism by which simply-typed values can safely be coerced to dependent types and, conversely, dependently-typed programs can defensively be exported to a simply-typed application. In this article, we give a semantic account of dependent interoperability. Our presentation relies on and is guided by a pervading notion of type equivalence, whose importance has been emphasized in recent work on homotopy type theory. Specifically, we develop the notions oftype-theoretic partial Galois connectionsas a key foundation for dependent interoperability, which accounts for the partiality of the coercions between types. We explore the applicability of both type-theoretic Galois connections and anticonnections in the setting of dependent interoperability. A partial Galois connection enforces a translation of dependent types to runtime checks that are both sound and complete with respect to the invariants encoded by dependent types. Conversely, picking an anticonnection instead lets us induce weaker, sound conditions that can amount to more efficient runtime checks.Our framework is developed in Coq; it is thus constructive and verified in the strictest sense of the terms. Using our library, users can specify domain-specific partial connections between data structures. Our library then takes care of the (sometimes, heavy) lifting that leads to interoperable programs. It thus becomes possible, as we shall illustrate, to internalize and hand-tune the extraction of dependently-typed programs to interoperable OCaml programs within Coq itself.

scholarly journals Transporting functions across ornaments

2014 ◽  
Vol 24 (2-3) ◽  
pp. 316-383 ◽  
Author(s):  
PIERRE-ÉVARISTE DAGAND ◽  
CONOR McBRIDE
Keyword(s):  
Type Theory ◽  
Structured Data ◽  
Dependent Types ◽  
Natural Numbers ◽  
Code Reuse ◽  
Domain Specific ◽  
Definition Of ◽  
The Relationship ◽  
Do So ◽  
Structural Invariants

AbstractProgramming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of datatypes: We can finally write correct-by-construction software. However, this extreme accuracy is also a curse: A datatype is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any attempt to reuse code across similarly structured data. In this paper, we capitalise on the structural invariants of datatypes. To do so, we first adapt the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornaments, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: The users can ask the definition of addition to be lifted to lists and they will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in the type theory with a universe of datatypes and all our constructions have been implemented as generic programs, requiring no extension to the type theory.

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

10.29007/tvpp ◽  
2018 ◽  
Author(s):  
Paventhan Vivekanandan
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Domain Specific ◽  
Front End ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Higher Dimensional ◽  
Type Specification ◽  
The Universe ◽  
Preliminary Application

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.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

scholarly journals Reviving the Philosophy of Geometry

2018 ◽  
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Great Part ◽  
Mathematical Physics ◽  
Relevant Today ◽  
Homotopy Type Theory ◽  
Philosophical Account ◽  
Hard Times ◽  
Anglophone World

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. This chapter argues that philosophy can come to a better understanding of mathematics by providing an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. It returns to the discussions of Weyl and Cassirer on geometry whose concerns are very much relevant today. A way of encompassing a great part of modern geometry via homotopy toposes is discussed, along with the `cohesive’ variant of their internal language, known as `homotopy type theory’. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.

Homotopy Types

2020 ◽  
pp. 77-106
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Definite Descriptions ◽  
Homotopy Types ◽  
Homotopy Type Theory

A further innovation is the introduction of an intensional type theory. Here one need not reduce equivalence to mere identity. How two entities are the same tells us more than whether they are the same. This is explained by the homotopical aspect of HoTT, where types are taken to resemble spaces of points, paths, paths between paths, and so on. This allows us to rethink Russell’s definite descriptions. Mathematicians use a ‘generalized the’ in situations where it appears that they might be talking about a multiplicity of instances, but there is essentially a unique instance. Taken together with the ‘univalence axiom’ there results a language in which anything that can be said of a type can be said of an equivalent type. This allows homotopy type theory to become the language of choice for a structuralist, avoiding the need for any kind of abstraction away from multiple instantiations.

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