Abstract Data Types without the Types. Dedicated to David Turner on the occasion of his 70'th birthday

The data abstraction mechanism of Miranda may be adapted to a dynamically typed programming language by applying ideas from gradual typing

Theoretical Pearl: ≐≃≡ Leibniz equality is isomorphic to Martin-Löf identity, parametrically

status: publishe

Parametricity in an Impredicative Sort

Reynold\u27s abstraction theorem is now a well-established result for a large class of type systems. We propose here a definition of relational parametricity and a proof of the abstraction theorem in the Calculus of Inductive Constructions (CIC), the underlying formal language of Coq, in which parametricity relations\u27 codomain is the impredicative sort of propositions. To proceed, we need to refine this calculus by splitting the sort hierarchy to separate informative terms from non-informative terms. This refinement is very close to CIC, but with the property that typing judgments can distinguish informative terms. Among many applications, this natural encoding of parametricity inside CIC serves both theoretical purposes (proving the independence of propositions with respect to the logical system) as well as practical aspirations (proving properties of finite algebraic structures). We finally discuss how we can simply build, on top of our calculus, a new reflexive Coq tactic that constructs proof terms by parametricity

Proof-relevant parametricity

Parametricity is one of the foundational principles which underpin our understanding of modern programming languages. Roughly speaking, parametricity expresses the hidden invariants that programs satisfy by formalising the intuition that programs map related inputs to related outputs. Traditionally parametricity is formulated with proofirrelevant relations but programming in Type Theory requires an extension to proof-relevant relations. But then one might ask: can our proofs that polymorphic functions are parametric be parametric themselves? This paper shows how this can be done and, excitingly, our answer requires a trip into the world of higher dimensional parametricity

A Relationally Parametric Model of Dependent Type Theory

Reynolds’ theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds’ original work exploited the typing discipline of the polymorphically typed -calculus System F, but there is now considerable interest in extending relational parametricity to type systems that are richer and more expressive than that of System F.This paper constructs parametric models of predicative and impredicative dependent type theory. The significance of our models is twofold. Firstly, in the impredicative variant we are able to deduce the existence of initial algebras for all indexed functors. To our knowledge, ours is the first account of parametricity for dependent types that is able to lift the useful deduction of the existence of initial algebras in parametric models of System F to the dependently typed setting. Secondly, our models offer conceptual clarity by uniformly expressing relational parametricity for dependent types in terms of reflexive graphs, which allows us to unify the interpretations of types and kinds, instead of taking the relational interpretation of types as a primitive notion. Expressing our model in terms of reflexive graphs ensures that it has canonical choices for the interpretations of the standard type constructors of dependent type theory, except for the interpretation of the universe of small types, where we formulate a refined interpretation tailored for relational parametricity. Moreover, our reflexive graph model opens the door to generalizations of relational parametricity, for example to higher-dimensional relational parametricity

An interpretation of system F through bar recursion

International audienceThere are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a fundamentally different computational behavior and their relationship is not well understood. We make a step towards a comparison by defining the first translation of system F into a simply-typed total language with a variant of bar recursion. This translation relies on a realizability interpretation of second-order arithmetic. Due to Gödel's incompleteness theorem there is no proof of termination of system F within second-order arithmetic. However, for each individual term of system F there is a proof in second-order arithmetic that it terminates, with its realizability interpretation providing a bound on the number of reduction steps to reach a normal form. Using this bound, we compute the normal form through primitive recursion. Moreover, since the normalization proof of system F proceeds by induction on typing derivations, the translation is compositional. The flexibility of our method opens the possibility of getting a more direct translation that will provide an alternative approach to the study of polymorphism, namely through bar recursion

Soporte para ARM en un compilador verificado

Tesis (Lic. en Cs. de la Computación)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2022.Fil: Arranz Olmos, Santiago. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.Este trabajo es un estudio de un lenguaje de programación, llamado Jasmin, utilizado para desarrollar criptografía eficiente y confiable, así como una propuesta de una extensión a esta herramienta para agregar soporte para nuevas arquitecturas de hardware como ARM Cortex M4. Se estudia el problema de desarrollar software crítico con aplicaciones en seguridad, en particular criptografía, y se describen brevemente algunos de los fundamentos y herramientas utilizados para especificar e implementar estos sistemas. Luego, se describen el lenguaje de programación Jasmin, su compilador y la verificación formal de este último; y por último una generalización del compilador para adecuarlo a nuevos casos de interés.This work is a study on the Jasmin programming language, used to develop high-speed high-assurance cryptography, as well as a proposal for an extension to add support for new hardware architectures such as ARM Cortex M4. We study the problem of developing critical security software, in particular cryptography, as well as some of the theoretic foundations and tools used to specify and implement these systems. Then, we describe the Jasmin programming language, its compiler and the formal verification of the latter; finally, we report on a generalization proposed by us to adapt the compiler to new cases of interest.Fil: Arranz Olmos, Santiago. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina