39 research outputs found

    Semantics for Homotopy Type Theory

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    The main aim of my PhD thesis is to define a semantics for Homotopy type theory based on elementary categorical tools. This led us to extend the study of this system in other directions: we proved a Normalisation theorem, and defined a generic syntax. All those results are obtained for a subset of the whole Homotopy type theory, which we called 1-HoTT theories. A 1-HoTT theory is composed by Martin-Löf type theory with generic inductive types, the axioms of function extensionality and univalence, truncation and generic 1-higher inductive types, which are a subset of the higher inductive types in which the higher constructor of a type T is limited to the type =T . For those theories we obtained some proof theoretic results; the main one is a Normalisation theorem, following Girard's reducibility candidates technique. The semantics is sound and complete, with the completeness result following from the existence of a canonical model, which is also classifying. Our conjecture is that our proof theory and semantics can be extended to every single higher inductive type. The dissertation shows that a very large amount of higher inductive types can be analysed inside our framework: what prevents to extend the results is the lack of a systematic treatment of the syntax of the higher inductive types, which is still an open issue in Homotopy type theory

    Gluing for Type Theory

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    A predicative variant of a realizability tripos for the Minimalist Foundation.

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    open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

    Parametricity, Automorphisms of the Universe, and Excluded Middle

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    Univalent Foundations and the UniMath Library. The Architecture of Mathematics.

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    We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

    Univalent Foundations and the UniMath Library. The Architecture of Mathematics.

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    We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

    Univalent Foundations and the UniMath Library. The Architecture of Mathematics.

    Get PDF
    We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

    Sheaf semantics of termination-insensitive noninterference

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    We propose a new sheaf semantics for secure information flow over a space of abstract behaviors, based on synthetic domain theory: security classes are open/closed partitions, types are sheaves, and redaction of sensitive information corresponds to restricting a sheaf to a closed subspace. Our security-aware computational model satisfies termination-insensitive noninterference automatically, and therefore constitutes an intrinsic alternative to state of the art extrinsic/relational models of noninterference. Our semantics is the latest application of Sterling and Harper's recent re-interpretation of phase distinctions and noninterference in programming languages in terms of Artin gluing and topos-theoretic open/closed modalities. Prior applications include parametricity for ML modules, the proof of normalization for cubical type theory by Sterling and Angiuli, and the cost-aware logical framework of Niu et al. In this paper we employ the phase distinction perspective twice: first to reconstruct the syntax and semantics of secure information flow as a lattice of phase distinctions between "higher" and "lower" security, and second to verify the computational adequacy of our sheaf semantics vis-\`a-vis an extension of Abadi et al.'s dependency core calculus with a construct for declassifying termination channels.Comment: Extended version of FSCD '22 paper with full technical appendice

    Truncation levels in homotopy type theory

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    Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type theory. We present several original results in homotopy type theory which are related to the truncation level of types, a concept due to Voevodsky. To begin, we give a few simple criteria for determining whether a type is 0-truncated (a set), inspired by a well-known theorem by Hedberg, and these criteria are then generalised to arbitrary n. This naturally leads to a discussion of functions that are weakly constant, i.e. map any two inputs to equal outputs. A weakly constant function does in general not factor through the propositional truncation of its domain, something that one could expect if the function really did not depend on its input. However, the factorisation is always possible for weakly constant endofunctions, which makes it possible to define a propositional notion of anonymous existence. We additionally find a few other non-trivial special cases in which the factorisation works. Further, we present a couple of constructions which are only possible with the judgmental computation rule for the truncation. Among these is an invertibility puzzle that seemingly inverts the canonical map from Nat to the truncation of Nat, which is perhaps surprising as the latter type is equivalent to the unit type. A further result is the construction of strict n-types in Martin-Lof type theory with a hierarchy of univalent universes (and without higher inductive types), and a proof that the universe U(n) is not n-truncated. This solves a hitherto open problem of the 2012/13 special year program on Univalent Foundations at the Institute for Advanced Study (Princeton). The main result of this thesis is a generalised universal property of the propositional truncation, using a construction of coherently constant functions. We show that the type of such coherently constant functions between types A and B, which can be seen as the type of natural transformations between two diagrams over the simplex category without degeneracies (i.e. finite non-empty sets and strictly increasing functions), is equivalent to the type of functions with the truncation of A as domain and B as codomain. In the general case, the definition of natural transformations between such diagrams requires an infinite tower of conditions, which exists if the type theory has Reedy limits of diagrams over the ordinal omega. If B is an n-type for some given finite n, (non-trivial) Reedy limits are unnecessary, allowing us to construct functions from the truncation of A to B in homotopy type theory without further assumptions. To obtain these results, we develop some theory on equality diagrams, especially equality semi-simplicial types. In particular, we show that the semi-simplicial equality type over any type satisfies the Kan condition, which can be seen as the simplicial version of the fundamental result by Lumsdaine, and by van den Berg and Garner, that types are weak omega-groupoids. Finally, we present some results related to formalisations of infinite structures that seem to be impossible to express internally. To give an example, we show how the simplex category can be implemented so that the categorical laws hold strictly. In the presence of very dependent types, we speculate that this makes the Reedy approach for the famous open problem of defining semi-simplicial types work
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