Inductive types and exact completion
نویسندگان
چکیده
منابع مشابه
Inductive types and exact completion
Using the theory of exact completions, we show that a specific class of pre-topoi, consisting of what we might call \realizability pretopoi", can act as categorical models of certain predicative type theories, including Martin-LL of type theory. Our main theoretical instrument for doing so is a categorical notion, the notion of weak W-types, an \intensional" analogue of the \exten-sional" notio...
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Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HI...
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Consider a movie recommendation system where apart from the ratings information, side information such as user’s age or movie’s genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the rati...
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Higher inductive-inductive types (HIITs) generalise inductive types of dependent type theories in two directions. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalising higher inductive types of homotopy type theory. Examples that make use of both features are the Cauch...
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In 1959, Kreisel introduced a notion of “modified” realizability that, among other things, provides an alternative technique to Gödel functional (dialectica) interpretation for establishing the connection between Peano Arithmetic and System T. While the dialectica interpretation has been widely studied in the literature, Kreisel’s technique, although remarkably simpler, has apparently been almo...
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ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 2005
ISSN: 0168-0072
DOI: 10.1016/j.apal.2004.09.003