Nax Paul Mendler. Predicative Type Universes and Primitive Recursion. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, pages 173--184. June 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be treated in this way, as endofunctors on certain slice categories Type D , where Type is a suitable category of types. In the context of dependent type theories, Eduardo Gimenez [37] has introduced an extension of the Calculus of Constructions with inductive and co inductive types. Mendler in [69] has also formulated such schemes in the context of Martin Lof s type theory with predicative universes. Mendler s formulations are inspired by categorical considerations. In the context of simple type theory, there has been a great deal of work, and some quite interesting formulations have been ....

N.P. Mendler. Predicative type universes and primitive recursion. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, pages 173--184. IEEE, Amsterdam, 1991.

....and not of the logical framework. The reader is referred to the literature on categorical semantics of dependent types for the latter, see for example Cartmell [4] Seely [31] Dybjer [11] or Hofmann [13] The categorical semantics of universes has previously been investigated by Mendler [20]. There he considers various universes which are all inductiverecursive definitions with D = set. Our approach goes further since we consider inductive recursive definitions with arbitrary D and characterize the collection of endofunctors which have initial algebras. 4.1 Strictly Positive ....

P. F. Mendler. Predicative type universes and primitive recursion. In Proceedings Sixth Annual Synposium on Logic in Computer Science. IEEE Computer Society Press, 1991.

....(X) P n (X) are interpretations of P 1 (X) P n (X) on a suitable category. Our task is to code up this construction in polymorphism as faithfully as we can. This is done in [19] by coding the corresponding initial T algebras. Our view is that the I algebras introduced in [15] are most suitable to the description of the encodings of Hagino s categorical data types. Suppose P 1 (X) Delta Delta Delta ; P n (X) are n type expressions in the second order calculus with the free variable X occurring positively in all of them. Definition 2.1 Let F 1 ; F n be ....

N. Mendler. Predicative type universes and primitive recursion. In Proceedings of the Sixth Symposium on Logic in Computer Science, pages 173--184. IEEE Computer Science Press, July 1991.

....Dybjer [9] and Paulin [18] A schema for inductive recursive definitions was introduced by Dybjer [7] Categorical semantics of inductive types and of universes. The categorical semantics of inductively defined dependent types has been discussed for example by Coquand and Paulin [5] and Mendler [14]. The latter article also discusses categorical semantics of universes in type theory. In a future article we plan to extend Mendler s work, by giving categorical semantics of inductive recursive definitions in terms of initial algebras on endofunctors in slice categories. We will also show how ....

P. F. Mendler. Predicative type universes and primitive recursion. In Proceedings Sixth Annual Synposium on Logic in Computer Science. IEEE Computer Society Press, 1991.

....definitions is obtained by a minor syntactic modification of the schema in Dybjer [20] This adds evidence to the fundamental nature of the schematic natural deduction formulation of inductive definitions in type theory. The idea to consider this generalisation was inspired by Nax Mendler s paper [32] on the categorytheoretic semantics of universes in type theory. Our analysis improves fundamentally on Mendler s, since the category theoretic machinery can be applied only if the rules for U 0 and T 0 already have been represented as an endofunctor on a category of families of sets. This ....

P. F. Mendler. Predicative type universes and primitive recursion. In Proceedings Sixth Annual Synposium on Logic in Computer Science. IEEE Computer Society Press, 1991.

....as initial T algebras is studied in [Ore92] The construction of functors from type constructors is used to code up some T algebras. In loc.cit. there are interesting examples of how to use this kind of inductive types. Categorical formulation of inductive (coinductive) types can be found in [Men90, Men91b]. Related to this is the work [Fre91] where some properties of algebraic complete categories are obtained. The model theory of the inductive types has been a research topic for some time. An old approach is to interpret an inductive type by the initial T algebra of a continuous functor on ....

N. Mendler. Predicative type universes and primitive recursion. In Proceedings of the Sixth Symposium on Logic in Computer Science, pages 173--184. IEEE Computer Science Press, July 1991.

....or just impossible. We propose a categorical notion of (primitive) recursion which can easily be added as computation rule to a typed lambda calculus and gives us a clear view on what the dual of recursion, corecursion, on coinductive types is. The same notion has, independently, been proposed by [Mendler 1991]. We look at how these syntactic notions work out in the simply typed lambda calculus and the polymorphic lambda calculus. It will turn out that in the syntax, recursion can be defined in terms of corecursion and vice versa using polymorphism: Polymorphic lambda calculus with a scheme for either ....

....recursion (just like initial algebra categorically represents the notion of iteration) One of the trade offs is that we can dualize all this to get a notion of corecursion on coinductive types. These categorical notions of recursion and corecursion have independently been found by Mendler (see [Mendler 1991]) who treats these constructions in Martin Lof type theory with predicative universes. What we define as (co)recursive (co)algebras are what Mendler calls (co)algebras that admit simple primitive recursion . We shall always use the term recursion , because, although the function definitionscheme ....

N.P. Mendler, Predicative type universes and primitive recursion. Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science. Amsterdam, The Netherlands, IEEE, pp 173-184

....inductive types internally. This is adopted in [Wra89, PPM91, Fu93b] This method does not pose any model theoretical problems in the traditional sense. The semantic interest here is about how to relate two levels of models. See [Pit87, Fu93a, Fu92a] for details. Categorical Inductive Types. In [Men91], a categorical formulation of recursion is given. It is interesting to see what this general definition means in concrete categories in which dependent typed calculus can be modeled. See [Fu92b] for some examples. 3 The Initial Algebraic Approach There is another way of formalizing the ideas ....

N. Mendler. Predicative type universes and primitive recursion. In Proceedings of the Sixth Symposium on Logic in Computer Science, pages 173--184. IEEE Computer Science Press, July 1991.

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Nax Paul Mendler. Predicative Type Universes and Primitive Recursion. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, pages 173--184. June 1991.

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P. F. Mendler. Predicative type universes and primitive recursion. In Proceedings Sixth Annual Synposium on Logic in Computer Science. IEEE Computer Society Press, 1991. 39

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N.P. Mendler. Predicative type universes and primitive recursion. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, pages 173--184. IEEE, Amsterdam, 1991.

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N.P. Mendler. Predicative type universes and primitive recursion. In: Proc. Sixth Annual Symp. on Logic in Computer Science, pp. 173 -- 184. IEEE Computer Society Press 1991.

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