Containers: Constructing strictly positive types
نویسندگان
چکیده
منابع مشابه
Constructing Strictly Positive Types
We introduce container functors as a representation of data types providing a new conceptual analysis of data structures and polymorphic functions. Our development exploits Type Theory as a convenient way to define constructions within locally cartesian closed categories. We show that container morphisms can be full and faithfully interpreted as polymorphic functions (i.e. natural transformatio...
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In order to represent, compute and reason with advanced data types one must go beyond the traditional treatment of data types as being inductive types and, instead, consider them as inductive families. Strictly positive types (SPTs) form a grammar for defining inductive types and, consequently, a fundamental question in the the theory of inductive families is what constitutes a corresponding gr...
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We begin by revisiting the idea of using a universe of types to write generic programs in a dependently typed setting by constructing a universe for Strictly Positive Types (SPTs). Here we extend this construction to cover dependent types, i.e. Strictly Positive Families (SPFs), thereby fixing a gap left open in previous work. Using the approach presented here we are able to represent all of Ep...
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Functional programs using foldable containers need reasoning tools as they are not equipped with laws. Moreover we want to allow any finite type to be foldable as well. Folding over all the values of a finite type is particularly interesting in a dependent type theory which features Π and Σ types. Our solution uses parametricity to show how foldable containers relate to monoid homomorphisms. Ou...
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Let H be any complex inner product space with inner product < ·, · >. We say that f : | C → | C is Hermitian positive definite on H if the matrix ( f(< z,z >) )n r,s=1 (∗) is Hermitian positive definite for all choice of z, . . . ,z in H, all n. It is strictly Hermitian positive definite if the matrix (∗) is also non-singular for any choice of distinct z, . . . ,z in H. In this article we prove...
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ژورنال
عنوان ژورنال: Theoretical Computer Science
سال: 2005
ISSN: 0304-3975
DOI: 10.1016/j.tcs.2005.06.002