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The Gentle Art of Levitation
"... We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a de ..."
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We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a description. Datatypegeneric programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently selfsupporting setup is achievable without paradox or infinite regress. 1.
The Nax Language: Unifying Functional Programming and Logical Reasoning in a Language based on Mendlerstyle Recursion Schemes and Termindexed Types
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System Fi a HigherOrder Polymorphic λCalculus with Erasable TermIndices
"... Abstract. We introduce a foundational lambda calculus, System Fi, for studying programming languages with termindexed datatypes – higherkinded datatypes whose indices range over data such as natural numbers or lists. System Fi is an extension of System Fω that introduces the minimal features need ..."
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Abstract. We introduce a foundational lambda calculus, System Fi, for studying programming languages with termindexed datatypes – higherkinded datatypes whose indices range over data such as natural numbers or lists. System Fi is an extension of System Fω that introduces the minimal features needed to support termindexing. We show that System Fi provides a theory for analysing programs with termindexed types and also argue that it constitutes a basis for the design of logicallysound lightweight dependent programming languages. We establish erasure properties of Fitypes that capture the idea that termindices are discardable in that they are irrelevant for computation. Index erasure projects typing in System Fi to typing in System Fω. So, System Fi inherits strong normalization and logical consistency from System Fω.
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, 2012
"... Hollow hydroxyapatite microspheres as devices for controlled delivery of proteins and as scaffolds for tissue engineering ..."
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Hollow hydroxyapatite microspheres as devices for controlled delivery of proteins and as scaffolds for tissue engineering
An Algebra of Dependent Data Types
"... Abstract. We extend the standard categorical approach to algebraic data types to dependent algebraic data types, so that dependency between two algebraic data types has natural semantics. Specifically, for two inductive data types S and A characterized by two Falgebra F and G, any natural transfor ..."
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Abstract. We extend the standard categorical approach to algebraic data types to dependent algebraic data types, so that dependency between two algebraic data types has natural semantics. Specifically, for two inductive data types S and A characterized by two Falgebra F and G, any natural transformation η: F → G gives rise to a dependency of S on A. This natural dependency is the initial object of what we call a Fηalgebra. The initiality further allows us to describe certain dependencies in functions that both involve S and A. We have used Objective Caml to write functional programs where dependencies among data types (and in the relevant functions) are made explicit. This is done by a systematic mapping of layers of categorical constructions to layers of Objective Caml modules. 1