93,623 research outputs found
CHAD for Expressive Total Languages
We show how to apply forward and reverse mode Combinatory Homomorphic
Automatic Differentiation (CHAD) to total functional programming languages with
expressive type systems featuring the combination of - tuple types; - sum
types; - inductive types; - coinductive types; - function types. We achieve
this by analysing the categorical semantics of such types in -types
(Grothendieck constructions) of suitable categories. Using a novel categorical
logical relations technique for such expressive type systems, we give a
correctness proof of CHAD in this setting by showing that it computes the usual
mathematical derivative of the function that the original program implements.
The result is a principled, purely functional and provably correct method for
performing forward and reverse mode automatic differentiation (AD) on total
functional programming languages with expressive type systems.Comment: Under review at MSC
A Note on Ontology and Ordinary Language
We argue for a compositional semantics grounded in a strongly typed ontology that reflects our commonsense view of the world and the way we talk about it. Assuming such a structure we show that the semantics of various natural language phenomena may become nearly trivial
Groupoid Semantics for Thermal Computing
A groupoid semantics is presented for systems with both logical and thermal
degrees of freedom. We apply this to a syntactic model for encryption, and
obtain an algebraic characterization of the heat produced by the encryption
function, as predicted by Landauer's principle. Our model has a linear
representation theory that reveals an underlying quantum semantics, giving for
the first time a functorial classical model for quantum teleportation and other
quantum phenomena.Comment: We describe a groupoid model for thermodynamic computation, and a
quantization procedure that turns encrypted communication into quantum
teleportation. Everything is done using higher category theor
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
- …