Does this mean anything?

[Boyish-Tigerlilly]

Abstract:

In this dissertation, I explore aspects of computable analysis and topology in the framework of relative realizability. The computational models are partial combinatory algebras with subalgebras of computable elements, out of which categories of modest sets are constructed. The internal logic of these categories is suitable for developing a theory of computable analysis and topology, because it is equipped with a computability predicate and it supports many constructions needed in topology and analysis. In addition, a number of previously studied approaches to computable topology and analysis are special cases of the general theory of modest sets.

In the first part of the dissertation, I present categories of modest sets and axiomatize their internal logic, including the computability predicate. The logic is a predicative intuitionistic first-order logic with dependent types, subsets, quotients, inductive and coinductive types.

The second part of the dissertation investigates examples of categories of modest sets. I focus on equilogical spaces, and their relationship with domain theory and Type Two Effectivity (TTE). I show that domains with totality embed in equilogical spaces, and that the embedding preserves both simple and dependent types. I relate equilogical spaces and TTE in three ways: there is an applicative retraction between them, they share a common cartesian closed subcategory that contains all countably based T0-spaces, and they are related by a logical transfer principle. These connections explain why domain theory and TTE agree so well.

In the last part of the dissertation, I demonstrate how to develop computable analysis and topology in the logic of modest sets. The theorems and constructions performed in this logic apply to all categories of modest sets. Furthermore, by working in the internal logic, rather than directly with specific examples of modest sets, we argue abstractly and conceptually about computability in analysis and topology, avoiding the unpleasant details of the underlying computational models, such as Gödel encodings and representations by sequences.


Does this actually mean anything? It sounds like a lot of jargon to me. I don't know.

[Keevan_Colton]

If it isnt complete nonsense it was written by someone who misunderstood the meaning of the word abstract in this context.

[Ra]

Is this related to that fake Computer Science paper generator? Because it sounds like the shit that was on there.
Just an idea that came up in my twisted mind.
- Ra

[Kuroneko]

Does this actually mean anything? It sounds like a lot of jargon to me. I don't know.

I believe so. The paper deals with the representation of mathematical objects through programs (realizability), based on a kind of crippled lambda calculus (partial combinatory algebra). One can intuitively think of a 'modest set' as a generalization of effectively computable sets of basic computability theory, only with arbitrary mathematical objects instead of natural numbers, if that helps. The underlying logic is intuitionist, which means that the existential quantifier really means "is constructible", or, in this jargon, "is realizable" (I really have no idea why intuitionistic mathematicians find the existence of non-constructible objects problematic, but I can see how that would be relevant in computer science). Whether the author succeeds in the stated goals, I have no idea, but it is meaningful.

[Boyish-Tigerlilly]

Well. At least it means something, I kept reading the report and at the end of it I had no idea what the hell they were talking about. It's supposed to be some PhD. Dissertation in "pure and applied logic."