12
votes
What are pertinent references to cite on Scott domains?
I asked Dana Scott who kindly responded. I am relaying his answer:
I think the paper “A type-theoretical alternative to ISWIM, CUCH, OWHY” answers the questions and gives the context of the discovery....
- 29.4k
10
votes
Accepted
What else (besides the usual) can be said about a Scott Information System if the constructed domain is required to be Hausdorff?
Since the least element $\bot$ of any Scott domain is a compactification point $-$ the only open set containing it is the whole space $-$ the Scott topology is never Hausdorff, unless it is trivial.
...
- 611
8
votes
What are pertinent references to cite on Scott domains?
First papers
Scott (1993), A type-theoretical alternative to ISWIM, CUCH, OWHY.
This 1969 manuscript was later published in TCS. The title is a bit odd but it seems to hide the very first written ...
7
votes
Accepted
Does the Category of CPOs have omega^op limits?
Here's an attempt (please check!).
We have that $\bot_D = d$, where
$$
d_i = \bigsqcup^{D_i} \{\bot_i,f_i(\bot_{i+1}),f_i(f_{i+1}(\bot_{i+2})),\ldots\}
$$
By construction (and monotonicity), the ...
- 688
7
votes
An analogue of Scott continuity for infinite-time-Turing-computable functions
The domain-theoretic considerations of the kind you are asking about can be carried out using synthetic domain theory. Related to it is syntehtic topology, and in fact the two share many common ideas.
...
- 29.4k
7
votes
Accepted
Is there an isomorphism between universal domains $\mathcal{P}\omega$ and the interval domain $\mathbf{I}\mathbb{R}$?
This is only half an answer, but allow me to clear up a constructive point about the interval domain.
The usual definition of the interval domain is
$$\mathrm{I}\mathbb{R} = \{[a,b] \mid a, b \in \...
- 29.4k
5
votes
Accepted
Given a domain, how do we build a language whose denotation is the domain?
As it turns out, the OP is interested in the specific case of the interval domain. Martín Escardó's PhD thesis "PCF extended with real numbers: a domain-theoretic approach to higher-order exact ...
- 29.4k
4
votes
Accepted
Commutativity of Clock Quantification and Disjunction/Existential Quantification in Guarded Type Theories
Let me first say that I did not look carefully at the second part of your question, nor your sketch of why the clock quantifier should commute with propositional existential quantification.
I will ...
3
votes
Accepted
Is $D_\infty$ a universal domain?
It has been a while, but if I recall correctly, the following paper should be useful in answering the question:
Andrej Bauer, Gordon Plotkin, Dana S. Scott: Cartesian closed categories of separable ...
- 29.4k
3
votes
Accepted
Denotational semantics of intersection types
Intersection types appear in typed programming languages to capture the idea that a given expression may carry multiple functionalities. For example, given a type $\mathsf{read}\;\alpha$ of readable ...
- 29.4k
3
votes
Denotational semantics of intersection types
Later edit: When I wrote the answer below, I was thinking of intersection types as they are understood in the context of the untyped $\lambda$-calculus. It is now clear that this is not the right ...
- 5,598
3
votes
Accepted
Do realizable systems always have some non-well-founded sets?
You are using the wrong definition of well-foundedness.
Let $R \subseteq A \times A$ be a relation. Consider the following definitions:
$R$ is inductive when for all $B \subseteq A$, if $\forall x \...
- 29.4k
2
votes
Do realizable systems always have some non-well-founded sets?
CZF includes the $\in$-induction axiom, which is the constructively sensible version of the foundation axiom. So everything in one of its models is well-founded in that sense.
However, while I'm no ...
- 1,021
2
votes
Commutativity of Clock Quantification and Disjunction/Existential Quantification in Guarded Type Theories
This question sounds related to Transfinite Iris, which proposes to change the Iris model from Nat-indexed propositions to Ordinal-indexed propositions to have "later" commute with ...
- 2,040
Only top scored, non community-wiki answers of a minimum length are eligible