# Tag Info

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### Is there any known CCC closed under a probabilistic powerdomain operation?

The following is an extended comment, it does not answer your question in the terms you posed it but does give a semantics for higher-order probabilistic calculi which you may find of interest. In ...
• 4,963
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### Uses of quasi-PERs/difunctional relations/zig-zag relations?

Makoto Takeyama and I sent the following to data-refinement@etl.go.jp on Jan 5, 1996: Subject: what is a data refinement relation? Dear all: anyone still interested in data refinement? ...
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### The precise definition of Normalization By Evaluation?

The difference you see between higher-order and closure-based representations is a lot smaller than it first seems: the closure-based representation arises as the defunctionalisation of the higher-...
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### In which posets is the set of compact elements downwards closed?

The only natural condition I can think of is Berry's "I condition" ([1], Sect. 12.3): (I) each compact element dominates finitely many elements. The above condition is the defining property of Berry'...
• 4,963
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### Precise definition of syntatic categories / syntatic domains in abstract syntax

Most people avoid giving precise descriptions of what a syntactic category is, because if you do it properly with all the details, the ratio of insight to necessary mathematical sophistication ends up ...
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### 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 ...
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### What is the relation/difference between axiomatic and denotational semantics one one side, and the data flow analysis(DFA) on the other sied?

The textbook that might be most relevant to your question is Principles of Program Analysis by Nielson, Nielson and Hankin. It does cover dataflow analysis and its relationship to denotational ...
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### Is there any known CCC closed under a probabilistic powerdomain operation?

The comment below is correct, but it's important to understand the meaning of "finite" or "compact" elements of a domain. These are the denotations of objects computable in finite time, so their ...

### Precise definition of syntatic categories / syntatic domains in abstract syntax

I never found an explicit definition either, but I have inferred the folowing. As I understand, you split the language into syntactic domains; with the addition that syntactic domains must be fully ...
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### Flat vs non-flat domains

With only flat domains, you cannot define limits to construct "infinite" structures, such as looping structures, for data or for programs. Fixpoint constructions in denotational semantics (since you ...
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### Categorical semantics for non-monotonic logics?

[My apologies for writing this as an answer, despite the fact that it is basically just a comment to the previous answer. But I am not allowed to post a comment up there, since I do not have enough "...

### Categorical semantics for non-monotonic logics?

Non-monotonic logic is kind of a wide area -- do you have any particular logics in mind? Anyway, defeasibly assuming :) that you are interested in any logic in which the principle of monotony fails, ...

### Books on programming language semantics

I would like to add two books not found on the answers given up to now: Aaron Stump, Programming Language Foundations David Schmidt, Denotational Semantics: A Methodology for Language Development ...
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### Books on programming language semantics

For a complete beginner studying operational semantics, I would suggest Programming Languages and Operational Semantics by Maribel Fernández. Everything is explained in a very simple manner which is ...
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### Flat vs non-flat domains

For what concerns the use of non-flat domains, babou already gave examples. I can add that sometimes it may even be useful to see integers as streams: there's ⊥, above which there are 0 and S⊥, above ...
• 4,963
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### What is contextual equivalence ignoring non-termination called?

(This is an extended comment). I may be misreading your definitions, but it seems to me that the relation you introduce, let us call it $\simeq$, is not an equivalence relation because it is not ...
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### Observational Equivalence of open terms in PCF

Got the answer, thanks to jonsterling on reddit for the insight. The error is that both M and N written as full judgements, ...
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### What are Zhang's molecules?

Restating the definition to make the quantifiers easier to understand: A molecule is a finite stable approximable mapping, such that there exists a largest pair $(a,p) \in m$, such that for all ...

### Is there any known CCC closed under a probabilistic powerdomain operation?

Well, Mislove's quote already contains a positive answer: the category of dcpos is carteisan closed and also closed under the probabilistic powerdomain. It can indeed be used to give a denotational ...
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### Name a set of program variables

This new version of the answer tries to take into account the changes in the question, and the information exchanged in the comments. This answer assumes that $S$ should be the set of variables that ...
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### Induction-recursion in models other than $\mathbf{Set}$
Define an IR system be a family $U \in \mathrm{Set}$ and a function $\phi \in U \to \mathrm{Set}$. IR systems form a domain. The least element is $\bot = (\emptyset, !)$. The order relationship \$(U, \...