# Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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### An analogue of Scott continuity for infinite-time-Turing-computable functions

$\newcommand{\tb}{2_{\!\bot}}\newcommand{\tbO}{\tb^{\,\omega}}$Let $2 = \{0,1\}$ and $\tb = \{0,\bot,1\}$. Scott continuity is important for defining models of lambda calculus, a formalism for ...
1 vote
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### Are there more learnable but undecidable cases except the halting problem

Per request, I cross post the question here which is original from math.stackexchange In the ICML 1996 paper, On the Learnability of the Uncomputables, by Richard Lathrop, he proved that halting ...
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### What are the application of Scott-Topology in theoretical computer science?

During a work I came across the Scott-Topology and I see that Scott-continuous functions show up in the study of models for lambda calculi. What I cannot understand is how this enrich the lambda-...
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### Understanding the construction of an uncomputable function

The following is from Arora and Barak's "Computational Complexity." I think one does not have to read the second paragraph of the proof to answer this question. Theorem 1.10 There exists a ...
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### How can I switch into computability theory when I am part way through my PhD in deep learning?

I am partway through my PhD studying deep learning. I chose it just because it's useful and would yield a lot of industry opportunities. However, I am really missing my previous coursework in ...
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### Is having a particular form equivalent to being computable for functions on Church numerals?

In SKI-combinator calculus, consider the following function which reduces an expression (involving SKI and variables) to a canonical form: ...
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### Expressive power of lambda-calculus with restricted application

Consider a syntactic restriction of the (untyped) $\lambda$-calculus in which an application cannot have another application as an immediate subterm. More precisely, restricted terms ($R,S,...$) and ...
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### Why are some classes (ALL, ELEMENTARY, R, etc) badly behaved as oracles?

Some classes, such as ALL, ELEMENTARY, and R, are very badly behaved when used as oracles. For instance, all three of these classes trivially collapse P and EXP, even though (by the Time Hierarchy ...
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### Is relativization statement-dependent or proof-dependent?

I'm relearning some computability theory, and have encountered the idea of relativization of results to arbitrary subsets of $\omega$ and the subtlety of figuring out what the correct relativized ...
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### Where, if any, is there currently any research being done on the subject of ternary computers? [closed]

I had the experience several years ago of working with a team that had developed a ternary computing system. It ran out of funding and was abandoned but I felt it was ahead of its time. Currently, ...
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### Proof and computational complexity

I couldn't find documents elaborating on this: if the Curry Howard correspondence is to be interpreted as establishing a strong relation between proofs and programs, should there not be a strong ...
1 vote
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### A proved computationally-irreductible function

In arXiv:1111.4121 and arXiv:1304.5247, Hervé Zwirn and Jean-Paul Delahaye propose a formal definition of computational irreducibility. Is there a (possibly artificial) function that can be proved to ...
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### What is the difference between a model of computation and a programming language?

https://en.wikipedia.org/wiki/Model_of_computation includes sequential models, functional models and concurrency models. Sequential models include finite state machine, Turing machines, random access ...
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### Is this variant of bitwise cyclic tag Turing-complete?

Cross-posted to MathOverflow. CT is an extremely minimalist programming language that can simulate arbitrary tag systems, and is therefore Turing-complete. A program consists simply of a string of 3 ...
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### Are all computable functions monoidal from a category theory POV? [closed]

I have been studying some category theory only to understand the "why" behind the various categorical constructs we use ("borrow") in the world of functional programming. One of ...
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### Is there a language of first-order logic such that every r.e. set is Turing-equivalent to some finitely axiomatizable theory in that language?

I hope that mathematical logic / recursion theory type questions are welcome here. I am sorry this question is so long and technical, but I believe that if you read it you will find that it is well-...
1 vote
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### How come Wikipedia says that Random Turing Machines can provide uncomputable output?

Wikipedia article mentioned : Hypercomputation The third paragraph starts off with: Technically, the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses ...
1 vote
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### What kind of computational model is the brain? [duplicate]

I was wondering what kind of computational model is the human brain (as it seems superior to a Turing machine). Another thing that should be a separate question, What would be a perfect computer model ...
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### Complexity relative to the graph of the Busy-Beaver function

This question is inspired by the comments made on this other question that I asked, and by an attempt to provide an explicit example of a complexity question beyond the Turing degree $\mathbf{0}$. (...
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### Has there been any development of complexity theory for other Turing degrees than 0?

(I'm not sure if this question is better suited for MathOverflow or here. I'll try here first, and move over to MO later if it appears to be more appropriate.) Complexity theory can be very broadly ...
1 vote
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### Given a program specification, S, what can be said about the size and efficiency of programs that exactly satsify S, with respect to the size of S?

Suppose we are given a program specification, $S$, and we want to reason about programs $P$ that satisfy $S$. One might like to think that if the specification is 'simple', the the program should be '...