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Results tagged with computability-theory
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user 1946
computable sets and functions, Turing degrees, c.e. degrees, models of computability, primitive recursion, oracle computation, models of computability, decision problems, undecidability, Turing jump, halting problem, notions of computable randomness, computable model theory, computable equivalence relation theory, arithmetic and hyperarithmetic hierarchy, infinitary computability, $\alpha$-recursion, complexity theory.
5
votes
Accepted
Turing degrees of lim infs of computable functions
You need a uniformity requirement in your limit characterization of $0'$, that is, the functions $f_k(n)$ need to be uniformly computable, in that there is a single computable function $f(k,n)$, such …
- 236k
1
vote
Accepted
Resource request (probability theory, computability theory, algebra)
Here are a few areas of overlap for those research topics.
Computable model theory is a nice overlap of computability theory and algebra, since one is looking at the nature of computably effective pr …
- 236k
1
vote
Accepted
What is the computational complexity to verify a P solution with a deterministic Turing mach...
In general it is computationally undecidable to determine if a given program $p$ solves a given decision problem, and also to decide whether it does so in polynomial time. For example, consider the tr …
- 236k
16
votes
Accepted
Finding the largest integer describable with a string of symbols of predefined length
My first remark is that if you allow players to pick their own theories, but only allow consistent theories to win, then you will not be able to compute the winner of the contest. The reason is that t …
- 328
15
votes
2
answers
871
views
Which are the hereditarily computably enumerable sets?
My question is about sets that are computably enumerable with respect to their hereditary membership structure. Specifically, let me define that a hereditarily computably enumerable (h.c.e.) set is on …
- 7,545
43
votes
4
answers
3k
views
What do we know about the computable surreal numbers?
The surreal numbers are built up in a natural iterative process, by which at any ordinal stage, if one has two sets of surreal numbers $L$ and $R$, with every number $x_L$ in $L$ strictly below every …
- 3,073
10
votes
What do we know about the computable surreal numbers?
Let me explicate fuller details about the computable surreal number operations. Let's start by showing that they form a ring.
Theorem. The computable surreal numbers form a ring.
Proof. We have to sho …
- 236k
17
votes
What do we know about the computable surreal numbers?
Here is some partial progress. I claim that the computable surreal numbers include some noncomputable real numbers, confirming my guess in connection with question 2.
For each TM program $e$ we can wr …
- 236k
7
votes
Splitting 0' into two 1-generic reals
Let us construct $x$ and $y$ in stages from oracle $0'$, so that they are both $1$-generic, but $x\oplus y$ will code $0'$. We specify the binary digits of $x,y$ in stages. At the first stage, we spec …
- 236k
10
votes
Accepted
Natural Numbers
Here is a positive way to interpret your question, and I think this is probably what you had intended to ask about.
Theorem. There is no computable model $M$ on domain $\mathbb{N}$ in a language $L$ i …
- 236k
1
vote
why $L=\{\langle M\rangle\mid M \text{ is a TM that accepts all even number}\} \notin \text{...
You have made an error. One cannot prove that a set is computably enumerable by reducing the halting problem to it. That would be like trying to show that someone is short by proving that they are at …
- 236k
7
votes
Accepted
Hermit H-machines
Your $H$-machine concept is essentially the same as the concept of oracle computation, due originally to Turing, which gave rise to the elaborate theory of Turing degrees in computability theory. The …
- 236k
5
votes
Accepted
Infinite multiplicity set of continuous functions
There is no such computable decision procedure, largely because of the same issues underlying Rice's theorem, as mentioned in the comment of Steven Stadnicki. The moral of Rice's theorem is that you c …
- 236k
8
votes
4
answers
938
views
Are there two computable binary trees such that each has a branch not computing any branch t...
It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build such …
CommunityBot
- 1
28
votes
Using Busy Beavers to prove conjectures
Although the other answers point out correctly that the exact value of $\text{BB}(n)$ is independent of ZF for large enough and even moderately sized values of $n$, nevertheless I should like to point …
- 236k