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Results tagged with computability-theory
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user 23648
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.
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Jump Inversion of Arithmetic
I suspect you were looking for the stuff answered above.
But, if you want arithmetic jump inversion you can have that too (Simpson ..nice presentation in Odifreddi II). Essentially what you would exp …
- 3,029
1
vote
1
answer
90
views
Downward density of w-REA sets under arithmetic reducibility?
Is the question of the downward density of the w-REA sets under $\leq_a$ still open? If not can anyone point me to a proof? That is do we know if for every $\omega$-REA set $X >_a 0_a$ there exists …
- 3,029
1
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0
answers
52
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Term for degrees realizing least possible first n jumps
Is there a term for (Turing) degrees which realize the least possible jump (in the following sense) for the first n jumps.
That is degrees which satisfy for all $0 < m \leq n$:
$$X^m \equiv_T X \oplus …
- 3,029
2
votes
0
answers
231
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Is computability theory less cumulative than other areas in mathematics?
I love compatibility theory, degree theory etc and I'm astonished by the advances that have been made in the field but it often seems like computability is less cumulative than other areas of mathemat …
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2
votes
Proof of the existence of hyperimmune-free degrees
You have to be careful here and realize that $Q(\sigma)$ is a function from $2^{<\omega}$ to $2^{<\omega}$ and thus $Q(\sigma)$ can be arbitrarily long even if $|\sigma|=n$.
When the construction mak …
- 3,029
4
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2
answers
464
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Infinite descending chain of Turing jumps with equality
How can one demonstrate there is no sequence $X_i$ of sets such that $X_{i+1}' = X_i$ (this is really equality as sets though Turing equivalence would be interesting too).
I know it fails if I relax e …
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4
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Meta-incomputability
It's worth noting that the answer above is answering the following question: is there a formula $\phi(x)$ in the language of ZFC such that ZFC can't prove either $\lbrace x \in \omega \mid \phi(x) \rb …
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0
votes
Do "seemingly impossible functional programs" work with arrow types interpreted as Turing ma...
Yes you can.
EDIT: That means find a computable program $H(e)$ taking an index $e$ for a predicate $P_e$ that returns $1$ (assuming $P_e$ satisfies above conditions) iff there is some code for an elem …
- 3,029
1
vote
Accepted
Is set of the indices of c.e.sets that cover a productive set also productive one?
No, it isn't. This is a consequence of the fact that we have a choice of infinitely many equivalent indices for each c.e. set. Let $E$ be a set of indices such that
$$(e, j \in E \land e \neq j) \i …
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7
votes
1
answer
155
views
Join Density in R.E. Degrees: Are there r.e. B, C with all r.e. X below B computable or C jo...
Are there r.e. sets $B >_T 0$ and $C >_T 0$, $C \not\geq_T B$ such that for all r.e. $W \leq_T B$ either $W \leq_T 0$ or $C \oplus W \geq_T B$. The explanation for the title is because one can think …
- 3,029
1
vote
1
answer
92
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If $X \leq_T Y + 0'$ does there exist $Z \leq_T Y$, $Z \leq_T X$ with $X \leq_T Z'$?
If $X \leq_T Y + 0'$ does there always exist $Z \leq_T Y$, $Z \leq_T X$ with $X \leq_T Z'$?
Obviously, we can find $Z \leq Y$ where the $y$-th column of $Z$ has a limit equal to $X(y)$. Just let $\la …
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2
votes
Accepted
If $X \leq_T Y + 0'$ does there exist $Z \leq_T Y$, $Z \leq_T X$ with $X \leq_T Z'$?
Ohh, I think I'm being dumb. The answer is no.
Given $X \not\leq_T 0'$ we build $Y$ using the finite extension method and modify the usual minimal pair construction by coding in the bits of $X$ into …
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1
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1
answer
181
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Using Ordinal Notations in Computability Theory Is There A Standard Notation For The Notatio...
I find I frequently have to refer to the set of ordinal notations below some given notation. For instance given a notation $\alpha$ I often need to refer to the set $\lbrace \beta \mid \beta <^{\math …
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8
votes
1
answer
395
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Good source for admissible set theory?
So I need to writeup some old results of Harrington's which imply various results about admissible ordinals. I've never really learned admissible recursion theory so what's a good reference?
- 3,029
2
votes
1
answer
122
views
Splitting $\Pi^0_2$ Singletons?
Given a (non-computable) $\Pi^0_2$ singleton $Y$ are there Turing incomparable $\Pi^0_2$ singletons $X_0, X_1$ with $Y \equiv_T X_0 \oplus X_1$?
What about the same question for arithmetic reducibilit …
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