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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.

6 votes
1 answer
226 views

Is there an injective homomorphism on the Turing degrees?

I know that the existence of a non-trivial automorphism of the Turing degrees is a long-standing open problem. I am curious to know if something is known about (non-trivial) injective homomorphisms (o …
Manlio's user avatar
  • 342
4 votes

Understanding the definition of a (computably / continuously) “transparent” function

The intuition for transparency functions is the following (then all the other notions are slight variations): if you are trying to apply some $f$ after $U$, this is indeed the same as applying some ot …
Manlio's user avatar
  • 342
3 votes
1 answer
170 views

Computing the halting problem with no computable bound on the use function

I would like to prove that there are two sets $A,B\subset \mathbb{N}$ such that $A |_T B$ $\emptyset' \equiv_T A\oplus B$ for every $e$, if $\{e\}^{A\oplus B}=\emptyset'$ then the map sending $(B,n)$ …
Manlio's user avatar
  • 342
1 vote
0 answers
97 views

Existence of limit computable map

Is there a limit computable function $\Phi$ with the following properties? Let $T\subset \mathbb{N}^{<\mathbb{N}}$ be a tree (coded as its characteristic function) and $f\in\mathbb{N}^\mathbb{N}$ be …
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  • 342
5 votes
0 answers
157 views

The set of homogeneous solutions of a clopen contains an hyperarithmetical set

In the context of Galvin-Prikry generalization of Ramsey's theorem, I read in a couple of papers ([1],[2]) that Solovay [3] proved that if $P$ is a clopen of $[\mathbb{N}]^{\mathbb{N}}$ then the set o …
Manlio's user avatar
  • 342