Questions tagged [computability-theory]
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.
1,102 questions
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If $X'$ computes $\mathcal{O}^{Y}$ must $X$ compute $Y$?
If $X'$ computes $\mathcal{O}^{Y}$ must $X$ compute $Y$? If not is there a function $\Gamma$ which guarantees that if $X'$ computes $\Gamma(Y)$ then $X$ computes $Y$?
It is easy enough to see that ...
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7
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1
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How to understand non-standard halting times?
Take any desired consistent c.e. theory $T$ asserting the basic facts of arithmetic such that $T$ can formalize the operation of Turing machines. There is some Turing machine program $p$ and input $n$ ...
- 1,820
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For which sets do Arthur and Nimue have a winning strategy in this game (“communicate a bit”)?
TL;DR: I define a three-player game (Arthur, Nimue, Merlin) where Nimue is shown a hidden bit $b$ chosen by Merlin and tries to communicate it to her ally Arthur, but Arthur must act computably while ...
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A universal ordering on the sets in a (Turing) degree
Do all the Turing degrees agree, in a definable (hyperarithmetic? arithmetic?) way, on an ordering of their representatives? I'll make this precise below but roughly the question is whether there is ...
- 4,019
3
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1
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Supernormal sequences
We call a binary sequence $s:\mathbb{N}\to\{0,1\}$ supernormal if
for every injective, increasing and computable function $\iota:\mathbb{N}\to\mathbb{N}$, the binary sequence $s\circ\iota:\mathbb{N}\...
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13
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How far does Cantor-Bendixson rank counting let us build computable isomorphisms between ordinals?
This is tangentially related to this old question of mine.
Say that a clean well-ordering is a computable well-ordering $\triangleleft$ of $\mathbb{N}$ such that the following additional data is ...
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7
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Admissibility spectrum and recursively large ordinals
Some people already have asked questions concerning Sy Friedman's results:
(1) For $x\in\mathbb{R}$ if every $x$-admissible ordinals are stable, then $0^\#\in L[x]$.
(2) There can be, by a class-...
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Is Exercise 5.3.5 of Chong and Yu's book Recursion Theory really correct?
In the book, Chi Tat Chong and Liang Yu, "Recursion Theory, Computational Aspects of Definability", De Gruyter 2015, Exercise 5.3.5 (p.98) quotes S.G.Simpson's result as follows:
If there ...
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694
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Are there any moderately difficult Collatz-type problems?
This question is inspired by a recent Quanta article, which explained that in order to compute BB(6), it is necessary to solve an "antihydra problem" which is somewhat similar to the ...
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6
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128
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Randomness in $\omega^\omega$ and other measure relativization
It is a result of Levin and (independently?) Kautz that if $X \in 2^\omega$ is 1-ML random relative to a computable measure $\mu$ the either $X$ is computable (it is an atom of $\mu$) or $X$ is Turing ...
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1
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Is choice needed to construct a free ultrafilter on the boolean algebra of computable sets?
Is choice needed to construct a free ultrafilter on the boolean algebra of computable sets?
Inspiration
I ask this purely out of curiosity and because it's a natural follow-up question to GVT's ...
- 1,751
8
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3
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773
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Model of ZFC In Which Martin's Cone Theorem Fails?
Is it known to be consistent with ZFC for there to exist a Turing degree invariant projective set which neither contains nor is disjoint from a cone? What about in $L$, i.e., is it known that (the ...
- 4,019
5
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318
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Hypercomputational explanatory strength
Lets consider two entities with different hyper-computational strength.
Entity A is able to comprehend the whole continuum and hence it decides every arithmetic sentence and, assuming Projective ...
- 489
2
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257
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Using dilators to formalize the intuitions about the size of small uncomputable ordinals
The least recursively inaccessible ordinal $I$, is, intuitively, the supremum of the ordinals that come from "recursively" iterating the function $\alpha\mapsto\omega^{CK}_\alpha$. For an ...
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What is known about realizability model of the real analysis?
What do "Realizability of the axiom of choice in HOL" and "Realizability and the Axiom of Choice" mean when they claim they realize a non extensional version of $\sf AC$?
Can they ...
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18
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838
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How to approximate the max and min of Hydra-Game?
Basic background:
A hydra is a finite rooted tree (with the root usually drawn at the bottom). The leaves of the hydra are called heads. Hercules is engaged in a battle with the hydra. At each step of ...
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2
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2
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576
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Computation Via Infinite Sums
Suppose I want to uniformly represent computation in an infinite series, what is the smallest/most natural set of operations I need to express the $n$-th term that allows me to capture arbitrary ...
- 4,019
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Does this specific 5-state Turing machine halt? [closed]
Basic setups:
A Turing machine M operates on a doubly infinite tape. The tape is divided into cells. Each cell can hold either a 0 or a 1 (meaning, we will consider only TM’s with two symbols). The ...
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Are there natural statements in primitive recursive arithmetic with only bounded quantifiers that are true but not provable?
Goodstein's "Recursive Number Theory" (1957) presents a "logic-free" version of primitive recursive arithmetic: all statements in the logic are equalities of expressions involving ...
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502
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What does “the” mean in “the first Kleene algebra”? (In what sense is it unique?)
Definition: “The” first Kleene algebra $\mathcal{K}_1$ is the set $\mathbb{N}$ of natural numbers endowed with the partial operation $(p,n) \mapsto p\bullet n := \varphi_p(n)$ where $\varphi_p$ is the ...
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10
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Is Noether's Problem undecidable?
I begin by recalling Noether's problem over $\mathbb{Q}$:
Let $G$ be a finite group that act faithfully by field automorphisms on $\mathbb{Q}(x_1,\ldots,x_n)$, with the action on $\mathbb{Q}$ trivial. ...
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7
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1
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337
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A topos for realizability under a variable oracle
I am looking for a topos that describes realizability by Turing machines with access to a “variable oracle”.
I think the construction I want is this. Start with Baire space $\mathcal{N} := \mathbb{N}^...
- 38.7k
9
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1
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582
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Alternate proofs of the undecidability of the homeomorphism problem for finite simplicial complexes
It is classical that the homeomorphism problem for finite simplicial complexes is unsolvable. All the sources I know for this actually prove something slightly different:
Theorem: For $n \geq 5$, ...
8
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1
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Does the axiom of choice enable shorter minimum proofs of non-halting than ZF alone is capable of?
It is well-known that ZFC cannot prove the non-halting behavior of any more Turing machines than ZF alone can. However, does the addition of AC permit shorter minimum proofs of non-halting behavior in ...
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4
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Are Category and Measure Special?
In logic, and I expect in mathematics more broadly, it seems like there is a special role played by notions like measure and (baire) category (as in meeting/avoiding dense sets). Obviously, these ...
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