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.
1,107 questions
3
votes
0
answers
82
views
Sets which inductively define themselves (with just "$+$")
This is related to the second part of this old question of mine.
For $A\subseteq\mathbb{N}$ let $\mathfrak{N}_A=(\mathbb{N};0,1,+,A)$ be the expansion of Presburger arithmetic with (a predicate naming)...
- 19.6k
-1
votes
0
answers
81
views
Do we have a hierarchy/generalisations of effectively axiomatised theories, just like we have a hierarchy of Turing machines?
The notion of an effectively axiomatised theory is based on computing abilities of a Turing machine. For instance, the wffs and proofs are required to be decidable, the theorems are required to be ...
- 107
2
votes
1
answer
201
views
Is the product of "non-coding" forcings also "non-coding"?
Say that a forcing notion $\mathbb{P}$ is slow iff there is some $f:\mathbb{R}\rightarrow\mathbb{R}$ (in $V$) such that for every $\mathbb{P}$-name for a real, $\nu$, we have $\Vdash_\mathbb{P}\exists ...
- 19.6k
16
votes
1
answer
493
views
What can be computed without collapsing $\omega_1$
Below work in $\mathsf{ZFC+CH}$ for simplicity.
Say that a (set) forcing notion $\mathbb{P}$ captures a map $f:\mathbb{R}\rightarrow\mathbb{R}$ iff there is some $\mathbb{P}$-name for a real $\nu$ ...
- 19.6k
6
votes
1
answer
368
views
Higher analogues of Gandy basis theorem
For $n\in\omega$ and $x$ a real let $C_n^x$ be the canonical $\Pi^1_n(x)$-complete set. E.g. $C_1^x=\mathcal{O}^x$, etc. I recall seeing long ago the fact that, assuming large cardinals (precisely: ...
- 19.6k
4
votes
1
answer
151
views
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 ...
- 4,039
5
votes
1
answer
421
views
What are some nice consequences of Martin's conjecture?
I was reading last evening a paper which talked a bit about behaviors on cones and I know for a fact that Martin's conjecture is one of the most active areas in computability and that it was settled ...
- 1,483
7
votes
1
answer
749
views
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 $n$ ...
- 1,860
11
votes
0
answers
448
views
Can we “encrypt” in the Turing degrees?
Definitions:
If $M,C\subseteq\mathbb{N}$, let us say that a function $f\colon\mathbb{N}\to\mathbb{N}$ is an encoding of $M$ by $C$ when $f^{-1}(C) = M$ (i.e., for all $m\in\mathbb{N}$ we have $m\in M$...
- 38.8k
11
votes
0
answers
406
views
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 ...
- 38.8k
12
votes
4
answers
461
views
Complete sets in the arithmetic hierarchy from outside of logic
What are some good examples of problems which are complete for higher levels of the arithmetic hierarchy and which come from parts of mathematics outside of logic? By "complete for higher levels ...
- 665
5
votes
1
answer
321
views
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 ...
- 4,039
7
votes
1
answer
209
views
Martin's Conjecture and Arithmetically Pointed Trees
One consequence of Martin's conjecture is that if $f$ is a Turing degree invariant Borel function from $2^\omega$ to $2^\omega$ then there is a pointed perfect tree $T$ such that either $f$ is ...
- 4,039
3
votes
1
answer
200
views
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}\...
- 54.4k
4
votes
1
answer
158
views
What ordering does Ershov's unbounded path through $\mathcal{O}$ have?
There is this theorem I found in an article by Stephen, Yang and Yu due to Ershov(theorem 4.6 in http://maths.nju.edu.cn/~yuliang/ssyy.pdf) which says the following:
(a) There exists an unbounded path ...
- 1,483
13
votes
1
answer
320
views
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 ...
- 19.6k
6
votes
0
answers
230
views
Have you seen this variation of Medvedev reducibility?
In what follows, $A, B$ will denote subsets of $\mathbb{N}^{\mathbb{N}}$, i.e., sets of total functions $\mathbb{N} \to \mathbb{N}$ (maybe assume them to be inhabited to avoid any headaches about the ...
- 38.8k
7
votes
1
answer
192
views
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-...
- 503
5
votes
1
answer
333
views
When does the choice of the fundamental sequence matter?
I am quoting a paragraph from the second paper of Jockush and Shore on REA operators about generating the REA sets recursively via a system of notations:
"We can then associate $R$-sets with this ...
- 1,483
21
votes
1
answer
2k
views
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 ...
- 503
17
votes
0
answers
740
views
How to compute A263996?
Consider the sequence
$$a_n:=\min_{\substack{A\subseteq\mathbb{Z}\\|A|=n}}|(A+A)\cup (A\cdot A)|.$$
This is A263996 in OEIS. (Actually, they restrict to $\mathbb{N}$, but it makes little difference to ...
- 8,150
16
votes
1
answer
723
views
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 ...
- 6,523
3
votes
1
answer
176
views
Models of Second Order Arithmetic with non-standard length and all subsets of $\omega$?
Are there any non-standard models of RCA$_0$ such that every subset of $\omega$ appears as a restriction of a set in the second-order part to it's standard initial segment? In other words, does ...
- 4,039
6
votes
0
answers
130
views
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 ...
- 4,039
20
votes
3
answers
2k
views
Is the set of theorems of a PA + “PA is inconsistent” equivalent to the halting set?
In what follows, we let $T$ be a consistent, recursively axiomatizable theory that includes $\mathsf{PA}$ (Peano arithmetic).
Definition: Let us say that the theory $T$ is creative when the set of ...
- 38.8k
2
votes
1
answer
274
views
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 ...
- 1,855
8
votes
2
answers
586
views
Is there a filter containing every computable set or its complement that is not an ultrafilter?
As in the title: let $\mathfrak{F}$ be a filter on $\mathbb{N}$ such that, for every computable set $A\subseteq \mathbb{N}$, either $A\in\mathfrak{F}$ or $\mathbb{N}\setminus A\in\mathfrak{F}$; is $\...
- 223
8
votes
3
answers
789
views
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 ...
- 4,039
11
votes
1
answer
522
views
Computably constructing a set that is not in a free ultrafilter
Let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$. Given a computable function $f : \mathbb{N} \to \mathbb{N}$ such that $f(0) < f(1) < \cdots$, suppose we have a set $S(f)$ such that $|\{...
- 1,057
5
votes
0
answers
324
views
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
5
votes
3
answers
401
views
Regarding the realizability topos on the computable part of Kleene's second algebra
Let:
$\mathcal{K}_1$ be the first Kleene algebra, meaning $\mathbb{N}$ endowed with the partial operation $(p,n) \mapsto p\bullet n := \varphi_p(n)$ where $\varphi$ is the $p$-th partial computable ...
- 38.8k
3
votes
0
answers
267
views
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 ...
- 1,802
8
votes
0
answers
196
views
Lower bounds on positive $\sin$-polynomials
I am aware that the structure $(\mathbb{R},+,\cdot,<,\sin)$ is extremely wild. Indeed, since the natural numbers are definable in such structure, one can define, via first-order formulas without ...
- 457
2
votes
0
answers
186
views
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 ...
- 1,465
3
votes
0
answers
108
views
Is there a structure properly contained in the $\Pi^0_2$ singleton classes, richer than the $\alpha-REA$ sets?
I am going through Jockush and Shore's paper on transfinite pseudo-jump operators from '84 and in it's introduction it is mentioned that the transfinite pseudo-jump operators are properly contained in ...
- 1,483
18
votes
1
answer
851
views
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 ...
- 71
7
votes
1
answer
286
views
A theorem of Harrington on $\Pi^1_1$ paths through $\mathcal{O}$
I am currently reading a subsection of Chong & Yu(2010), specifically section 11.2 which is dedicated to a theorem of Harrington and I quote:
Theorem 11.2.1 For any ordinal $ \beta < \omega_{\...
- 1,483
2
votes
2
answers
577
views
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 ...
- 4,039
2
votes
0
answers
100
views
Determinacy and $\omega$ branching subtrees on which a functional is either partial or total
All trees discussed here are fully pruned subsets of $\omega^{< \omega}$ closed under substring and containing $\langle \rangle$ (the empty string).
Definition: T is completely $\omega$ branching ...
- 4,039
-2
votes
1
answer
569
views
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 ...
- 71
7
votes
1
answer
380
views
Problems on explicit embeddings of recursive groups into finitely presented groups
By this post I would like to ask the Community to kindly share information about existing problems on explicit embeddings of recursive groups into finitely presented groups.
Every recursive group (i.e....
- 199
6
votes
0
answers
268
views
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 ...
- 489
3
votes
1
answer
193
views
On cone avoidance
The following questions may be well-known in which case I would be happy to be directed to a published reference.
Q1. Let $d \in 2^{\omega}$ be noncomputable and $A \subseteq \omega$. For each $C \...
- 91
8
votes
1
answer
507
views
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 ...
- 38.8k
5
votes
1
answer
484
views
How to understand this inductive definition over a non well-founded set of ordinals
$\newcommand\ZF{\mathrm{ZF}}\newcommand\KPU{\mathrm{KPU}}$I am trying to read Barwise's Admissible Sets and Structures and I am just starting, so pardon if I may be missing something basic here. The ...
- 235
10
votes
0
answers
323
views
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. ...
- 3,756
10
votes
2
answers
823
views
Is it possible to add a new axiom schema to classical propositional logic?
Let IPL mean the intuitionist propositional calculus. One can add a great diversity of axiom schemas to obtain intermediate logics between IPL and CPL, where CPL is the classical propositional ...
- 391
7
votes
1
answer
342
views
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}^...
- 38.8k
4
votes
1
answer
266
views
Hard to relativize results in computablity theory
There is this paper by Khan and Miller titled "FORCING WITH BUSHY TREES"(here is a free source:https://people.math.wisc.edu/~jsmiller8/Papers/bushy_trees.pdf) in which the main results ...
- 1,483
9
votes
1
answer
587
views
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 $n \geq 5$, ...
