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Results tagged with computability-theory
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user 8628
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
1
answer
423
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Is the set of generalized Fermat triples computable?
Is $\;\big\{(a,b,c)\in\mathbb{N}^3: \big(\exists m,n,\ell \in (\mathbb{N}\setminus\{0,1,2\})\big): a^m + b^n= c^\ell\big\}\;$ computable?
- 48.9k
3
votes
1
answer
125
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The sequence of the power chromatic numbers $(\chi(G^n))_{n\in\mathbb{N}}$
For any finite, simple, undirected graphs $G, H$ we denote by $G\times H$ their categorical product. For any graph $G$ we let $G^1 = G$ and for $n\geq 1$ we let $G^{n+1} = G \times G^n$.
It is easy to …
- 48.9k
17
votes
1
answer
577
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Aperiodic monotile in $\mathbb{R}$
Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower d …
- 48.9k
2
votes
1
answer
144
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Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls
Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:
$(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s …
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2
votes
0
answers
73
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Is parquetability decidable?
Let $T\neq \emptyset$ be a finite subset of $\mathbb{Z}\times\mathbb{Z}$. We say that $\mathbb{Z}^2 = \mathbb{Z}\times\mathbb{Z}$ is parquettable by $T$ if there is a partition $\frak P$ of $\mathbb{Z …
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2
votes
1
answer
241
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"Rule 30" in the infinite setting
This question tries to get right what went wrong in an earlier question.
Let $\{0,1\}^\mathbb{Z}$ denote the set of all functions $x:\mathbb{Z}\to \{0,1\}$. Let $+$ denote addition modulo $2$ on $\{0, …
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1
vote
1
answer
515
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Possible finite periodicities of "Rule 150" in the infinite setting
"Rule 150" is a fascinating one-dimensional and simple cellular automaton giving raise to some quite chaotic behaviour. This is the starting point of this question.
Let $\{0,1\}^\mathbb{Z}$ denote the …
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0
votes
1
answer
151
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Realizability of metric matrices
We call an $n\times n$-matrix ${\bf A}\in \text{Mat}(n\times n, \mathbb{R})$ a metric matrix if
${\bf A}_{ii} = 0$ for all $i\in \{1,\ldots,n\}$,
${\bf A}_{ij} = {\bf A}_{ji}$ for all $i,j \in \{1,\l …
- 48.9k
0
votes
1
answer
115
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Non-isomorphic graphs with identical iterated degree matrix
If $G = (V, E)$ is a simple, undirected graph and $T \subseteq V$, let $$N(T) = \{v \in V: \{v, t\}\in E \text{ for some }t\in T\}.$$
Given $v\in V$ we let $N_0(v) = \{v\}$ and $N_{k+1}(v) = N_k(v) \c …
- 48.9k
4
votes
2
answers
277
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Checking for finite fibers in hash functions
Let $\{0,1\}^{<\omega}$ denote the collection of finite binary sequences. By a hash function we mean a computable map $$h: \{0,1\}^{<\omega} \to \{0,1\}^n$$ for some fixed $n\in\omega$. Define $\text{ …
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2
votes
0
answers
199
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Integers $n$ such that $n^d + (n+1)^d$ is never prime
Let us call an integer $n>0$ pure if for all integers $d>0$ we have that $n^d + (n+1)^d$ is not prime. Is the set of pure integers non-empty? Is it computable?
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4
votes
2
answers
467
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Is the collection of primitive recursive functions a lower set in the poset of computable fu...
If $g:\mathbb{N}\to\mathbb{N}$ is primitive recursive and $f:\mathbb{N}\to\mathbb{N}$ is computable such that $f(n) \leq g(n)$ for all $n\in \mathbb{N}$, does this imply that $f$ is primitive recursiv …
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34
votes
9
answers
5k
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Decision problems for which it is unknown whether they are decidable
In computability theory, what are examples of decision problems of which it is not known whether they are decidable?
0
votes
1
answer
79
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Normal $0,1$-sequence with infinitely many frequent finite substrings
Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$.
G …
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4
votes
1
answer
160
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Is sum-balanceability computable?
Let $\mathbb{N}$ denote the set of positive integers, and let $G=(V,E)$ be a finite simple, undirected graph. Given $f:V\to \mathbb{Z}$ we define the neighborhood
sum function $\mathrm{nsum}_f:V\to\ma …
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