Questions tagged [computability]
Questions about Turing computability and recursion theory, including the halting problem and other unsolvable problems. Questions about the resources required to solving particular problems should be tagged (computational-complexity).
2,547 questions
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Can countably many giraffes guess the color of their own scarf correctly if each may be mistaken about the color of finitely many scarves?
This question is based on the question Is it possible to formulate the axiom of choice as the existence of a survival strategy? (MathOverflow).
Consider the following "computable giraffes, lion &...
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answer
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Is the Collatz map algorithmically decidable?
Question: Has it been proven that the following decision problem is algorithmically decidable?
$$
P_\text{Collatz} := \text{Given $n \in \mathbb{N}_+$, does the Collatz-Iteration of $n$ eventually ...
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2
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Recursively enumerable sets and the natural numbers
The standard definition of recursive enumerability is as follows: a set of natural numbers is recursively enumerable if there exists an algorithm which halts precisely on inputs which are elements of ...
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0
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93
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Can probability theory be made fully computable?
I’ve been reading about objective Bayesian theories lately and came upon the concept of universal priors and specifically, the Solomonoff prior. This seemed to answer my initial query about whether a ...
5
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1
answer
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Upward closure of the PA degrees
In the book I read, PA degrees are defined as containing a completion of Peano arithmetic.
I'm searching for a proof that PA degrees are closed upwards (which is non-trivial with this definition). ...
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5
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1
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Theorem 3.4 of Soare
I am reading Robert I. Soare's "Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets" for a directed reading course. I am having trouble ...
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0
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Proving Inf is 1-reducible to Tot
I am currently working on a proof that $\mathrm{Inf}\leq_1\mathrm{Tot}$, where
$$\mathrm{Inf}=\{e:\mathrm{Dom}\,\varphi_e \text{ is infinite}\} \quad \mathrm{Tot}=\{e:\varphi_e\text{ is total}\}$$
(...
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1
answer
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Is function defined by infinite cases partial recursive?
I am trying to prove that a certain function is partial recursive. Suppose we have fixed primitive recursive $g:\mathbb{N} \rightarrow \mathbb{N}$ and for each $d$ let $f_d:\mathbb{N}\rightarrow \...
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Where is the relativized uniform complementor on r.e. indices stated explicitly (index-operator form)—existence for $\emptyset'$ and any minimality?
Let $\langle W_e : e\in\mathbb{N}\rangle$ be the standard effective enumeration of recursively enumerable (r.e.) sets, where
$$
n\in W_e \;\Longleftrightarrow\; \exists s\;\big(\varphi_e(n)\ \text{...
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1
answer
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Can superposition alone generate the Kalmár elementary function $x^y$ from $⟨x + y, x \bmod y, 2^x⟩$?
In Mihai Prunescu, Lorenzo Sauras-Altuzarra, and Joseph M. Shunia (2025), A Minimal Substitution Basis for the Kalmar Elementary Functions, the authors define a minimal generating set for the Kalmár ...
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Computational content of axiom of countable choice in HoTT
The question is possibly related to this, but I don’t find a satisfactory answer there.
Let $X$ be a set and $P(n, x)$ be a mere proposition for all $n: \mathbb{N}$ and $x: X$. The axiom of countable ...
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vote
1
answer
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Recursive functions are $\Sigma_1$ in PA? [duplicate]
I am working with recursive functions on $\omega$. It is well known that these functions are representable, in the sense of the following definition:
A function $f:\omega\to\omega$ is representable in ...
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Proof without diagonalization that Kleene's $\mathcal{O}$ is not computable
I wanted to prove that Kleene's $\mathcal{O}$ is not computable without using the diagonalization trick, and was wondering if the following proof would work?
Note: ordering = well-ordered set.
Suppose ...
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When are $\ell$-adic betti numbers of varieties computable
Let $X$ be a proper variety over a global field $k$. Then we can find a model $\mathfrak{X}$ over a Dedekind domain $R$ of finite type. Every special fiber of $\mathfrak{X}$ is over a finite field, so ...
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What is the Turing degree of an infinite subset of the Halting problem?
I am not a professional mathematician, so please forgive me for my inaccurate description.
$H$ is a sequence that expands the Chaitin's constant into binary form.
Program $A$ takes $k$ as input and ...
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59
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Confused about a quadratic Julia set, Jordan curve and "noncomputable"
I was reading here :
https://mathworld.wolfram.com/JuliaSet.html
And it said, if I am not mistaken :
Consider
$$z_{n+1} = z_n^2 + c$$
for small $c$, then the Julia set $J_c$ is also a Jordan curve, ...
- 18.4k
1
vote
0
answers
117
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Is the solution to the Halting Problem non-finitary?
I find Hilbert's appeal to finitism (restriction to finitary arguments) appropriate for metamathematical reasoning about formal systems. The proofs to Gödel's Incompleteness Theorems that can be ...
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2
answers
136
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Is there a computable sequence of Diophantine equations whose minimal height of integer solutions grows faster than any computable function?
This question is inspired by the busy beaver function and MRDP theorem. It is known that the busy beaver function grows faster than any computable function, that is,
$$
\lim_{n \to \infty} \frac{\...
- 650
3
votes
1
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On the definition of $\Pi^0_n$-classes
Apparently according to Downey&Hirschfeldt, there are $\Pi^0_2$ classes which are not $\Pi^0_1$ relative to $0'$ (since they state that there are weakly 1-random relative to $0'$ sets which are ...
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0
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How to informally understand `hash reducibility`?
I was reading a paper on a certain reducibilities in Polynomial classes. And I stumbled across this definition: For sets $A,B$ computable in polynomial time, we write $A \leq^{\#} B$ via a polynomial $...
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49
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Accessible introductions to Type 2 computability
Is there a good introduction to Type-2 computability (including its connections to effective topological spaces) that is more accessible than Weihrauch's Computable Analysis, An Introduction? I find ...
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1
answer
75
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One-to-one reduction of $K_1$ to $K$
Let $K_1 := \left\{ x : W_x \neq \emptyset \right\}$ with $W_x := \operatorname{Dom}(\varphi_x)$. Let $K = \{x : \varphi_x(x) \text{ halts}\}$.
Intuitively, $K_1$ shuold be many-one reducible to $K$, ...
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2
votes
0
answers
70
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Undecidability of halting problem and pattern matching
Let $K$ be the set of indexes such that $\varphi_k(k)$ converges for any $k \in K$. In other words, $K$ is the set of "programs" (indexes of Turing machines) that halt when taking their own ...
- 3,595
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2
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116
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$\Pi^1_1$ Uniformization by HYP (when total)
I think I'm being dumb but suppose that $P(f, g)$ for $f, g \in \omega^\omega$ is $\Pi^1_1$ and that $\forall f \exists g P(f, g)$. It should be true that for every $f$ there is a $g$ hyperarithmetic ...
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2
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313
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Since the Discrete Logarithm has an only algorithmic structure, is it possible that the limit point will not always be computable when $n\to\infty$?
I have a problem that I am completely stuck on.
Let
$$f_1(n)^{\rm DL_1(n)}\equiv g_1(n) \!\!\!\!\!\pmod {h_1(n)}$$
and
$$f_2(n)^{\rm DL_2(n)}\equiv g_2(n) \!\!\!\!\!\pmod {h_2(n)}$$
The functions $f_i(...
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1
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101
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Decidability of there is a word that is accepted by a TM in fewer steps than the length of the word
I was presented with this question:
L = {<$M$> | There is a word $x$ that is accepted by $M$ in fewer than $|x|$ steps}
I think that this is not decidable, and my immediate thought was to make a ...
8
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2
answers
384
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When constructing primitive recursive functions, can you modify the parameter?
Let $g(a), h(a,n,b), p(a)$ be primitive recursive functions. Consider the function $\phi(a,n)$ defined recursively by
$$
\phi(a,0) = g(a) \\
\phi(a,sn) = h(a,n,\phi(p(a), n))
$$
Question: Is $\phi$ ...
- 7,694
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1
answer
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notion of category of computable functions
I was learning about one-way functions in cryptography and it occurred that it might make sense to consider a category of computable functions. Is this a thing?
For example, we could define a category ...
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Clarification on the meaning of "any program" in Sipser's explanation of the Recursion Theorem
I'm reading Michael Sipser's Introduction to the Theory of Computation, and I came across the following statement in the section on the Recursion Theorem:
"The recursion theorem provides the ...
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2
answers
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Is there a "true" value of BB(745)?
I was reading this reddit thread and I got confused by one part. I always thought that there is always a "true" value of BB(n), even though it might not be provable or findable.
There is a ...
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Is it possible to express $\sqrt2$ as a convergent infinite summation using only integer arithmetic and nested finite sums? [closed]
I’m exploring a restricted arithmetic framework and have arrived at the following conjecture:
Tyler’s 2025 Conjecture:
There exists no infinite summation of the form
$$\sum_{k=0}^{\infty} f(k)$$
that ...
- 87
1
vote
1
answer
98
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Using the construction in Kleene's recursion theorem to build a quine
Kleene's recursion theorem guarantees that, for every computable function $f$, there is a program $e$ such that $e$ and $f(e)$ compute the same function, or equivalently, $\varphi_e(x) \simeq \varphi_{...
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0
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66
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Can paraconsistent logic be used to define classes of total recursive functions?
I have studied subrecursive hierarchies for fun. Perhaps the most known examples of them being Peano Arithmetic and its fragments, and Grzegorczyk-hierarchy within primitive recursive functions. The ...
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53
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Is there a machine / grammar that exactly computes *total* recursive programs? [duplicate]
I was curious about the implications of the Halting problem for writing real world programs. The main question being "Is there a system/machine/grammar that expresses all halting programs while ...
2
votes
1
answer
93
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A Weakly 2-Random Set that is both Hyperimmune and Generalized Low
In a 2007 paper, Nies, Montalbán and Lewis build a weakly 2-random set that is not generalized low, hence separating weak 2-randomness from 2-randomness. This is done by constructing a 1-random ...
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1
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generalization of immune set
A set is called immune if it is infinite and contains no infinite $\Sigma_1$ i.e. r.e. set.
Let's call a set $k$-immune if it is infinite but contains no infinite $\Sigma_k$ set. The question is does $...
2
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1
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a property between compactness and non-compactness in logic
Background and motivation:
Without specifying exactly what "a logic" is (since any one definition would be unnecessarily restrictive for the purposes of this question), we can say that a ...
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1
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98
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Neither Recursively Enumerable nor co-Recursively Enumerable Set?
A set $S \subset \mathbb{N}$ is said to be recursively enumerable (RE) if there is a existential formula $(x \in S) \iff \exists x_1 \exists x_2 ... \exists x_n P(x,x_1,x_2,...x_n)$ and we assume ...
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What does it mean to "determine" the time constant for first passage percolation?
A fundamental question in FPP is to determine the time constant for some non-trivial distribution on $\mathbb{Z}^2$ (https://arxiv.org/pdf/1511.03262, question 1). We would all be very happy if there ...
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Least fixpoint of monotone maps in alternating turing machines
I am currently reading Dexter Cozen's book on the theory of computation, and at chapter 6 contains materials about fix point, prefix point operators, Knaster-Tarski theorem as well as other things, ...
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3
votes
1
answer
108
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Computationally weak set with computationally strong jump
There exists a set $A \subseteq \mathbb{N}$ with the following properties:
$A \ngeq_T \emptyset'$,
$A' \geq_T \emptyset''$.
(Take, for example, any high set that is not $\Delta^0_2$-complete.) This ...
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3
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485
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Is there a concept or tool that measures the "density" of infinite sets, that includes countably infinitely many elements?
It is difficult to ask questions when we do not know some mathematical concepts. Since I do not know the concepts, I will ask my question by giving an example.
Let me write some examples.
Define $S$ ...
- 672
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1
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75
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Clarity on proof insolubility of decision problem for logical implication
I'm currently reading Computability and Logic by Boolos, Burgess, and Jeffrey — specifically Chapter 11, pages 126–134 — where they give two proofs showing that the decision problem for logical ...
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11
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4
answers
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Why doesn't Cantor's diagonal argument prove only that the reals are not recursively enumerable?
In this answer to the question Why are the total functions not enumerable? the following argument is made:
Because of diagonalization. If $(f_e: e \in \mathbb{N})$ was a computable enumeration of all ...
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1
vote
1
answer
108
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Eliminate bounded quantifier in arithmetic by adding finitely many rudimentary functions
In set theory, we can add function symbols of Godel functions and a constant symbol of the empty set, and extend the theory with their rudimentary (that is, $\Delta_0$) definitions as axioms. The ...
- 147
1
vote
0
answers
22
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Reducing the number of special symbols in mapping reduction of Modded PCP to post's correspondence problem
Background: there exists a proof for a mapping reduction of modified PCP(MPCP) to PCP (in MPCP you always start the answer sequence with the first domino). This proof (can be found online, from ...
- 11
1
vote
0
answers
125
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Is the set of primitive recursive functions recursive?
This question has probably been asked elsewhere, but I cannot for the life of me find the answer. I understand as follows: set of primitive recursive functions is not enumerable by some primitive ...
- 972
3
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0
answers
104
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Names of Computable Approximation Conditions?
I came up with some conditions for dense subsets of topological sets and I assume the conditions have names already but I have no idea what they are.
Suppose I have a topological set $X$ and a subset $...
- 426
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votes
1
answer
117
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Missing Multiplier
For this plumbing job, you are tasked to create a water pipe connection which MULTIPLIES the water volume.
There are 3 pipes pumping water into the site: One pipe is connected to the city water supply,...
5
votes
2
answers
509
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Entire function that grows faster than any iteration of exponentials
Define the class $\mathcal{F}$ of entire functions satisfying, for some integer $n$: $$\limsup_{r\rightarrow \infty}\frac{\log _{n}M(r)}{\log r}<\infty,$$where $M(r)$ is the maximum of the function ...
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