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.
972
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Does this hierarchy of fragments of $I \Sigma_1$ collapse?
Does anyone know whether the following hierarchy of fragments of
$\mathrm{I} \Sigma_1$ (or rather
$\mathrm{I} \Pi_1$) collapses or not?
Let $\Sigma^b_n$ denote formulas in the language of arithmetic ...
7
votes
2
answers
600
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Ideals generated by Turing independent sets
Recall that $X \subseteq 2^{\omega}$ is Turing independent if no $y \in X$ is computable from the Turing join of any finite subset of $X \setminus \{y\}$.
Question 1. Can we construct a Turing ...
2
votes
0
answers
66
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Reverse mathematics on lightface $\Pi^1_1$-uniformization for unary relation
It is known that the following form of $\Pi^1_1$-uniformization is equivalent to $\Pi^1_1$-Comprehension over $\mathsf{ATR}_0$ (cf. VI.2.6 of Simpson's book) :
(Kondo's uniformization theorem) For ...
4
votes
1
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Does $A \leq_{\alpha} B$ imply $A \leq_{\beta} B$ for admissible ordinals $\alpha < \beta$?
My very superficial intuition of $\alpha$-recursion is that it replaces the tape in a Turing machine with $L_{\alpha}$ for some admissible $\alpha$, so that $L_{\alpha}$ functions as working memory. ...
3
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0
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156
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Are all "reasonable" Gödel encodings isomorphic in some sense?
It is clear that many different Gödel numberings can work in Gödel's proof. Yet for the proof one just needs a few properties of how the numberings of related sentences are related, and I'm wondering ...
1
vote
1
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102
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Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
3
votes
0
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133
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Lindström's theorem part 2 for non-relativizing logics
By "logic" I mean the definition gotten by removing the relativization property from "regular logic" — see e.g. Ebbinghaus/Flum/Thomas — and adding the condition that for every ...
1
vote
0
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62
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EF-games with scrambling
This question is motivated both by the notion of zero-knowledge proofs and by general curiosity about versions of the infinitely-long Ehrenfeucht-Fraisse game which don't trivialize (= Duplicator win ...
4
votes
1
answer
198
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Is every compact, sober, second-countable space the image of $2^\omega$?
As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$?
As a further bonus, can we strengthen "image" to "quotient"?
My motivation for ...
5
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0
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130
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What is known about propositional realizability for the second Kleene algebra and related PCAs?
Short version: Various things are known about realizability of propositional formulas for Kleene's “first algebra” (i.e., $\mathbb{N}$), like examples of realizable but unprovable formulas, and some ...
4
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1
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228
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What is the theory of statements with a provably *bounded* realizer (according to PA)?
$\let\T\mathrm\def\kr{\mathrel{\mathbf r}}$This is a follow up to Kleene realizability in Peano arithmetic.
We can summarize the results from Emil Jeřábek's answer as follows:
\begin{gather*}
T_1 = \{ ...
3
votes
2
answers
222
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Question regarding $W$ as not hyperarithmetic
Consider the indexes of all ordinary programs generating functions from $\mathbb{N}^2$ to $\{0,1\}$. If we let $W$ be the set of exactly of all those indexes $e$ such that $\phi_e$ computes a total ...
0
votes
1
answer
115
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Integer quadratic representation subject to discriminant minimization algorithm
Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...
8
votes
1
answer
994
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Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?
Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...
33
votes
3
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Using Busy Beavers to prove conjectures
I've been pondering some stuff on Shtetl Optimized where Yedidia and Aaronson construct Turing machines that will only halt if (e.g.) the Riemann Hypothesis is false, or Goldbach's conjecture is false....
2
votes
0
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118
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Notes on Lachlan's monster
I have been trying to look for a reference I have seen in a paper called "R. Soare, Notes on Lachlan’s Monster Theorem" without success. I was wondering if anyone had a digital copy of them ...
14
votes
2
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Church–Turing thesis for higher order functions
The Church–Turing thesis states that, simply speaking, any reasonable definition of "effectively computable functions" $\mathbb{N} \to \mathbb N$ agrees with the definition using Turing ...
6
votes
0
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205
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What are these non-classical versions of ZFC defined by realizability?
See Kleene realizability in Peano arithmetic for a similar question, but about PA instead of ZFC.
In the context of constructive set theory, consider two ways of defining realizability.
The first is $\...
6
votes
1
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274
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Need help in trying to understand an argument by V. A. Yankov on the nonrealizability of Scott's axiom
(This is really long because I give a lot of context, but you can skip right to the end where the excerpt I'm trying to make sense of is copied and translated.)
Background: I'm trying to understand ...
3
votes
1
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265
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Root finding algorithm for an analytic function
Given an analytic function $f(x)$. What is the best algorithm to find roots on the interval $[a,b]$ inside the radius of convergence> What is its complexity with respect to the length of input of ...
3
votes
1
answer
106
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Kleene normal form theorem for r.e. relations proven in arithmetical theories
After reading the relevant chapters of Classical Recursion Theory (freely available from here), I have the following questions concerning Theorem II.1.10 (Normal form theorem) and Theorem IV.1.9 (...
3
votes
1
answer
264
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When is an upper bound on the longest irreducible program outputting something computable?
Given some way to to encode programs to strings with a finite alphabet, which we assume has a computable translation to/from Turing machines, a program is irreducible if no subsequence of it has the ...
4
votes
0
answers
188
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Computational complexity of zeros of an analytic function
The work of Friedman and Ko, page 342, Corollary 4.3.1
states that all zeros of analytic polynomial time computable function are polynomial time computable, but for me that is not clear how it could ...
14
votes
1
answer
472
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How exactly are realizability and the Curry-Howard correspondence related?
Consider, on the one hand:
the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...
7
votes
1
answer
307
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Proving finiteness in Reverse Mathematics
In (second-order) Reverse Mathematics, a (code for an) open set $U\subset \mathbb{R}$ is given by two sequences of rationals $(a_n)_{n \in \mathbb{N}}, (b_n)_{n \in \mathbb{N}}$. The idea is that $U$ ...
4
votes
3
answers
363
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Hyperarithmetically least elements in $\Pi^1_1$ sets
My question is: Do we have a hyperarithmetically $\le_H$-least real in any $\Pi^1_1$ set? That is
Question. Suppose that $A$ is a non-empty $\Pi^1_1$ set. Then can we find a real $a\in A$ such that $...
5
votes
0
answers
179
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Complexity implications on computability
Are there any known links between complexity theory and computability theory by which I mean non-trivial theorems of the form: If NP $\neq$ co-NP then there is no strong minimal pair of r.e. sets or ...
3
votes
0
answers
90
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Comparing computable structures via Kleene and Skolem
Below, by "structure" I mean "computable structure in a finite language with domain $\omega$," and by "sentence" I mean "finitary first-order sentence containing no ...
0
votes
1
answer
95
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Is there a canonical mapping between countable transfinite ordinals and $\omega$? What about recursive ordinals?
Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built.
I was wondering if there is a ...
1
vote
0
answers
110
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Sudden drop in complexity class due to the more general correlations
Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum''...
1
vote
1
answer
85
views
How large can a subset of computable reals, whose comparison function is computable, grow?
How large can a subset of computable reals, whose comparison function is computable, grow?
For example, rational numbers are computable reals, and its comparison function is computable. As another ...
0
votes
0
answers
166
views
Can a model of "true computation" exist? What would be its consequences?
Analogous to the model of True Arithmetic, the model of "True Computation" is defined to be the set of all true first-order statements about Turing machines i.e. answers to elementary ...
4
votes
0
answers
106
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Decidability of whether two polynomial bijections generate a free group
I am wondering about the decidability of the following question:
Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
1
vote
0
answers
55
views
Are the lower elementary functions closed under limited recursion?
The lower elementary functions (also called Skolem elementary functions) are functions generated from the successor, modified subtraction, projection functions by the operations of composition and ...
8
votes
2
answers
548
views
Turing degrees of sets separating two computably inseparable sets (theorems and antitheorems)
Let $A\subseteq\mathbb{N}$ be the set of Gödel codes of theorems of Peano arithmetic, and $B\subseteq\mathbb{N}$ be the set of codes of antitheorems (i.e, refutable statements, statements whose ...
10
votes
2
answers
426
views
Is the set of permissible numbers of models of various cardinalities computable?
This question arose in the comments to this question.
Let $X$ be the set of pairs $(m,k)$ such that there is some (consistent complete countable first-order) theory $T$ with exactly $m$ models of size ...
2
votes
1
answer
120
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Splitting $\Pi^0_2$ Singletons?
Given a (non-computable) $\Pi^0_2$ singleton $Y$ are there Turing incomparable $\Pi^0_2$ singletons $X_0, X_1$ with $Y \equiv_T X_0 \oplus X_1$?
What about the same question for arithmetic ...
4
votes
1
answer
216
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Can a halting oracle determine if a Turing machine is an ordinal?
For the sake of clarity, I am regarding a computable relation on $\mathbb{N}$ as a $2$-symbol ($0$ and $1$) Turing machine $T$ which halts on any initial binary string (which are interpreted as some ...
6
votes
1
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272
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What is the power of the “anti-halting” oracle?
Let me first ask the question, and then, as it may seem a bit cryptic, explain how it comes up (and whence the “anti-halting oracle” in the title):
Notations: we write $\langle m,n\rangle$ for a ...
3
votes
1
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120
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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 ...
3
votes
1
answer
124
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A $\Pi^0_1$ class of $\alpha$-subgenerics for all $\alpha < w_1^{CK}$
In Harrington's mimeographed notes (see here) solving McLaughlin's conjecture he builds reals $f \in \omega^\omega$ which have the property of being $\alpha$-subgeneric defined as follows. He does ...
2
votes
1
answer
128
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Harrington's notes on McLaughlin/Arithmetically incomparable singletons
At one point I had copies of the handwritten notes Leo created about the McLaughlin conjecture and I know a similar set of notes exist titled Arithmetically incomparable arithmetic singletons. I've ...
4
votes
0
answers
167
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Status of Problems in 102 problems in mathematical logic
Is there any location that records the current status of the problems in 102 problems in mathematical logic? Or, better yet, serves as a status board for open problems in mathematical logic? ...
7
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0
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102
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How tightly are decidability and "induction-completeness" linked?
It is known that there are a number of expansions of the structure $\mathfrak{N}:=(\mathbb{N};+)$ which are decidable (= have computable theories); one such example is the expansion by a predicate ...
6
votes
0
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121
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An analogue of Scott sentences in the (mostly) computable realm?
Below, "structure" means "computable structure in a computable language." In particular, we do distinguish between isomorphic copies of the same structure.
Let $\mathcal{L}_{\...
4
votes
1
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164
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Let $\pi$ be a $ℍ𝑌𝑃_𝔐$-recursive projection of $ℍ𝑌𝑃_𝔐$ into 𝔐. What does $ℍ𝑌𝑃_{(𝔐, Domain(\pi))}$ contain?
Let the structure $\mathfrak{A} = (A, R_1, ..., R_n)$ be strongly acceptable iff $\mathfrak{A}$ is an acceptable structure (in the sense of Moschovakis' Elementary Induction on Abstract Structures), $\...
2
votes
1
answer
116
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The counterpart of productive set with polynomial computational complexity
For definition of productive set, see here and here, that is defined with computability, or computable function. Restricting computable function as function of polynomial computational complexity, is ...
5
votes
0
answers
205
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Is there a transfinite version of Post's Theorem?
Let $\emptyset^{(n)}$ denote the $n$th Turing jump of the empty set. Post's theorem states:
A set $B$ is $\Sigma^0_{n+1}$ if and only if $B$ is recursively enumerable by an oracle Turing machine with ...
5
votes
1
answer
203
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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 (...
5
votes
1
answer
99
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Understanding the definition of a (computably / continuously) “transparent” function
The following definitions of a “transparent function” are essentially taken from references [1] (where it is called a “jump operator”), [2] and [3], except that the variation “primitively recursively ...