# Tagged Questions

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### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

**2**

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**1**answer

49 views

### A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...

**0**

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**0**answers

72 views

### Linkage between homotopy equivalence and identification of algorithms

I vaguely recall that someone says there is linkage between homotopy equivalence and identification of algorithms which may be isomorphic or morphism or something like that,the algorithm may be ...

**2**

votes

**3**answers

167 views

### How to define the input of computable function or Turing machine over real numbers

Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...

**7**

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**4**answers

523 views

### (reference request) Chaitin's constant is incompressible

I've been looking for a full, detailed proof that Chaitin's constant is incompressible, i.e. there is a universal constant $c$ such that every program writing first $n$ digits of $\Omega$ has length ...

**3**

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**3**answers

735 views

### An established proof in Wang Tile which I doubt

When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf
I could not ...

**7**

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**1**answer

350 views

### Demuth's theorem in set theory

I am quite sure the following fact must have been known for set theorists, though I could not find it anywhere.
If $r$ is random over $L$ and $x\in L[r]\setminus L$, then there must be some real ...

**6**

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**2**answers

222 views

### Only admissibles start gaps in clockable ordinals

This is a question about ITTM model introduced by Hamkins et al. In this paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I ...

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103 views

### Graph theoretical representation of Wang Tile

We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...

**5**

votes

**1**answer

116 views

### Attribution of an equivalence of the existence of omega-models of RCA0

There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of ...

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90 views

### Comparing two non-deterministic Turing equivalents as basis for Logic, request for references

I am designing a logic, that is simpler than FOL + PA. And I like to know if there already exists something in this direction.
First of all a non-deterministic Turing equivalent is defined by ...

**3**

votes

**1**answer

199 views

### Reverse mathematics, Ramsey theorem and mass problem

If we look at reverse mathematics statements as mass problems, considering the class of solutions of an instance, it is known that Weak König's lemma has a maximal instance in the sense that there is ...

**8**

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**2**answers

347 views

### Where should I learn about Kolmogorov complexity of overlapping substrings?

I would like to know more about the relationship between the Kolmogorov complexity of a string and that of its substrings. The relation that up to an additive constant, $K(x,y) = K(x) + K(y\ |\ x, ...

**9**

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**2**answers

386 views

### Reverse mathematics below RCA

I'm sure this is a fairly basic question, but I can't seem to find a solid answer:
My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to ...

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**0**answers

799 views

### Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his ...

**18**

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**1**answer

775 views

### Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal

Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were ...

**5**

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**1**answer

210 views

### Notion of independence of Turing degrees

I've been thinking for a while about different ways two Turing degrees might be "independent" of each other (from the point of view of computability theory). The simplest such notion would be to say ...

**6**

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**1**answer

269 views

### Status of the Isomorphism problem for automatic groups?

I only ask because I don't know how to look for the answer.

**3**

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**0**answers

143 views

### Alternate proof of van de Wiele's theorem in E-recursion

Hello, all
I'm currently trying to understand $E$-recursion theory, which is a generalization of classical recursion theory to arbitrary sets. One of the difficulties I'm having with understanding ...

**4**

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**1**answer

210 views

### Reference for a “recursive” fragment of infinitary logic?

Does anyone know of any texts or papers out there concerning properties of the fragment of $ \mathcal{L}_{\omega_1, \omega} $ in which the only admissable infinite conjunctions are those which are ...

**4**

votes

**1**answer

445 views

### Infinite monkeys computing … triangle area?

I wonder if it is possible to specialize the question:
(a) What is the probability that a random Turing Machine program
will halt?, to: (b) What is the probability that a random Turing Machine
...

**12**

votes

**1**answer

615 views

### Are wild problems related to undecidable ones?

In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...

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votes

**1**answer

379 views

### Enumerating levels of Grzegorczyk-hierarchy

Grzegorczyk has divided the class of primitive recursive functions to Grzegorczyk-hierarchy by their rate of growth. In this hierarchy $E_i\subset E_{i+1}$ and the subset-relation is strict. Also ...

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1k views

### Decidable but nonrecursive sets

Until recently, I believed that recursive=decidable,
subscribing to this Wikipedia quote:
"In computability theory, a set is decidable, computable, or recursive if there
is an algorithm that ...

**12**

votes

**3**answers

908 views

### Has there ever been a weaker Church-like thesis?

Background. The Church-Turing thesis, in one of its many equivalent formulations, states that the intuitively computable arithmetical functions are exactly those computed by Turing machines.
...

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875 views

### Is there a “primitive-recursively enumerable” set whose complement is not such?

Call a subset of $\mathbb{N}$ primitive-recursively enumerable (p-r.e.) if it is empty or an image of a primitive recursive function. I feel like a lot must be known about the poset of such sets ...

**11**

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**3**answers

741 views

### Alive dynamical system

Intuitively, one can say that a dynamical system is alive if one can build a universal Turing machine inside.
So, Conway's Game of Life is alive and shift space should be dead.
I fail to make this ...