**3**

votes

**0**answers

133 views

### Recursively Pointed Sacks Forcing and Preserving $\omega_1$

Let $\mathbb{P}$ denote recursively pointed Sacks forcing. This is forcing with recursively pointed perfect trees ordered by inclusion. A tree $T \subseteq {}^{<\omega}2$ is recursively pointed if ...

**4**

votes

**0**answers

70 views

### Upper bound on ranks of well-founded trees in $SKI\Omega$ calculus

All ideas explained below are due to A.P.Goucher, and defined here.
First of all, $SKI\Omega$ calculus is an extension of standard SKI calculus, with additional type of combinator, called oracle ...

**9**

votes

**1**answer

301 views

### Sets computable from enough hints

Is there a non-computable set $X\subset\omega$ such that, for some $Y\subset\omega$, any infinite subset or cosubset (=subset of the complement) of $Y$ computes $X$?
More generally, call a set $X$ ...

**3**

votes

**1**answer

204 views

### On fast-growing hierarchy

Is there exists a recursively enumerable set of computable total fast-growing functions $(\mathbb N \rightarrow \mathbb N)$ such, that this set has no upper boundary in the set of all such functions ...

**2**

votes

**2**answers

81 views

### Background for Kierstead terms

I was looking at some slides of John Longley's here, where he mentions "the Kierstead functional"
$$\lambda f.f(\lambda x.f(\lambda y.x)) \ ,$$
(where $f$ should be of type $2$, and $x,y$ of ground ...

**1**

vote

**1**answer

93 views

### Total conditional complexity

By $C(|)$ denote conditional complexity.
By $CT(|)$ denote total conditional complexity.
For every n there exist two strings $x$ and $y$ of length $n$ such that $C(x|y) = O(1)$
but $CT(x|y) \ge n $.
...

**-2**

votes

**1**answer

81 views

### What are the formula of representation of quasicrystals and the law or mechanism of the formation [closed]

I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics ...

**1**

vote

**0**answers

93 views

### reference on aperiodicity and cluster [closed]

From this image:
I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)

**-3**

votes

**1**answer

260 views

### When are two algorithms essentially the same? [closed]

Inspired by Blass/Dershowitz/Gurevich's paper When are two algorithms the same? (which was referenced in another context here) I tried to boil down the question to the following situation:
Consider ...

**2**

votes

**0**answers

97 views

### 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}= ...

**1**

vote

**2**answers

268 views

### Rationale behind an requirement on Turing machines

Hopcroft and Ullman's definition of a Turing machine seems to be standard. This definition defines a Turing machine to be a 7-tupel $T = \langle Q,\Gamma,b,\Sigma,\delta,q_0,F \rangle$ obeying some ...

**1**

vote

**1**answer

51 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 ...

**5**

votes

**1**answer

110 views

### Aperiodic set of corner Wang Tile [closed]

There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with ...

**0**

votes

**1**answer

125 views

### Definability of arithmetic functions and relations

Motivation: Many "weak" arithmetic functions and/or relations ("relations" for short) are equivalent with relations explicitly definable by relations which were recursively defined by them beforehand ...

**0**

votes

**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

173 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**

votes

**5**answers

625 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 ...

**2**

votes

**1**answer

159 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises ...

**2**

votes

**0**answers

146 views

### Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

**3**

votes

**1**answer

55 views

### Are lightface \Delta-1-1 classes of reals describable with hyperarthmetic formulae?

I'd appreciate if someone could check my reasoning. Suppose $S$ is a lightface $\Delta^1_1$ class of reals. I want to argue that there is a computable $\Delta^0_\alpha$ formula $\phi(Y)$, for ...

**8**

votes

**1**answer

256 views

### Infinite decreasing sequence by the Turing jump

I saw in Wikipedia the existence of an infinite sequence of Turing degrees $\bf{a_0}, \bf{a_1}, \dots$ such that $\bf{a}_{i+1}' \leq_T \bf{a}_i$
where $\bf{a}_{i+1}'$ is the Turing jump of $\bf{a}_i$.
...

**3**

votes

**1**answer

450 views

### Some types of diophantine equations and their decidability

The MDRP theorem – which answers Hilbert's tenth problem in the negative – says:
There is no algorithm for determining whether an
arbitrary diophantine equation has a solution.
In ...

**8**

votes

**2**answers

220 views

### Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...

**-3**

votes

**1**answer

127 views

### Randomness about coefficients of series

$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary.
Now,the question :if ...

**10**

votes

**1**answer

432 views

### Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined.
In "Turing ...

**12**

votes

**1**answer

380 views

### Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?

Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...

**10**

votes

**1**answer

296 views

### Higher recursion theory and reverse mathematics: What is to $\Pi^1_1-CA_0$ as $RCA_0$ is to $ACA_0$?

There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1_1$" - indeed, this is the starting point of metarecursion theory, and $\alpha$-recursion theory in ...

**4**

votes

**1**answer

316 views

### Uncountable time Turing machines

When writing with a friend of mine today we came up with idea of extending ITTM concept of Hamkins and Kidder. First of all, I am familiar with one of Hamkins and Lewis results saying that every ...

**4**

votes

**1**answer

275 views

### Essential incompleteness via diophantine formulas?

Work in the first order language of number theory, consisting of the symbols $\mathbf{0}$, $\mathbf{S}$, $\boldsymbol{+}$, and $\boldsymbol{\cdot}$, and let $Q$ denote Robinson's arithmetic.
By a ...

**2**

votes

**3**answers

740 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**

votes

**1**answer

352 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 ...

**7**

votes

**0**answers

179 views

### Infinite time game of life

Today in a talk with a friend of mine I had an idea of extending cellular automatons to transfinite working time. I know it has already been considered, but, as far as I can tell, GoL extended to ...

**4**

votes

**1**answer

178 views

### Necessity of omega-models in second order arithmetic

Are there examples of independence results over subsystems of true second order arithmetic that cannot be established using omega-models? To rule out trivial examples, let us assume that the base ...

**1**

vote

**1**answer

35 views

### How to select a subset of points from a universal to minimize the distance from outside to inside?

Here is the detailed problem.
I have a set of N points in K-dimension space, called U, and I want select M points of them, called S. For each point p in U, we define the distance from p to S as
$$ ...

**6**

votes

**2**answers

225 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 ...

**5**

votes

**1**answer

186 views

### Let Abit$(x,y,n)$ be the $n$th bit of Ack$(x,y)$ (the Ackermann function). Is the function “Abit” primitive recursive?

Example of "Abit": Ack$(2,3)=9=1001_2$ (base 2). Thus Abit(2,3,3)=1
(the leftmost bit of $1001$. The index of the rightmost bit is 0)
Question 1: Is the function "Abit" primitive recursive (PR)?
...

**4**

votes

**4**answers

465 views

### Are there natural, small, and total recursive functions that are not primitive recursive?

In a sense the Ackermann function is not primitive recursive (PR)
because it grows too fast.
Are there total recursive, not PR, small functions?
Using a diagonal argument,
we may define a total ...

**19**

votes

**1**answer

630 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...

**14**

votes

**0**answers

247 views

### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

**15**

votes

**3**answers

553 views

### Which distributions can you sample if you can sample a Gaussian?

Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this ...

**0**

votes

**0**answers

125 views

### Basis of periodic tiling of Wang tile

Given a set of Wang tile,
Given 3 periodic tiling: A, B, C
We define 3 vector F[A], F[B], F[C]
each vector correspond to the appearing frequency of each type of tiles in the tiling.
Now, we ...

**10**

votes

**1**answer

275 views

### Is this property equivalent to Lusin's property (N) for continuous functions?

A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...

**6**

votes

**1**answer

566 views

### Is rule 30 Turing complete? Is there a proof that it isn't?

It is well known that the elementary cellular automaton known as rule 110 is Turing complete.
Its cousin rule 30 also produces complicated behaviour. When I read Wolfram's a New Kind of Science (in ...

**1**

vote

**1**answer

157 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**1**

vote

**0**answers

104 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 ...

**3**

votes

**1**answer

365 views

### Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...

**3**

votes

**2**answers

188 views

### Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...

**8**

votes

**1**answer

203 views

### Martin-Löf randomness relative to a $\Delta^0_2$-representation of a real

I have a question which I already asked on a more specialized site (http://logicblogfrontend.hoelzl.fr/), but perhaps M.O. will allow me to reach a wider range of experts.
Suppose that $X$ is ...

**5**

votes

**1**answer

223 views

### Interaction between Turing and many-one reducibility

This is a question about two reducibility notions in computability theory. I suspect the answer is a fairly simple construction, and I'm just not seeing it.
For sets $X, Y\subseteq\omega$, we say $X$ ...

**2**

votes

**2**answers

180 views

### Absolutely algorithmically random infinite sequence

Let's call an infinite sequence of bits $f:N\rightarrow \{0,1\}$ absolutely random if any computably constructed subsequence is not computable, i.e. there aren't monotonic computable function $g:N ...