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 ...

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4
votes
2answers
153 views

Relation between Turing degrees and functions computable with them

Suppose $A<_T B$ ($A$ is a set computable from $B$ but not vice versa). Is it always the case that there exists a $B$-computable function which eventually outgrows all $A$-computable functions? Of ...
-1
votes
2answers
392 views

Can an algorithm decide whether a program computes all strings? [closed]

I am interested in the type of program, which is given as input to a Universal Turing Machine (UTM) with language $L$, and for which it holds that every possible finite string $s$ of symbols in $L$ ...
3
votes
1answer
276 views

Are there sets which are computable in one model, but uncomputable in another?

Suppose we have two models of set theory, $U$ and $V$ which have the same $\Bbb N$. Is it possible that there is a set $A\subseteq\Bbb N$ such that, in $U$, this set is computable, i.e. there is a ...
3
votes
1answer
57 views

The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...
1
vote
1answer
168 views

The definition of computational complexity or complexity measure of computing reals [closed]

A real $r$ is computable if given any $i\in \mathbb{N}$, the $i$th bit can be outputed by a Turing Machine or an algorithm. So, what is computational complexity or complexity measure of computing the ...
4
votes
0answers
149 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
0answers
71 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
1answer
314 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
1answer
210 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
2answers
88 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
1answer
96 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
1answer
86 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
0answers
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
1answer
269 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
0answers
102 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
2answers
272 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
1answer
57 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
1answer
118 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
1answer
128 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
0answers
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
3answers
186 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 ...
8
votes
5answers
639 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 ...
4
votes
1answer
191 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
0answers
150 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
1answer
57 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
1answer
261 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
1answer
464 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
2answers
230 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
1answer
128 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
1answer
439 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
1answer
387 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
1answer
322 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
1answer
325 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
1answer
278 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
3answers
745 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
1answer
353 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 ...
8
votes
0answers
184 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
1answer
186 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
1answer
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
2answers
226 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
1answer
188 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
4answers
476 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
1answer
636 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
0answers
249 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
3answers
562 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
0answers
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
1answer
287 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
1answer
580 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
1answer
165 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
0answers
107 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 ...