**4**

votes

**1**answer

296 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

**1**answer

123 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 ...

**7**

votes

**0**answers

133 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

191 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

144 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 ...

**9**

votes

**2**answers

377 views

### What is the precise notion of “enough arithmetic” in Godel's first Incompleteness theorem?

I'm trying to reconstruct the proof of Godel's first theorem (Rosser's strong version) from the uncomputability of the Halting function. If we just started with the language $\mathcal{L}=\{0, S, +, ...

**0**

votes

**1**answer

107 views

### What is the probability that a randomly chosen number from set of c.e.number is period(number)?

What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)?
What is the probability that a randomly chosen number from the set of computable numbers is ...

**3**

votes

**1**answer

199 views

### What is the relation between KC and height of rational number?

Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...

**6**

votes

**5**answers

516 views

### practical algorithms for np complete problems

Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...

**3**

votes

**1**answer

103 views

### Decidability of prime gap sequences

Is the following problem undecidable?
Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ?
If not, is ...

**11**

votes

**0**answers

256 views

### Do all linear orders in this class have computable copies?

This is a question which has been bothering me now for quite some time. I've talked to a number of people about it, and we've shown that a few basic ideas can't work, but other than that haven't made ...

**6**

votes

**2**answers

198 views

### Is every non-recursive set in $\Sigma_1$ complete in $\Sigma_1$ (relatively to many-to-one reductions)?

Most well known sets in $\Sigma_1 \setminus\Delta_0$, such as the
Halting problem, are complete in $\Sigma_1$, relatively to the
many-to-one reduction. In fact I don't know any example of a (non ...

**1**

vote

**1**answer

150 views

### Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?

I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...

**5**

votes

**1**answer

106 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 ...

**4**

votes

**2**answers

435 views

### What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...

**7**

votes

**2**answers

443 views

### What is the name of this type of groups?

Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as:
$$G=\langle ...

**1**

vote

**1**answer

85 views

### Every infinite C.E.language is infinite or finite union of regular languages including at least one infinite regular language?

Is Every infinite C.E.language infinite or finite union of regular languages including at least one infinite regular language?
And is every infinite C.E.language that is not indexed language(that may ...

**8**

votes

**3**answers

298 views

### Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
...

**6**

votes

**2**answers

952 views

### Is Turing degree actually useful in real life? [closed]

In theoretical computer science, we classify problems according to their Turing degree. Is there any practical application of this?
Edit: Given that we cannot explicitly and mechanically understand ...

**9**

votes

**4**answers

426 views

### Are there two computable binary trees such that each has a branch not computing any branch through the other?

It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build ...

**4**

votes

**1**answer

184 views

### N^2 and two counter machines

I asked this question on cstheory a few months ago, but I didn't receive an answer, so I'm posting it here to see if there are original ideas from the "math world" to solve it. The original question ...

**4**

votes

**1**answer

181 views

### Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...

**4**

votes

**1**answer

115 views

### Self-similarity in the theory of computability

Let $M = w_0w_1... \in \{0,1\}^*$.
For any computable function $f$ define $M_f = w_{f(0)}w_{f(1)}...$
Let for any computable strictly increasing function $f$ there is continuous
computable mapping ...

**4**

votes

**2**answers

353 views

### Reverse Math of High Sets?

Is there a standard principle in reverse math that is known to be equivalent (over $RCA_0$) to the existence of a set of high (Turing) degree? I'm interested in the general case, but would be happy to ...

**2**

votes

**0**answers

86 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 ...

**4**

votes

**0**answers

78 views

### Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in ...

**3**

votes

**2**answers

198 views

### “Rice (like) Theorem” for primitive recursive functions?

As primitive recursive (PR) functions seem to be so important
(see for instance Kleene normal form Theorem) we may expect that
many decision questions related to PR functions are undecidable.
...

**3**

votes

**1**answer

161 views

### Are there proofs of Rice Theorem without using the undecidability of some problem?

Most proofs of Rice theorem seem to be based on the undecidability of
the halting problem. They are "reduction-based".
Are there "direct" elementary proofs, perhaps based on diagonalization?
I think ...

**7**

votes

**0**answers

271 views

### “Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...

**44**

votes

**1**answer

844 views

### How feasible is it to prove Kazhdan's property (T) by a computer?

Recently, I have proved that Kazhdan's property (T) is theoretically provable
by computers (arXiv:1312.5431,
explained below), but I'm quite lame with computers and have
no idea what they actually ...

**6**

votes

**1**answer

86 views

### Is 0' of PA degree relative to a non-low set?

Definitions:
A set $X$ is of PA degree relative to a set $Y$ if every infinite $Y$-computable binary tree has an infinite $X$-computable path.
A set $X$ is low if $X'$ is computable from ...

**2**

votes

**0**answers

91 views

### What are natural examples of non-relativizable proofs? [duplicate]

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles).
Virtually all proofs seem to be relativizable, though.
What are good examples of ...

**3**

votes

**1**answer

88 views

### Is there a Turing degree which is a strong minimal cover and does not have itself a strong minimal cover?

We say that a Turing degree $a$ is a strong minimal cover of $b$ if $a$ is strictly above $b$ and if any $c$ strictly below $a$ is (not necessarily strictly) below $b$. It is known that some degrees ...

**3**

votes

**1**answer

66 views

### Jump of strongly hyperhyperimmune degrees and DNR relative to 0'

A function f is diagonaly non-recursive (DNR) if for every Turing index $e$, $f(e) \neq \Phi_e(e)$.
A set is strongly hyperhyperimmune if there is no r.e. set of disjoint r.e. set intersecting it.
...

**22**

votes

**10**answers

2k views

### Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ ...

**10**

votes

**1**answer

666 views

### Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made:
Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened.
Define ...

**3**

votes

**1**answer

100 views

### Productive sets and indices of constant functions

Let $\phi_0,\phi_1,\phi_2,\ldots$ be an acceptable programming system. Recall that a set $S$ is productive if there exists a recursive function $p$ such that $(\forall x)(W_x\subseteq S\Rightarrow ...

**2**

votes

**0**answers

68 views

### Equivalence of LOOP (primitive recursive functions) and of SRL (reversible transformations) programs

This is a question about the decidability of program equivalence.
Primitive recursive functions correspond exactly to the functions
that can be implemented on a specific register machine usually
...

**4**

votes

**0**answers

198 views

### About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...

**5**

votes

**2**answers

379 views

### Why is there no product type in simply typed lambda-calculus?

Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped ...

**9**

votes

**2**answers

311 views

### Is equivalence of functions built from nested exponentiations a decidable problem?

Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that
The symbol $x$ is in $\mathcal{E}$, and
If expressions $P,Q\in\mathcal{E}$, then the ...

**3**

votes

**3**answers

163 views

### Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...

**4**

votes

**3**answers

391 views

### Infinite Partitions of the Primes and Sums of Reciprocals (Revised)

I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO ...

**9**

votes

**1**answer

298 views

### Ackermann's function over the reals

Ackermann's function is defined over integers $x$, $y$, $A(x,y)$,
with conditions for when $x{=}0$ or $y{=}0$, and otherwise uses recursive
definitions involving arguments $x{-}1$ and $y{-}1$.
Is ...

**3**

votes

**1**answer

92 views

### Disobedience of some complete r.e. set to some additive cost function

An additive cost function is defined as $c: \omega\times \omega \to \mathbb{Q}_2$ such that it is recursive, monotonic (i.e. $c(x+1,y)\leq c(x,y)\leq c(x,y+1)$ and $c(x,y)=0$ whenever $x\geq y$, the ...

**4**

votes

**1**answer

206 views

### Nondeterministic Turing machines and the recursion theorem

This is almost certainly a silly question, but:
I am currently reading Moschovakis' article "Kleene's amazing second recursion theorem" (http://www.math.ucla.edu/~ynm/papers/1602-002-1.pdf) and there ...

**3**

votes

**2**answers

241 views

### Is There An Algorithmic Complexity Of A Random Distribution

Has anyone studied an equivalent to algorithmic complexity for probability distributions?
This would be a measure which was similar to Kolmogorov complexity but look at the complexity of a (discreet ...

**2**

votes

**0**answers

68 views

### Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background
By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that
$$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...

**2**

votes

**0**answers

238 views

### Sorting of countabe set [closed]

Let $X$ be a countable ordered set. My question is very simple - Can we sort $X$ in countable number of steps?
When $X$ is finite, the answer is obviously yes. But what is the answer when $X$ is ...

**19**

votes

**4**answers

762 views

### Kolmogorov complexity is the strongest noncomputable function

Yury I. Manin says that Kolmogorov complexity (in some nontrivial sense) is the strongest noncomputable function ("Колмогоровская сложность... невычислима... она во многих интересных смыслах ...