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.

Filter by
Sorted by
Tagged with
12 votes
1 answer
803 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, ...
Vladimir Reshetnikov's user avatar
15 votes
1 answer
993 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 $\alpha$-...
Noah Schweber's user avatar
9 votes
1 answer
839 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 ...
Wojowu's user avatar
  • 27.4k
9 votes
1 answer
601 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 ...
Dave Albert's user avatar
2 votes
3 answers
977 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 ...
user40780's user avatar
  • 867
7 votes
1 answer
451 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 $r_0$...
喻 良's user avatar
  • 4,191
11 votes
0 answers
366 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 ...
Wojowu's user avatar
  • 27.4k
4 votes
1 answer
443 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 ...
Ashutosh's user avatar
  • 9,771
1 vote
1 answer
264 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 $$ d(...
npbool's user avatar
  • 563
7 votes
2 answers
415 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 ...
Wojowu's user avatar
  • 27.4k
6 votes
2 answers
717 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)? ...
Armando Matos's user avatar
9 votes
5 answers
2k 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 ...
Armando Matos's user avatar
28 votes
1 answer
2k 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 ...
Scott Aaronson's user avatar
15 votes
1 answer
578 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, $k$. ...
Noah Schweber's user avatar
19 votes
3 answers
1k 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 ...
Alex Zorn's user avatar
  • 902
14 votes
2 answers
2k 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 ...
Linda Brown Westrick's user avatar
26 votes
1 answer
3k 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 ...
N. Virgo's user avatar
  • 1,316
1 vote
1 answer
612 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 ...
user40780's user avatar
  • 867
2 votes
0 answers
158 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 ...
user40780's user avatar
  • 867
3 votes
1 answer
503 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 ...
user40780's user avatar
  • 867
3 votes
2 answers
285 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 ...
user40780's user avatar
  • 867
8 votes
1 answer
308 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 Martin-...
Laurent Bienvenu's user avatar
5 votes
1 answer
630 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 $X$ ...
Noah Schweber's user avatar
2 votes
2 answers
420 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 \...
Dan's user avatar
  • 1,288
10 votes
2 answers
1k 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, +, \...
JAN's user avatar
  • 169
-1 votes
1 answer
417 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 ...
XL _At_Here_There's user avatar
3 votes
1 answer
357 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 ...
XL _At_Here_There's user avatar
5 votes
6 answers
1k 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 ...
user40780's user avatar
  • 867
3 votes
1 answer
284 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 ...
Marzio De Biasi's user avatar
12 votes
0 answers
492 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 ...
Noah Schweber's user avatar
6 votes
2 answers
706 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 ...
Armando Matos's user avatar
2 votes
1 answer
581 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 ...
Wolphram jonny's user avatar
5 votes
1 answer
221 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 ...
Benedict Eastaugh's user avatar
9 votes
3 answers
2k 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 ...
Wolphram jonny's user avatar
6 votes
2 answers
489 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 ...
Sh.M1972's user avatar
  • 2,183
2 votes
2 answers
613 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 ...
XL _At_Here_There's user avatar
9 votes
3 answers
808 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 ...
user avatar
10 votes
2 answers
2k 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
922 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 ...
Joel David Hamkins's user avatar
4 votes
1 answer
389 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 ...
Marzio De Biasi's user avatar
5 votes
1 answer
387 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 ...
user40780's user avatar
  • 867
4 votes
1 answer
154 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 ...
Alexey Milovanov's user avatar
5 votes
2 answers
502 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 ...
Eric Astor's user avatar
4 votes
0 answers
118 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 $M$....
Jing Zhang's user avatar
  • 3,138
8 votes
3 answers
1k 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. ...
Armando Matos's user avatar
6 votes
2 answers
2k 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 ...
Armando Matos's user avatar
9 votes
0 answers
523 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 ...
Noah Schweber's user avatar
101 votes
4 answers
5k 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 can ...
Narutaka OZAWA's user avatar
6 votes
1 answer
278 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 $\emptyset'$....
Ludovic Patey's user avatar
2 votes
0 answers
120 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 ...
Sai's user avatar
  • 179

1
12 13
14
15 16
20