**2**

votes

**0**answers

248 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 ...

**18**

votes

**4**answers

803 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 ...

**9**

votes

**1**answer

308 views

### Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction?
The answer does ...

**2**

votes

**1**answer

194 views

### String transformer : Polynomial time approximation schemes?

A program P takes a string as an input and returns a string of same length as output.
Q Given two strings A and B how fast can a program tell weather string B cannot be obtained by a recursive ...

**6**

votes

**1**answer

162 views

### A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$

The question has relevance for constructing Scott sets with certain extra desirable properties.
Suppose that $\mathfrak X$ is a countable arithmetically closed family of subsets of $\mathbb N$: ...

**8**

votes

**3**answers

305 views

### If an oracle Turing machine halts with every infinite arithmetic oracle, can it fail to halt with some non-arithmetic oracle?

Let $e$ be an index of an oracle Turing machine program and $k$ be some natural number. Let us say that a subset of $\mathbb N$ is arithmetic if it is definable in the model $\langle \mathbb ...

**3**

votes

**1**answer

206 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 ...

**10**

votes

**2**answers

388 views

### What Turing-Complete models of computation carry a notion of time complexity that “agrees” with that of Turing Machines?

Certain models of computation are technically Turing-Complete, but cannot feasibly simulate a Turing Machine within the usual time constraints we hope for. One example of this is Godel's recursive ...

**1**

vote

**3**answers

147 views

### unbounded complexity

If a language L is decidable, does that imply that the is a computable function f such that L is in O(f(n)) ?
For example what would be the complexity class of the language of "provably halting ...

**8**

votes

**2**answers

352 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, ...

**1**

vote

**0**answers

120 views

### Is there an easy decision algorithm for the inhabitation problem for simple types?

Consider the basic system of simple types usually known as $TA_\lambda$. One can prove that (as a consequence of the Subject Reduction Property and the fact that any typable term is strongly ...

**8**

votes

**1**answer

179 views

### Cohesive set with degree below non-high Martin-Löf random reals

A set A is cohesive if $A\subseteq ^* W_e$ or $A\subseteq^* \bar{W_e}$ for each $e\in \omega$ (standard enumeration of r.e. sets). By Jockusch and Stephan's 1993 paper 'A cohesive set which is not ...

**9**

votes

**1**answer

291 views

### Concerning Silver's result

Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists.
I wonder whether various weaker or stronger versions of Silver's result ...

**28**

votes

**3**answers

2k views

### Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$,
decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective,
respectively, injective? --
And ...

**18**

votes

**1**answer

421 views

### Busy Beaver modulo 2

There is well-known Rado's "Busy Beaver" sequence — the maximal number of marks which a halting Turing machine with n states, 2 symbols (blank, mark) can produce onto an initially blank two-way ...

**6**

votes

**1**answer

276 views

### Where does the deterministic simulation of non-deterministic ω-Turing machines fail?

An $\omega$-Turing machine is just a usual Turing machine $T=(Q,\Sigma,\Gamma,\delta,q_0,F)$ where $Q$ is the finite set of states, $\Sigma$ is the input alphabet, $\Gamma\supset\Sigma$ is the tape ...

**2**

votes

**1**answer

121 views

### $\mu$-recursive definitions for the complexity classes P, NP, etc

The standard complexity classes such as P, NP are usually defined using Turing Machines. In finite model theory those classes can be defined via the classical first-order/second-order logics.
I am ...

**2**

votes

**1**answer

297 views

### Problem to a solution

Consider an NP hard problem $\frak P$ which takes an input of length n
$\frak P$ can be solved partially by a factor $ p_i = p(n,i)\in$ [0,1)... by a polynomial time algorithm $\mathcal A(i)$ ...

**0**

votes

**1**answer

147 views

### Interaction-based approximation for HP-complete λ-theory?

We are looking for a proof or counter-examples for the following hypothesis.
Two combinators $M$ and $N$ are solvable and equivalent in the HP-complete sensible $\lambda$-theory iff either
$$
\exists ...

**1**

vote

**1**answer

176 views

### Nontrivial, partially uncomputable function

is there any example of function which is computable on some set and uncomputable on other set? That is for example function f(n) which is computable on some (finite, or for example for even numbers) ...

**3**

votes

**1**answer

163 views

### A computability-theoretic preorder on reals

My question is about a fairly artificial preorder on functions from $\omega$ to $\omega$, which for simplicity I'll call "reals."
For $r, s\in {}^\omega\omega$, write $r\le_E^*s$ if for each real $f$ ...

**4**

votes

**3**answers

388 views

### What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?

Let me begin with an example.
Consider the computable function $f(x) = 2x$. A Turing machine can implement this function in $O(|n|)$ steps: simply walk to the end of the input string, write a $0$, ...

**4**

votes

**1**answer

106 views

### About infinite subset of halting probability and 1-random set

Let $\Omega$ be the halting probability (see (http://en.wikipedia.org/wiki/Chaitin's_constant) and R. Downey, and D. Hirschfeldt (2010), Algorithmic Randomness and Complexity for reference). If A is ...

**0**

votes

**2**answers

488 views

### Game of Chess and axiomatic systems [closed]

Consider a deterministic, perfect information, abstract strategy, finite game , with absurdly large state space, say ...chess
Q1 Is the game translatable to an axiomatic system?
Q2 Can all ...

**0**

votes

**3**answers

321 views

### Random infinite sequence : Can machines generate truly random sequences. [closed]

Test : "A True Random Sequence Source and a computer producing a certain sequence of numbers are kept in separate rooms and judges try to tell them apart by conducting a series of tests on the ...

**23**

votes

**1**answer

708 views

### Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
...

**0**

votes

**1**answer

190 views

### Random infinite sequences

An Algorithm/Turing machine
Produces a symbol from a finite alphabet, and continues doing so
infinitely.
Another algorithm gets a copy of this symbol,
...

**3**

votes

**1**answer

116 views

### What is the Arithmetical complexity of determining whether a 2-ary computable predicate has exactly one infinite column

Let $W_e$ be the $e$th computably enumerable set in a standard enumeration. For $A\subseteq \omega$, let $A^{[i]}:=${ $ a : \langle a,i\rangle \in A$}. What is the arithmetical complexity of {$e : ...

**2**

votes

**1**answer

129 views

### Question about undecidable consequences of Con, learnability and arithmetical complexity of logical consequence

Let$\:$ $T=\{\varphi \in \Pi_1: PA+Con(PA) \vdash \varphi\:\:and\:\: PA\nvdash \varphi \}$. $\:$By the facts presented here Are undecidable consequences of Con recursively enumerable? by Andreas ...

**5**

votes

**1**answer

464 views

### What can be done with computability logic that previous logic systems can't?

I've been reading a lot about computability logic lately and I'm superficially aware that it unifies classical, intuitionistic and linear logics.
What I'm seeking to know is:
Can computability logic ...

**3**

votes

**2**answers

237 views

### Computability complexity of the first-order theory of arithmetic?

Hello,
It's well known that Kleene's O is $\Pi^1_1$-complete. Does the same thing go for the first-order theory of arithmetic? (I'm talking specifically without set quantifiers---the theory of ...

**18**

votes

**3**answers

1k views

### Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?

Assume for this question that ZF set theory is sound.
Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF proof that M runs ...

**14**

votes

**1**answer

693 views

### Lawvere's fixed point theorem and the Recursion Theorem

Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi_e\text{ is total }\implies ...

**5**

votes

**1**answer

259 views

### Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no ...

**5**

votes

**1**answer

341 views

### First order consequence of a combinatorial principle

(Base theory $RCA_0$)The principle says there exists a function g such that g dominates any X-recursive function for any X in the model.
i.e. For any $f\le_T X$, $\exists b\in M$ such that ...

**12**

votes

**5**answers

847 views

### Are the two meanings of “undecidable” related?

I am usually confused by questions of the type "could such and such a problem be undecidable", because as far as I know there are two distinct possible meanings of "undecidable". I regard the ...

**2**

votes

**0**answers

134 views

### Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...

**11**

votes

**0**answers

145 views

### Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

Background
I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...

**3**

votes

**2**answers

194 views

### Smallest base to reach partial recursive functions as a closure of unbound search

It is customary to define the class of partial recursive functions by taking the set of primitive recursive functions $PR$ and taking closure over unbound search operation.
Do we need the "whole" set ...

**22**

votes

**1**answer

653 views

### Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...

**1**

vote

**1**answer

158 views

### Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?

I failed to get an answer at http://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In ...

**4**

votes

**2**answers

319 views

### A question about primitive recursive functions

I have a question about primitive recursive functions. Maybe it's trivial, if it is I will move it into math.stackexchange.
Is there a primitive recursive function $f$ which is a bijection of $N$ ...

**9**

votes

**2**answers

976 views

### Categories of recursive functions

I have a couple of conjectures on recursive functions, that I feel must have been proved or refuted by someone else, but I don't know where to look. In short:
1. The primitive recursive functions ...

**9**

votes

**2**answers

396 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 ...

**20**

votes

**0**answers

823 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 ...

**1**

vote

**1**answer

148 views

### Grzegorczyk-hierarchy, growth-rate and functions with finite image

Grzegorczyk-hierarchy divides primitive recursive functions in distinct classes with respect to their growth-rate. It seems that the higher we go the hierarchy, the more tools we have to define ...

**2**

votes

**3**answers

326 views

### Indices of r.e. sets

The last part of the paper Located Sets and Reverse Mathematics [Journal of Symbolic Logic 65 (1999), 1451–1480] by Giusto and Simpson involves a proof as follows:
Given $A$ an effectively ...

**3**

votes

**2**answers

165 views

### Disjoint sets of fixed points 2

Let $\phi$ be an acceptable programming system. For every recursive function $f$, let $(f)=\{x:\phi_x=\phi_{f(x)}\}$ the set of fixed points of $f$. Now, let $S$ be a set and suppose that there exist ...

**9**

votes

**1**answer

419 views

### New research on coding in reverse mathematics?

Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey ...

**2**

votes

**2**answers

129 views

### Size-limited oracles

I am interested in complexity of algorithms which have access to the following peculiar sort of oracle:
Suppose that an invocation of an algorithm f with an input of size n has access to an oracle ...