**4**

votes

**1**answer

116 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

529 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

355 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 ...

**24**

votes

**1**answer

734 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

209 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

120 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

137 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

493 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

248 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 ...

**19**

votes

**3**answers

2k 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 ...

**15**

votes

**1**answer

892 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

305 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

360 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 ...

**14**

votes

**5**answers

1k 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

138 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

153 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

203 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 ...

**34**

votes

**3**answers

1k 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

172 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

372 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 $N$ ...

**11**

votes

**2**answers

1k 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

423 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 ...

**24**

votes

**0**answers

884 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

157 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

333 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

176 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

434 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

139 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 ...

**18**

votes

**1**answer

835 views

### Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal

Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were ...

**3**

votes

**1**answer

218 views

### $\Sigma_1^0-COH$?

In reverse mathematics, $COH$ is a statement that there is a cohesive set for any uniform array of sets. Here uniform array of sets means that there exists a set $B$ such that $x\in B_e ...

**3**

votes

**1**answer

160 views

### Complexity of winning strategies for open games (for open player)

If $G\subseteq\omega^{<\omega}$ is a computable clopen game, then $G$ has a winning strategy which is hyperarithmetic $(\Delta^1_1)$, by an inductive ranking process. The key observation here is ...

**4**

votes

**1**answer

110 views

### Computable images of differences of r.e. sets

Suppose f is a computable function from a recursively enumerable set U to the natural numbers and that L,K are r.e. subsets of U. Is f(L-K) a difference of r.e. subsets? The motivation comes from
...

**8**

votes

**1**answer

841 views

### Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets?

Hamkins showed that his infinite time Turing machine has the power to decide some $\Delta_2^1$ sets. I wonder if some modifications of the machine could be made to reach level $\Sigma_1^2$ sets, or, ...

**8**

votes

**1**answer

290 views

### Cohesive sets with degree below some non-high 1-generic degrees?

Terminology:
Cohesive sets: $A\subset \omega$, for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite.
Non-high degrees: Degree $a$ such ...

**6**

votes

**1**answer

233 views

### Disjoint sets of fixed points

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, suppose that $f$ and $g$ are recursive ...

**10**

votes

**0**answers

297 views

### Automorphism group of the Turing degrees

It is conjectured that the automorphism group of the Turing degrees, $Aut(\mathcal{D})$, is trivial. However, to the best of my knowledge, the current state-of-the-art is that $Aut(\mathcal{D})$ is ...

**4**

votes

**1**answer

170 views

### When do substructures have computable copies?

Say that a class $\mathcal{C}$ of countable first-order structures in some finite signature has the effective substructure property if $\mathcal{C}$ is closed under isomorphism and whenever $A\in ...

**16**

votes

**1**answer

636 views

### Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...

**9**

votes

**1**answer

199 views

### Ensuring nonempty lightface Borel sets have elements via theories of second-order arithmetic

This question is an outgrowth of this MathSE question: http://math.stackexchange.com/questions/276068/members-of-lightface-borel-sets.
A Borel set $X\subseteq 2^\omega$ is a member of the smallest ...

**4**

votes

**1**answer

399 views

### Definition of HYP in $L_{\omega_1^{CK}}[a]$?

The structure $L_{\omega_1^{CK}}$ consists of only HYP sets (I believe) and HYP in this structure is the same as the actual hyperaritmetic sets. Now if I move to the structure $L_{\omega_1^{CK}}[a]$ ...

**6**

votes

**0**answers

175 views

### $\omega$-models of $\mathbf{\Sigma^1_1}-DC$ and $\mathbf{\Delta^1_1}-CA$

So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the ...

**5**

votes

**1**answer

224 views

### Notion of independence of Turing degrees

I've been thinking for a while about different ways two Turing degrees might be "independent" of each other (from the point of view of computability theory). The simplest such notion would be to say ...

**3**

votes

**0**answers

91 views

### Weak classes of diophantine functions

From a well-known work(s) by Putnam, Davis, Robinson and Matiyasevich, we know that every partially recursive function is diophantine.
Now it seems a natural question to ask: can we say something ...

**6**

votes

**1**answer

278 views

### Status of the Isomorphism problem for automatic groups?

I only ask because I don't know how to look for the answer.

**3**

votes

**2**answers

102 views

### How would one characterize a PR-complete language?

The complexity class $PR$ is the set of all formal languages that can be decided by a primitive recursive function. Is there any language $l$ known to be complete for this class, i.e., for every ...

**3**

votes

**1**answer

204 views

### Various notions of Turing reduction for partial functions

If $f$ and $g$ are partial functions $\mathbb{N} \to \mathbb{N}$, define six preorder relations $f \preceq g$ as follows:
$f \mathop{\preceq_{\mathrm{S}}} g$ ("$f$ is strict/Sasso reducible to $g$") ...

**14**

votes

**0**answers

588 views

### Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

**3**

votes

**2**answers

343 views

### Recursively enumerable sets as range sets of functions in Grzegorczyk-hierarchy

It is well known that recursively enumerable sets can be defined (among many other equivalent alternatives) as the range sets of primitive recusive functions (except for the trivial case of the empty ...

**8**

votes

**1**answer

263 views

### The equality problem between conjugate group elements

The Novikov--Boone Theorem, which is perhaps the archetypal local unsolvability result in group theory, states existence of a finitely presented group whose word problem is recursively unsolvable. ...

**2**

votes

**2**answers

302 views

### Second-order undecidability

Hi,
The idea of undecidability in computability theory seems to be along the lines of:
There can't be an effective procedure, that decides all instances of input, but a single instance can still be ...