**8**

votes

**1**answer

188 views

### Groups whose word problem can be solved in constant time

Given a finitely generated group $G$, define an encoding of $G$ to be a one-to-one function $\Phi:G\to \bigcup_n \{0,1\}^n$ that sends each group element to a unique finite word. For $a,b\in G$, ...

**1**

vote

**0**answers

97 views

### Efficient deterministic algorithms of factorizing

My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$.
Are there such algorithms that use poly$(n, \log q)$ bit operations?
I know ...

**11**

votes

**1**answer

422 views

### An efficient isomorphism between finite fields

Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...

**5**

votes

**2**answers

356 views

### Consistent sentences with no arithmetically definable models

I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent ...

**2**

votes

**0**answers

150 views

### Banach spaces: A ball being a subset of the interior of the union of two balls

Let $X$ be a separable reflexive Banach space and let $A$, $B_1$, and $B_2$ be three closed balls in $X$. Is there a `handy' necessary and sufficient condition for checking whether $A \subseteq (B_1 \...

**2**

votes

**1**answer

130 views

### Model existence and consistency conditions for $\Pi_1^0$ oracles

Let a $\Pi_1^0$ sentence be a sentence asserting that some given Turing machine never halts at the empty input tape. Let Q1 be a (potentially consistently lying) oracle for deciding $\Pi_1^0$ ...

**3**

votes

**1**answer

96 views

### Relationship between computational undecidability and axiomatic undecidability

On surface, these seem two completely different class of problems. One class represent statements which can't be proved or disproved in an axiomatic theory. For example
One can write down a ...

**12**

votes

**2**answers

271 views

### Can noncomputable sets be distinguishable in $RCA_0$?

Say that a set $X\subseteq\omega$ is distinguishable if there is some Turing machine $\Phi_e$ which, when given two sets exactly one of which is $X$, can determine which set is $X$. Formally, $X$ is ...

**2**

votes

**2**answers

204 views

### Linear systems of equations with singular coefficient matrix [closed]

Consider a consistent system of linear equations $Ax=b$. Let's assume for simplicity that $A$ is square $n \times n$. We are looking for an effectively computable approximate solution $\hat{x}$ in the ...

**13**

votes

**1**answer

367 views

### Turing degree of finding independent formulas

In this Paper of D. Isaacson, it is proved that the true arithmetic($Th(\mathbb{N}$)) is the only $\omega$-consistent and complete extension of $Q$ (Robinson's arithmetic). This, together with the ...

**2**

votes

**1**answer

171 views

### What is the extension of the truth-table degrees to Baire Space called?

Recall that for sets $A, B \in 2^\omega$ that we say $A \leq_{tt} B$ if there is a total Turing functional $F \colon 2^\omega \to 2^\omega$ such that $F(B)=A$. This is called truth-table reducibility....

**3**

votes

**0**answers

42 views

### Finite realization of irrational transfer functions

In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function $h:\mathbb{Z}\to\...

**1**

vote

**1**answer

117 views

### Computable function [closed]

Let $f(n)$ be
$$
f(n)=\begin{cases}
1,&\small{\text{if there are digits 1 in the constant $\textit{e}$ $\textit{n}$ times in a row}}\\
0,&\small{\text{otherwise.}}\\
\end{cases}
$$
Is it true ...

**10**

votes

**1**answer

369 views

### What Turing degree is this function?

Over at http://www.scottaaronson.com/blog/?p=2725#comment-1089004 we had a discussion of intermediate Turing degrees.
The following function came up:
Take Chaitin’s constant, and rearrange its ...

**-3**

votes

**1**answer

139 views

### Turing and Many one reductions in computability versus complexity

What are some non-trivial (please exclude poly time definitional difference) differences between Turing versus Many-one reductions in computability theory and those that occur in complexity theory?

**7**

votes

**2**answers

722 views

### Complexity of Turing Machine behavior

If one looks at the code for a Turing Machine (TM) with
$q$ states and, let's say, $2$ symbols, they all look
pretty much the same:
A list of $5$-tuples:
$$
< state, symbol{-}read, symbol{-}to{-}...

**6**

votes

**0**answers

107 views

### A way to smooth out the log* function?

I have seen here and there discussions about what is the "correct" way of extending the Ackermann function to the reals (the same way the Gamma function extends the factorial function to the reals). ...

**15**

votes

**4**answers

2k views

### Languages beyond enumerable

A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...

**7**

votes

**2**answers

397 views

### Difference between constructive Dedekind and Cauchy reals in computation

If the Axiom of Countable Choice (ACC)
$$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: \mathbb{N} \longrightarrow X . \forall n \in \mathbb{N} . \varphi [n, f(n)] $$
...

**3**

votes

**3**answers

177 views

### Coproducts and “Error Conditions” in Math vs CS

First, some background: recently in learning more about functional programming I saw one use for coproducts that surprised me a little bit: A function $f: A \rightarrow B \coprod C$ may result when ...

**6**

votes

**1**answer

333 views

### Current status of computable spectral theorem and interpretation of quantum mechanics

The spectral theorem states if $A$ is a Hermitian operator acting on an $n-$dimensional Hilbert space space $H$, and $\lambda_1, ... \lambda_m$ are $m \leq n$ distinct eigenvalues of $A$, then
$$ H=\...

**2**

votes

**0**answers

122 views

### When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...

**0**

votes

**0**answers

34 views

### Comparing product of positive affine functions over integers

Problem
Let $f_i: \mathbb Z^n \mapsto \mathbb Z$ and $g_i: \mathbb Z^n \mapsto \mathbb Z$ affine functions and $\mathcal D \subseteq \mathbb Z^n$ a set on which they are all positive. Let $P$ and $Q$ ...

**4**

votes

**1**answer

103 views

### Are these two definitions of arithmetical hierarchy of real numbers equivalent?

Zheng and Weihrauch (http://www-sst.informatik.tu-cottbus.de/~wwwti/zheng/publications/1999/mfcs99.pdf) define a real number $x$ to be $\Sigma_n$ if and only if there is a computable function $f:\...

**8**

votes

**1**answer

291 views

### How long does the slow inefficient algorithm for computing the product in classical Laver tables take?

Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let
$X_{n}$ be the set of all finite sequences of elements from $A_{n}$.
Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting
$E_{n}((...

**11**

votes

**1**answer

304 views

### Is ordinal arithmetic more complicated than classical arithmetic?

Consider the first-order language $\mathcal{L}_{\text{OA}}:=(+,\cdot,0,1)$; in this language, we can formulate statements of ordinal arithmetic. Clearly, the theory $T_{\text{OA}}$ of $(\text{On},+,\...

**2**

votes

**1**answer

519 views

### Can Turing machines clarify mathematical, philosophical, and physical existence?

From Harvey Friedman's manuscript on "Order Invariant Relations and Incompleteness":
DEFINITION 4.4. A $\Pi_1^0$ sentence is a sentence asserting that some given Turing machine never halts at the ...

**3**

votes

**1**answer

166 views

### Theories of arithmetic from recursively inseparable sets

Edit: all sets / theories considered below are supposed to be recursively enumerable, although I'd also be interested in any possible generalizations to non-enumerable theories.
In the comments on ...

**5**

votes

**1**answer

169 views

### Deciding isomorphism between structures which interpret in the pure set

I am interested in the following decision problem:
Given descriptions of two relational structures $A,B$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $A$ and $B$...

**6**

votes

**2**answers

158 views

### a variant of the Kleene tree

The (a?) Kleene tree is a computable (a.k.a. decidable) sub-tree of the full binary tree with no computable path. It is well-known.
I need a variant. (For those in the know, I need a c-bar which is ...

**5**

votes

**1**answer

129 views

### “Partial-computably isomorphic” sets

For $A,B \subseteq \mathbb{N}$, define $A\sim B$ when there exist partial computable functions $f,g\colon \mathbb{N}\rightharpoonup \mathbb{N}$ such that $f$ is defined at least on all of $A$ and $g$ ...

**6**

votes

**1**answer

259 views

### Finding limit-nondecreasing sets for certain functions

This is a question that arose a while ago in work with Damir Dzhafarov on some pieces of reverse mathematics. As far as I know, it has no deep significance; however, it feels like the sort of thing we ...

**2**

votes

**2**answers

525 views

### Can we compute every definable number with knowledge of the halting problem?

Suppose we knew the answer to the halting problem, and the halting problem for this new system with the old halting problem solved. And so on. Would this allow us to compute every definable number?

**16**

votes

**1**answer

1k views

### What is the relationship between Turing Machines and Gödel's Incompleteness Theorem?

In this article, Scott Aaronson talks about using Turing Machines for proving the Rosser Theorem.
What is the relationship between the numbering that Gödel used in his proof of incompleteness and ...

**0**

votes

**0**answers

104 views

### Distribution of definable integers

Consider the distribution of all formulas of length less then n which define an integer in PA.
So for instance f(7,n)=number of formulas of length less then n which output 7.
Or the number of steps ...

**3**

votes

**1**answer

226 views

### Can we use this symbol? [closed]

We consider the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ and the language $L=\{+, \cdot , \frac{d}{dx} , 0, 1\}$.
The ring consists of elements of the form $$\sum_{i=0}^N \...

**16**

votes

**1**answer

1k views

### Is it possible to make an algorithm that could predict the likelihood that a program will halt?

Today I began to read about computability theory. I do not even have an elementary understanding of the topic but it certainly got me thinking. I know there is there is no 'one-for-all' algorithm that ...

**0**

votes

**0**answers

150 views

### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the (...

**2**

votes

**2**answers

274 views

### Time Hierarchy Theorem and P vs NP

One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...

**8**

votes

**2**answers

577 views

### Did Bishop, Heyting or Brouwer take partial functions seriously?

The partial μ-recursive functions which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one ...

**15**

votes

**2**answers

496 views

### Can a stochastic Turing machine output a consistent extension of PA with positive probability?

Suppose that we interpret the output tape of a Turing machine as an assignment of true or false to all sentences of PA, taking the $n$th output bit as the truth value of the sentence with Goedel ...

**5**

votes

**0**answers

150 views

### List of finitely presented groups with undecidable word problem

Is there any reasonably updated list of (representative) examples of finitely presented groups with undecidable word problem?
By "representative" I mean "avoiding obvious redundancy", i.e. examples ...

**3**

votes

**2**answers

237 views

### Comparing really big numbers

Is there an intractability theorem that says that in any sufficiently rich system for defining really big numbers, there will be two numbers for which it's very, very, ... very difficult to decide ...

**4**

votes

**2**answers

166 views

### Does the notion of provably total function depend on the chosen representation?

A typical definition of "provably total function in a theory $T$" goes like this (paraphrased from Odifreddi, Classical Recursion Theory II):
A function $f : \mathbb{N}^n \to \mathbb{N}$ is ...

**2**

votes

**1**answer

226 views

### Effectively non-recursiveness of some sets

A set $A$ is completely productive if there exists a computable function $f$ such that for every $e$, $f(e)\in (A-W_e)\cup (W_e-A)$. A set is effectively non-recursive if it is r.e. and its ...

**5**

votes

**0**answers

241 views

### Rice-like Theorems

Let $\varphi$ be an acceptable programming system. Consider the set $S=\{x\in\mathbb{N}:\varphi_x=\varphi_{x+1}\}$. By using Rogers' Fixed Point Theorem (RFPT) it can be proved that $S$ is a non-...

**9**

votes

**0**answers

229 views

### (A little bit) Beyond the E-recursive

The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see http://www.math.harvard.edu/~...

**1**

vote

**0**answers

84 views

### A reference for “Borel Sets and Circuit Complexity”

Is there any pdf version of M.Sipser's "Borel Sets and Circuit Complexity" or , since I am unable to get this paper, is there other reference closely related to theory in that paper?

**4**

votes

**1**answer

78 views

### What class of probability distributions do probabilistic turing machines induce? [closed]

What class of probability distributions is induced by the class of probabilistic turing machines? Is there a precise characterization?

**17**

votes

**1**answer

437 views

### Are compact topological $n$-manifolds recursively enumerable?

Earlier this year it was asked on MO, "Are there only countably many compact topological manifolds?" Thanks to Cheeger and Kister, the answer is yes. On the other hand, Manolescu recently debunked ...