**3**

votes

**2**answers

344 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

265 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 ...

**2**

votes

**1**answer

308 views

### A question about recursively enumerable sets of rational numbers

Let (Q*,<) denote the ordered set in which the elements of Q* are just the positive rational numbers
less than 1 and "<" is the ordering relation of the ordered field (of all rational numbers) ...

**9**

votes

**0**answers

296 views

### Various definitions of recursion from ordinal machines

Background: I'm trying to get an intuitive understanding of α-recursion and related concepts in higher recursion theory. Once nice book is Peter Hinman's Recursion-Theoretic Hierarchies, available ...

**3**

votes

**0**answers

162 views

### Alternate proof of van de Wiele's theorem in E-recursion

Hello, all
I'm currently trying to understand $E$-recursion theory, which is a generalization of classical recursion theory to arbitrary sets. One of the difficulties I'm having with understanding ...

**4**

votes

**1**answer

767 views

### Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows:
build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...

**2**

votes

**1**answer

160 views

### The set of $\Delta_1$ indices

Is the set of Godel numbers of $\Delta_1$ formulae itself $\Delta_1$ definable (i.e., computable)?

**5**

votes

**1**answer

208 views

### Companion of the pointclass of inductive sets

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures.
I am interested in Spector classes on $\mathbb{R}$, ...

**8**

votes

**1**answer

420 views

### Fast-growing hierarchy and Turing machines

Is it possible to get an estimate of the size of a Turing machine computing $f_\alpha(n)$, for a given $\alpha$ (I am especialy interested in moderately large $\alpha$ like the ordinal of ...

**2**

votes

**2**answers

222 views

### Computability of finding roots in holomorphic functions.

Consider a holomorphic function $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$ (not necessarily simply connected). Is it then possible to determine if $f$ contains a ...

**24**

votes

**1**answer

825 views

### Can a string's sophistication be defined in an unsophisticated way?

This question is about sophistication, a way of measuring the amount of "interesting, non-random information" in a binary string, which was proposed by Kolmogorov and others in the 1980s. I'll define ...

**13**

votes

**2**answers

524 views

### Minimal degree of polynomial vanishing on the variety of small degree.

My question is assume that we know that the degree of some irreducible variety is small does it possible to conclude that there exists polynomial of small degree vanishing on this variety.
Let us ...

**2**

votes

**1**answer

357 views

### Turing-Complete Cellular Automata and Sym(Z)

Does there exist a Turing complete, cellular automata with universe and alphabet $\mathbb{Z}$ such that the only allowable configurations are permutations of $\mathbb{Z}$? Formally, consider $\tau : ...

**1**

vote

**2**answers

286 views

### Satisfiability problem for FOL[<,R]

Let FOL[<,R] be the fragment of first-order logic enriched with two relational symbols < and R and the first-order axioms that say:
< is a strict partial order and R is an irreflexive and ...

**3**

votes

**1**answer

258 views

### Proof of the existence of hyperimmune-free degrees

In Classical Recursion Theory Vol.I by P.Odifreddi, section V.5 on the Tree Method, the proof for the existence of hyperimmune-frees involves the construction of a series of trees.
Some definitions ...

**4**

votes

**3**answers

321 views

### Existential quantification over regular predicates

A regular language over an alphabet $\Sigma$ is a subset of the set of all words over $\Sigma$ that can be accepted by some finite automaton. A regular language identifies a certain property of ...

**4**

votes

**1**answer

216 views

### Reference for a “recursive” fragment of infinitary logic?

Does anyone know of any texts or papers out there concerning properties of the fragment of $ \mathcal{L}_{\omega_1, \omega} $ in which the only admissable infinite conjunctions are those which are ...

**20**

votes

**4**answers

1k views

### Algorithmically unsolvable problems in topology

This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...

**3**

votes

**2**answers

1k views

### Is there an algorithm that can “reverse engineer” a Regular Expression?

Given a Regular language (represented as a black box to which one can apply inputs and get 0/1) Is there an algorithm that can find a finite deterministic automaton that produces that language?

**6**

votes

**3**answers

689 views

### computational complexity of primitive recursive functions

If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...

**3**

votes

**1**answer

311 views

### Diagonalization and classes of computable functions

Fix a standard effective listing $(\phi_e)_{e\in\omega}$ of the partial computable functions from $\omega$ to $\omega$. Let $\mathcal{C}$ be a class of computable total functions $\omega\rightarrow ...

**6**

votes

**1**answer

395 views

### Probability that a Turing machine will nontrivially reduce a real

For a fixed Turing machine $\Phi_e$, what is the probability that it will reduce a given real to some less complex, yet still non-computable real?
More precisely: It is known that the set of reals ...

**3**

votes

**2**answers

347 views

### Representation of μ-recursive functions

Can every μ-recursive function be defined using a single instance of the μ operator applied to a primitive recursive function?
According to Wikipedia, any μ-recursive function can be expressed as the ...

**23**

votes

**4**answers

20k views

### How large is TREE(3)?

Friedman, in http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdf, shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman ...

**5**

votes

**1**answer

576 views

### Countable admissible ordinals

Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...

**3**

votes

**0**answers

101 views

### Is a parametric family which is universally consistent for multiple quantiles impossible?

Suppose I am dead-set on using Bayesian inference on independent and identically distributed data, but I'm lazy and insist on using a parametric likelihood function come what may. I'd be reassured to ...

**5**

votes

**1**answer

288 views

### Higher computability : Constructive ordinal and $\Delta^1_1$ predicates

Everything I know on this subject comes from Sacks book : "Higher recursion theory"
Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$.
We should have the result that $A ...

**6**

votes

**2**answers

930 views

### Busy Beaver - Proof for BB(2) = 4

Hi,
I need to prove the above claim.
I can show that $BB(2)\ge 4$ by building a turing machine,
but how can i show that $BB(2) \le 4$?
Searched a lot over the web, and saw that Rado proved it in ...

**4**

votes

**0**answers

293 views

### Difference between lambda-calculus with well-formed formulas vs properly-formed formulas

In S.C. Kleene's 1935 paper "$\lambda$-definability and recursiveness," he proves that all $\lambda$-definable functions are general recursive in the Herbrand-Godel sense and vice-versa. However, the ...

**3**

votes

**2**answers

761 views

### Kleene's fixed point theorem on recursive subsets of computable functions

I have a question about the possibility to apply/restate the Kleene fixed point theorem on recursive subsets of computable functions. I don't know if this is trivial and/or if related questions have ...

**4**

votes

**1**answer

457 views

### Infinite monkeys computing … triangle area?

I wonder if it is possible to specialize the question:
(a) What is the probability that a random Turing Machine program
will halt?, to: (b) What is the probability that a random Turing Machine
...

**10**

votes

**2**answers

496 views

### What is the computational-complexity-theoretic analogue of computable inseparability? For example, if P is not NP, are there disjoint NP sets with no separation in P?

Disjoint sets $A$ and $B$ are computably inseparable, if there
is no computable separating set, a computable set $C$ containing $A$ and disjoint from $B$. The
existence of c.e. computably inseparable ...

**12**

votes

**1**answer

642 views

### Does every feasible partial order relation on the natural numbers extend to a feasible linear order relation?

It is well known that every partial order on a set can be extended
to a linear order on that set. That is, for every partial order
$\lhd$ on a set $X$, there is a linear order $\prec$ on $X$ such
that ...

**55**

votes

**3**answers

6k views

### Can a group be a universal Turing machine?

This question was inspired by this blog post of Jordan Ellenberg.
Define a "computable group" to be an at most countable group $G$ whose elements can be represented by finite binary strings, with the ...

**2**

votes

**1**answer

187 views

### Is this c.e. set obtained via the Recursion Theorem?

Reading through a text on computability I came across a c.e. set defined as follows: Let $K=\lbrace x \in W_x \rbrace$ and let $f$ be a computable function. Then there exists $n \in \omega$ such that ...

**1**

vote

**1**answer

99 views

### Turing code numbers and c.e. sets.

Let $W_e$ be the c.e. set which is the domain of the p.c. function $\varphi_e$ and consider the equivalence $\sim$ such that $x \sim y$ if and only if $\varphi_x=\varphi_y$. I am wondering if $W_e$ ...

**1**

vote

**1**answer

363 views

### Infinite set with/without infinite c.e. subsets

Let $\varphi_e$ denote the p.c. function computed by the Turing Machine with code number $e$. I am looking at the set $M = \lbrace x : \neg (x < y)[\varphi_x=\varphi_y] \rbrace$. This set is ...

**3**

votes

**2**answers

261 views

### $\Delta_2$-inseparability?

Since long ago it is known the existence of non-recursive sets (i.e., non-$\Delta_1$), e.g., Halting problem. It is also known (firstly noticed by Trakhtenbrot, and deeply studied by Smullyan) the ...

**2**

votes

**1**answer

751 views

### What about the fastest-growing non-computable function ?

The Busy-Beaver trick provides a nice example of non-computable functions (let say from $\mathbb{N}$ to $\mathbb{N}$) which grows faster than any computable functions. But what can we say when we do ...

**9**

votes

**2**answers

352 views

### Can we represent computable functions by r.e. sets ?

As we know, if $f$ is a computable function, then every pre-image of a r.e. set under $f$ is also a r.e. set, i.e. $f^{-1}(X)$ is a r.e.set if $X$ is a r.e. set. So I want to know that if a function ...

**5**

votes

**4**answers

2k views

### Explicit expression for recursively defined functions

Consider a function $w(i)$, $i \in \mathbb{N}$, defined recursively by:
$w(0)=w(1)=1$, and
$w(i)={i}^{n}-\sum_{j=1}^{i-1}{i \choose j}w(j)$ for $i>1$.
Is it possible to write $w(i)$ out ...

**9**

votes

**3**answers

826 views

### Effective topos and computability in topological spaces

The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a ...

**6**

votes

**2**answers

480 views

### Natural statements independent from true $\Pi^0_2$ sentences

I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...

**2**

votes

**1**answer

327 views

### On families of finite graphs with undecidable first-order theory

It is well known that the first order theory of graphs (i.e., an irreflexive and symmetric "edge" relation on a set) is undecidable. The same holds for the first order theory of finite graphs.
I am ...

**0**

votes

**1**answer

151 views

### any given c.e.set has number M whose power bounds the corresponding elements of S?

For S ,any given c.e.set,does there exist a M (integer) and a partially computable function outputing every element of S the c.e.set ,such that $\forall x\in S,\exists n x=f(n)$ and $x=f(n)\leq ...

**8**

votes

**3**answers

1k views

### Undecidable problems in geometry

Are there any (many) algorithmically undecidable problems in computational (combinatorial/discrete) geometry?
Update: the Wang tiles answer the question with "any". (I have somewhat overlooked to ...

**19**

votes

**3**answers

2k views

### Inverse Ackermann - primitive recursive or not?

I wanted to put this originally on math.stackexchange, since I considered it to be a straightforward question and probably a fairly known fact. After I failed to solve the problem, I browsed through ...

**1**

vote

**3**answers

442 views

### Variant of the usual proof method for undecidability of the halting problem

This is a largely a question of pedagogy/references, though I may have overlooked some nuance of actual mathematics.
I am planning to introduce the concept of Turing machines and the halting problem ...

**25**

votes

**3**answers

1k views

### Is it decidable whether or not a collection of integer matrices generates a free group?

Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$. Is there an algorithm to decide whether ...