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 ...

learn more… | top users | synonyms

3
votes
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
3answers
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
1answer
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
4answers
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
2answers
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
3answers
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
1answer
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
1answer
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
2answers
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
4answers
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
2answers
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
1answer
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
2answers
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
4answers
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
3answers
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
2answers
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
1answer
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
1answer
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
3answers
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
3answers
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
3answers
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
3answers
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 ...