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 ...

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3
votes
2answers
336 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 ...
22
votes
4answers
15k 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
534 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
94 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
277 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
886 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
256 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
652 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
451 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
489 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
591 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 ...
51
votes
3answers
5k 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
179 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
98 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
329 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
256 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
689 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
332 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
774 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
471 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
321 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
150 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
1k 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
431 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 ...
23
votes
15answers
4k views

What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...
13
votes
1answer
633 views

Are wild problems related to undecidable ones?

In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...
0
votes
1answer
634 views

When may Function (meromorphic) be expanded as power series with coefficients of integers

Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form: $$F=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M ...
3
votes
1answer
292 views

Paths in Kleene's O and deciding $\Pi^0_1$ sentences

This question comes from the Wikipedia article on Kleene's O and a previous Math Overflow question. The claim in Wikipedia that I have a question about is the second sentence in the following quote. ...
5
votes
1answer
497 views

Machine model for primitive recursion?

General computable functions can be described either functionally (in terms of closure of the coordinate functions, constant functions, composition, primitive recursion, and $\mu$-recursion), or in ...
3
votes
1answer
423 views

Hyperarithmetic statements decidable by induction up to a recursive ordinal

The first version of this question received a helpful answer but was too vague to fully convey what I intended. I hope this version remedies that problem. For any hyperarithmetic set of integers $S$, ...
6
votes
4answers
258 views

What sets are “decidable from competing provers”?

Let $n$ be a member of $\omega$, which the omnipotent provers $Y$ and $N$ know. A Turing machine will be run starting with the $n$ already inputted, and the machine can have a natural number inputted ...
2
votes
3answers
391 views

Code universal arithmetical sets by a hyperarithmetical set?

For each n, there is a (lightface) Σ0n set Sn ⊆ ω2 that's universal for the Σ0n subsets of ω. Since {n} × Sn is Σ0n, there is a union R of arithmetical ...
7
votes
3answers
1k views

Can the twin prime problem be solved with a single use of a halting oracle?

It occurred to me that if it were possible to determine whether a given program halts, that could be used to answer the twin primes conjecture A) Write a program which takes input n and then counts ...
2
votes
1answer
1k views

How many cpus needed to check a 100 million digit prime number efficiently? [closed]

If I had access to potentially unlimited CPUs and wanted to quickly check 100 million digit numbers for primality using a map-reduce architecture, how many CPUs would be necessary? Each of the mapped ...
4
votes
1answer
286 views

Hyperarithemtic statements decidable by induction up to a recursive ordinal

Kleene's O is a $\Pi_1^1$ complete set that decides every hyperarithmetic statement. A Turing Machine that uses this set as an oracle to decide a hyperarithmetic question can only look at a finite ...
10
votes
0answers
467 views

Diagonal lemma from recursion theorem?

Does Gödel's diagonal lemma follow from Kleene's recursion theorem? I believe the converse is true, by an argument like the following. Let e ↦ θe be a bijection between ω and ...
15
votes
1answer
704 views

Undecidable theories easier than $Q$

Most proofs of undecidability for various theories (pure logic with binary relation, group theory, etc.) show that the natural numbers and Robinson's $Q$, in one form or another, can be encoded ...
4
votes
2answers
500 views

Existence of a set of valid Busy-Beaver entries.

In reference to 1961 paper "On Non Computable Functions" by T. Rado. Motivation - Scott Aaronson's Who Can Name the Bigger Number?. M is an n-state binary Turing machine. A valid BB-n entry is a ...
9
votes
1answer
2k views

Kolmogorov Complexity and Proof Techniques

I'm interested in examples of theorems that employ the proof techniques that are utilized in the proof of the undecidability of Kolmogorov Complexity. Definition:(Sipser) Let x be a binary string. ...
5
votes
2answers
593 views

Models of computation with decidable halting problem?

There are numerous examples of models of computation in which all programs halt, for example primitive recursion. Are there (non-trivial) examples of models in which only some programs halt, but the ...
0
votes
3answers
732 views

Proofs that use Infinite/Finite Priority Injury Method

Can anyone point me to any proofs (pref. interesting ones!) that make good (or bad) use of the Finite or Infinite Priority Injury Argument? Edit: I would suppose that my question could be put this ...
9
votes
4answers
3k views

Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?

Let ZF1 = ZF, ZFk+1 = ZF + the assumption that ZF1,...,ZFk are consistent, ZFω = ZF + the assumption that ZFk is consistent for every positive integer k, ... and similarly define ZFα ...
2
votes
1answer
279 views

Given a PDA M such that L(M) is in DCFL construct a DPDA N such that L(N) = L(M)

Is it possible to construct an algorithm which takes as input a pushdown automaton $M$ along with the information that the language accepted by this automaton $L(M)$ is a deterministic context-free ...
2
votes
0answers
208 views

A polytime feasible subuniverse of the Effective Topos

The effective topos is a well known universe of sets suitable for abstract computability, as it is build "from the ground up" via the classical notion of realisability by Kleene. I have found a few ...
8
votes
2answers
899 views

Why can Diophantine equations represent exponential growth?

The wikipedia page on Matiyasevich's theorem challenges: Unfortunately there seems to be as yet no short intuitive explanation as to why Diophantine equations can represent exponential growth only ...
23
votes
2answers
1k views

Are any natural examples of Gödel speed-up known?

In 1936 Gödel announced a theorem to the effect that proofs of certain theorems $T_1,T_2,\ldots$ become dramatically shorter when one passes from a formal system, such as Peano arithmetic PA, to a ...
4
votes
2answers
380 views

Jump Inversion of Arithmetic

I seem to recall once hearing a result to the effect that $\emptyset^{(\omega)}$ was the double jump of some other degree, but could not be the triple jump of any degree. However I'm unable to find ...