Proof that dynamical systems with bounded Kolmogorov complexity can't emulate...
Motivation:During a discussion with neuroscientists the question arose as to whether the human brain may emulate any Turing machine. If we assume that animal brains may be modelled as deterministic...
View ArticleIs Steiner symmetrization "Turing complete"?
This question stems from intuition so it is a little soft. It concerns performing computation using transformations on sets. The idea is that a rearrangement like Steiner symmetrization might be...
View ArticleWhat sorts of extra axioms might we add to ZFC to compute higher Busy Beaver...
First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that...
View ArticleAre ITTM's necessary to compute Turing's "computable numbers" and what does...
In his celebrated paper, "On Computable Numbers, With An Application To the Entscheidungsproblem", Turing defines a "computable number" as follows:The "computable" numbers may be described briefly as...
View ArticleComputing the halting problem with no computable bound on the use function
I would like to prove that there are two sets $A,B\subset \mathbb{N}$ such that$A |_T B$$\emptyset' \equiv_T A\oplus B$for every $e$, if $\{e\}^{A\oplus B}=\emptyset'$ then the map sending $(B,n)$ to...
View ArticleGame with Turing machines
IntroductionThe following game is quite nice: Alice has, in secret, constructed a polynomial $P \in \mathbb{Z}[x]$.On day $n=1,2,3,...$, she secretly writes down $P(n)$ on a piece of paper.Each day,...
View ArticleTuring degrees inside the $\Pi_1^0$ class with top Medvedev degree
I'm sure i have read that the following (or something that implies this) is trueLet $X$ be a $\Pi_1^0$ class with top Medvedev degree. Then for every$x\in X$, there is $y\in X$ with $y<_T x$.But i...
View ArticleA variant of the Busy Beaver function
Set $BB(k,n)$ to be the same definition as the Busy Beaver but where one is looking at all $n$-state machines, and the transition graph has at most $k$"write 1" instructions. This may be a natural...
View ArticleWhy do almost all points in the unit interval have Kolmogorov complexity 1?
Re-posted from math.stackexchange as I did not get any answers there.I am readingJin-yi Cai, Juris Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line,...
View Article"Natural" undecidable problems not reducible to the halting problem
There is a lot of known examples of undecidable problems, a large amount of them not directly related to turing machines or equivalent models of computations, for example here:...
View ArticleIs there a well defined subset of the integers that cannot be defined as a...
Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?I have long been intrigued by the observation that much of mathematics can...
View ArticleIf the join of two degrees compute one of their jumps, what can we say about...
Let $\mathbf{a}$ and $\mathbf{b}$ be two Turing degrees such that $\mathbf{a'} = \mathbf{a} \oplus \mathbf{b}$. Must it be the case that $\mathbf{a'} \leq \mathbf{b'}$? What if in addition, we know...
View ArticleCan a halting oracle determine if a Turing machine is an ordinal?
For the sake of clarity, I am regarding a computable relation on $\mathbb{N}$ as a $2$-symbol ($0$ and $1$) Turing machine $T$ which halts on any initial binary string (which are interpreted as some...
View ArticleIs there a "halting machine" which halts on itself?
The crux of the halting problem is that there can be no Turing machine $M$ such that $\text{Halt}(M(N))=\neg \text{Halt}(N(N))$ for all Turing machines $N$, since $\text{Halt}(M(M))=\neg...
View ArticleUsing Busy Beavers to prove conjectures
I've been pondering some stuff on Shtetl Optimized where Yedidia and Aaronson construct Turing machines that will only halt if (e.g.) the Riemann Hypothesis is false, or Goldbach's conjecture is false....
View ArticleIs there a known Turing machine which halts if and only if the Collatz...
Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture.Goldbach's conjecture asserts that every even...
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