Tagged Questions

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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practical algorithms for np complete problems

Inspired by: Conjecture on NP-completeness of tesselation of Wang Tile up to finite size And the practicality of this topic (solving tessellation on a lattice): coloring in lattice Computational ...
154 views

Decidability of prime gap sequences

Is the following problem undecidable? Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ? If not, is ...
327 views

Do all linear orders in this class have computable copies?

This is a question which has been bothering me now for quite some time. I've talked to a number of people about it, and we've shown that a few basic ideas can't work, but other than that haven't made ...
277 views

Is every non-recursive set in $\Sigma_1$ complete in $\Sigma_1$ (relatively to many-to-one reductions)?

Most well known sets in $\Sigma_1 \setminus\Delta_0$, such as the Halting problem, are complete in $\Sigma_1$, relatively to the many-to-one reduction. In fact I don't know any example of a (non ...
225 views

Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?

I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...
135 views

Attribution of an equivalence of the existence of omega-models of RCA0

There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of ...
749 views

What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
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Sorting of countabe set [closed]

Let $X$ be a countable ordered set. My question is very simple - Can we sort $X$ in countable number of steps? When $X$ is finite, the answer is obviously yes. But what is the answer when $X$ is ...
855 views

Kolmogorov complexity is the strongest noncomputable function

Yury I. Manin says that Kolmogorov complexity (in some nontrivial sense) is the strongest noncomputable function ("Колмогоровская сложность... невычислима... она во многих интересных смыслах ...
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Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction? The answer does ...
199 views

String transformer : Polynomial time approximation schemes?

A program P takes a string as an input and returns a string of same length as output. Q Given two strings A and B how fast can a program tell weather string B cannot be obtained by a recursive ...
166 views

A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$

The question has relevance for constructing Scott sets with certain extra desirable properties. Suppose that $\mathfrak X$ is a countable arithmetically closed family of subsets of $\mathbb N$: ...
Let $e$ be an index of an oracle Turing machine program and $k$ be some natural number. Let us say that a subset of $\mathbb N$ is arithmetic if it is definable in the model $\langle \mathbb ... 1answer 231 views Reverse mathematics, Ramsey theorem and mass problem If we look at reverse mathematics statements as mass problems, considering the class of solutions of an instance, it is known that Weak König's lemma has a maximal instance in the sense that there is ... 2answers 458 views What Turing-Complete models of computation carry a notion of time complexity that “agrees” with that of Turing Machines? Certain models of computation are technically Turing-Complete, but cannot feasibly simulate a Turing Machine within the usual time constraints we hope for. One example of this is Godel's recursive ... 3answers 176 views unbounded complexity If a language L is decidable, does that imply that the is a computable function f such that L is in O(f(n)) ? For example what would be the complexity class of the language of "provably halting ... 2answers 383 views Where should I learn about Kolmogorov complexity of overlapping substrings? I would like to know more about the relationship between the Kolmogorov complexity of a string and that of its substrings. The relation that up to an additive constant,$K(x,y) = K(x) + K(y\ |\ x, ...
Consider the basic system of simple types usually known as $TA_\lambda$. One can prove that (as a consequence of the Subject Reduction Property and the fact that any typable term is strongly ...