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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5
votes
6answers
822 views

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 ...
3
votes
1answer
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 ...
12
votes
0answers
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 ...
6
votes
2answers
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 ...
1
vote
1answer
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 ...
5
votes
1answer
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 ...
8
votes
3answers
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 ...
7
votes
2answers
466 views

What is the name of this type of groups?

Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as: $$G=\langle ...
1
vote
1answer
133 views

Every infinite C.E.language is infinite or finite union of regular languages including at least one infinite regular language?

Is Every infinite C.E.language infinite or finite union of regular languages including at least one infinite regular language? And is every infinite C.E.language that is not indexed language(that may ...
9
votes
3answers
399 views

Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation: coloring in lattice Reference for Wang Tile Computational approach deciding whether a set of Wang Tile could tile the space up to some size ...
7
votes
2answers
1k views

Is Turing degree actually useful in real life? [closed]

In theoretical computer science, we classify problems according to their Turing degree. Is there any practical application of this? Edit: Given that we cannot explicitly and mechanically understand ...
9
votes
4answers
575 views

Are there two computable binary trees such that each has a branch not computing any branch through the other?

It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build ...
4
votes
1answer
233 views

N^2 and two counter machines

I asked this question on cstheory a few months ago, but I didn't receive an answer, so I'm posting it here to see if there are original ideas from the "math world" to solve it. The original question ...
4
votes
1answer
225 views

Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
4
votes
1answer
136 views

Self-similarity in the theory of computability

Let $M = w_0w_1... \in \{0,1\}^*$. For any computable function $f$ define $M_f = w_{f(0)}w_{f(1)}...$ Let for any computable strictly increasing function $f$ there is continuous computable mapping ...
4
votes
2answers
408 views

Reverse Math of High Sets?

Is there a standard principle in reverse math that is known to be equivalent (over $RCA_0$) to the existence of a set of high (Turing) degree? I'm interested in the general case, but would be happy to ...
4
votes
0answers
94 views

Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in ...
4
votes
3answers
484 views

“Rice (like) Theorem” for primitive recursive functions?

As primitive recursive (PR) functions seem to be so important (see for instance Kleene normal form Theorem) we may expect that many decision questions related to PR functions are undecidable. ...
3
votes
2answers
414 views

Are there proofs of Rice Theorem without using the undecidability of some problem?

Most proofs of Rice theorem seem to be based on the undecidability of the halting problem. They are "reduction-based". Are there "direct" elementary proofs, perhaps based on diagonalization? I think ...
9
votes
0answers
358 views

“Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...
63
votes
2answers
2k views

How feasible is it to prove Kazhdan's property (T) by a computer?

Recently, I have proved that Kazhdan's property (T) is theoretically provable by computers (arXiv:1312.5431, explained below), but I'm quite lame with computers and have no idea what they actually ...
6
votes
1answer
115 views

Is 0' of PA degree relative to a non-low set?

Definitions: A set $X$ is of PA degree relative to a set $Y$ if every infinite $Y$-computable binary tree has an infinite $X$-computable path. A set $X$ is low if $X'$ is computable from ...
2
votes
0answers
93 views

What are natural examples of non-relativizable proofs? [duplicate]

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles). Virtually all proofs seem to be relativizable, though. What are good examples of ...
3
votes
1answer
116 views

Is there a Turing degree which is a strong minimal cover and does not have itself a strong minimal cover?

We say that a Turing degree $a$ is a strong minimal cover of $b$ if $a$ is strictly above $b$ and if any $c$ strictly below $a$ is (not necessarily strictly) below $b$. It is known that some degrees ...
3
votes
1answer
73 views

Jump of strongly hyperhyperimmune degrees and DNR relative to 0'

A function f is diagonaly non-recursive (DNR) if for every Turing index $e$, $f(e) \neq \Phi_e(e)$. A set is strongly hyperhyperimmune if there is no r.e. set of disjoint r.e. set intersecting it. ...
26
votes
10answers
2k views

Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not. Are there some large $n$ ...
10
votes
1answer
800 views

Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made: Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened. Define ...
3
votes
1answer
131 views

Productive sets and indices of constant functions

Let $\phi_0,\phi_1,\phi_2,\ldots$ be an acceptable programming system. Recall that a set $S$ is productive if there exists a recursive function $p$ such that $(\forall x)(W_x\subseteq S\Rightarrow ...
2
votes
0answers
123 views

Equivalence of LOOP (primitive recursive functions) and of SRL (reversible transformations) programs

This is a question about the decidability of program equivalence. Primitive recursive functions correspond exactly to the functions that can be implemented on a specific register machine usually ...
4
votes
0answers
307 views

About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , ‎of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...
5
votes
2answers
670 views

Why is there no product type in simply typed lambda-calculus?

Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped ...
10
votes
2answers
358 views

Is equivalence of functions built from nested exponentiations a decidable problem?

Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that The symbol $x$ is in $\mathcal{E}$, and If expressions $P,Q\in\mathcal{E}$, then the ...
3
votes
3answers
249 views

Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...
3
votes
3answers
425 views

Infinite Partitions of the Primes and Sums of Reciprocals (Revised)

I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO ...
12
votes
1answer
403 views

Ackermann's function over the reals

Ackermann's function is defined over integers $x$, $y$, $A(x,y)$, with conditions for when $x{=}0$ or $y{=}0$, and otherwise uses recursive definitions involving arguments $x{-}1$ and $y{-}1$. Is ...
3
votes
1answer
95 views

Disobedience of some complete r.e. set to some additive cost function

An additive cost function is defined as $c: \omega\times \omega \to \mathbb{Q}_2$ such that it is recursive, monotonic (i.e. $c(x+1,y)\leq c(x,y)\leq c(x,y+1)$ and $c(x,y)=0$ whenever $x\geq y$, the ...
4
votes
1answer
226 views

Nondeterministic Turing machines and the recursion theorem

This is almost certainly a silly question, but: I am currently reading Moschovakis' article "Kleene's amazing second recursion theorem" (http://www.math.ucla.edu/~ynm/papers/1602-002-1.pdf) and there ...
3
votes
2answers
289 views

Is There An Algorithmic Complexity Of A Random Distribution

Has anyone studied an equivalent to algorithmic complexity for probability distributions? This would be a measure which was similar to Kolmogorov complexity but look at the complexity of a (discreet ...
1
vote
0answers
85 views

Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that $$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...
2
votes
0answers
254 views

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 ...
18
votes
4answers
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 ...
9
votes
1answer
359 views

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 ...
2
votes
1answer
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 ...
6
votes
1answer
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 $\mathbb N$: ...
8
votes
3answers
353 views

If an oracle Turing machine halts with every infinite arithmetic oracle, can it fail to halt with some non-arithmetic oracle?

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 ...
3
votes
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 ...
10
votes
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 ...
1
vote
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 ...
9
votes
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, ...
1
vote
0answers
140 views

Is there an easy decision algorithm for the inhabitation problem for simple types?

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 ...