4
votes
1answer
300 views

Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
3
votes
1answer
143 views

Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature: The left color and ...
5
votes
1answer
191 views

Interaction between Turing and many-one reducibility

This is a question about two reducibility notions in computability theory. I suspect the answer is a fairly simple construction, and I'm just not seeing it. For sets $X, Y\subseteq\omega$, we say $X$ ...
9
votes
2answers
378 views

What is the precise notion of “enough arithmetic” in Godel's first Incompleteness theorem?

I'm trying to reconstruct the proof of Godel's first theorem (Rosser's strong version) from the uncomputability of the Halting function. If we just started with the language $\mathcal{L}=\{0, S, +, ...
6
votes
5answers
520 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 ...
11
votes
0answers
257 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
199 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 ...
5
votes
1answer
106 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 ...
4
votes
2answers
437 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
443 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 ...
8
votes
3answers
299 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 ...
6
votes
2answers
953 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
428 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
182 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
2answers
353 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 ...
2
votes
0answers
86 views

Comparing two non-deterministic Turing equivalents as basis for Logic, request for references

I am designing a logic, that is simpler than FOL + PA. And I like to know if there already exists something in this direction. First of all a non-deterministic Turing equivalent is defined by ...
4
votes
0answers
78 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 ...
7
votes
0answers
271 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 ...
6
votes
1answer
88 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
91 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
66 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. ...
22
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 $n$ ...
4
votes
0answers
198 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
380 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 ...
9
votes
2answers
311 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
163 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 $\Sigma_2$ ...
9
votes
1answer
298 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
92 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
206 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
241 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 ...
2
votes
0answers
238 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 ...
9
votes
1answer
257 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 ...
8
votes
3answers
257 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 ...
1
vote
0answers
98 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 ...
8
votes
1answer
145 views

Cohesive set with degree below non-high Martin-Löf random reals

A set A is cohesive if $A\subseteq ^* W_e$ or $A\subseteq^* \bar{W_e}$ for each $e\in \omega$ (standard enumeration of r.e. sets). By Jockusch and Stephan's 1993 paper 'A cohesive set which is not ...
9
votes
1answer
272 views

Concerning Silver's result

Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists. I wonder whether various weaker or stronger versions of Silver's result ...
0
votes
1answer
142 views

Interaction-based approximation for HP-complete λ-theory?

We are looking for a proof or counter-examples for the following hypothesis. Two combinators $M$ and $N$ are solvable and equivalent in the HP-complete sensible $\lambda$-theory iff either $$ \exists ...
3
votes
1answer
160 views

A computability-theoretic preorder on reals

My question is about a fairly artificial preorder on functions from $\omega$ to $\omega$, which for simplicity I'll call "reals." For $r, s\in {}^\omega\omega$, write $r\le_E^*s$ if for each real $f$ ...
23
votes
1answer
659 views

Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether ...
2
votes
1answer
120 views

Question about undecidable consequences of Con, learnability and arithmetical complexity of logical consequence

Let$\:$ $T=\{\varphi \in \Pi_1: PA+Con(PA) \vdash \varphi\:\:and\:\: PA\nvdash \varphi \}$. $\:$By the facts presented here Are undecidable consequences of Con recursively enumerable? by Andreas ...
5
votes
1answer
431 views

What can be done with computability logic that previous logic systems can't?

I've been reading a lot about computability logic lately and I'm superficially aware that it unifies classical, intuitionistic and linear logics. What I'm seeking to know is: Can computability logic ...
3
votes
2answers
208 views

Computability complexity of the first-order theory of arithmetic?

Hello, It's well known that Kleene's O is $\Pi^1_1$-complete. Does the same thing go for the first-order theory of arithmetic? (I'm talking specifically without set quantifiers---the theory of ...
18
votes
3answers
1k views

Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?

Assume for this question that ZF set theory is sound. Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF proof that M runs ...
14
votes
1answer
603 views

Lawvere's fixed point theorem and the Recursion Theorem

Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi_e\text{ is total }\implies ...
5
votes
1answer
214 views

Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no ...
5
votes
1answer
324 views

First order consequence of a combinatorial principle

(Base theory $RCA_0$)The principle says there exists a function g such that g dominates any X-recursive function for any X in the model. i.e. For any $f\le_T X$, $\exists b\in M$ such that ...
12
votes
5answers
811 views

Are the two meanings of “undecidable” related?

I am usually confused by questions of the type "could such and such a problem be undecidable", because as far as I know there are two distinct possible meanings of "undecidable". I regard the ...
11
votes
0answers
142 views

Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

Background I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...
3
votes
2answers
182 views

Smallest base to reach partial recursive functions as a closure of unbound search

It is customary to define the class of partial recursive functions by taking the set of primitive recursive functions $PR$ and taking closure over unbound search operation. Do we need the "whole" set ...
1
vote
1answer
145 views

Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?

I failed to get an answer at http://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here. In ...