4
votes
1answer
303 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
148 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 ...
6
votes
6answers
555 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 ...
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 ...
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 ...
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 ...
10
votes
2answers
333 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
133 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 ...
2
votes
1answer
108 views

$\mu$-recursive definitions for the complexity classes P, NP, etc

The standard complexity classes such as P, NP are usually defined using Turing Machines. In finite model theory those classes can be defined via the classical first-order/second-order logics. I am ...
2
votes
1answer
290 views

Problem to a solution

Consider an NP hard problem $\frak P$ which takes an input of length n $\frak P$ can be solved partially by a factor $ p_i = p(n,i)\in$ [0,1)... by a polynomial time algorithm $\mathcal A(i)$ ...
3
votes
1answer
267 views

What new primitive recursive functions are needed to reconcile Turing time complexity with Godel time complexity?

Let me begin with an example. Consider the computable function $f(x) = 2x$. A Turing machine can implement this function in $O(|n|)$ steps: simply walk to the end of the input string, write a $0$, ...
4
votes
1answer
98 views

About infinite subset of halting probability and 1-random set

Let $\Omega$ be the halting probability (see (http://en.wikipedia.org/wiki/Chaitin's_constant) and R. Downey, and D. Hirschfeldt (2010), Algorithmic Randomness and Complexity for reference). If A is ...
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 ...
1
vote
1answer
146 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 ...
2
votes
2answers
126 views

Size-limited oracles

I am interested in complexity of algorithms which have access to the following peculiar sort of oracle: Suppose that an invocation of an algorithm f with an input of size n has access to an oracle ...
16
votes
1answer
510 views

Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...
3
votes
2answers
95 views

How would one characterize a PR-complete language?

The complexity class $PR$ is the set of all formal languages that can be decided by a primitive recursive function. Is there any language $l$ known to be complete for this class, i.e., for every ...
5
votes
3answers
551 views

computational complexity of primitive recursive functions

If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...
3
votes
1answer
232 views

Diagonalization and classes of computable functions

Fix a standard effective listing $(\phi_e)_{e\in\omega}$ of the partial computable functions from $\omega$ to $\omega$. Let $\mathcal{C}$ be a class of computable total functions $\omega\rightarrow ...
4
votes
1answer
438 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
457 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
536 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 ...
9
votes
1answer
1k 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. ...
6
votes
1answer
236 views

computing abelianizations

Suppose I have a finitely presented group $G,$ and a subgroup $H$ of $G$ given by its finite generating set (given as words in the generators of $G.$ I want to know whether $H/[H, H]$ is finite. Is ...
5
votes
2answers
452 views

A Query regarding the Halting Problem (Omega): Halting Probability for Given Input Size

I was studying the Halting Problem in context of the Probability and had a few doubts regarding it. Hope someone could help me out. I am aware of the probability of a Random program halting on a ...
1
vote
0answers
165 views

Constructing hard inputs for the complement of bounded halting

If there is always a hard input for the complement of bounded halting, can that input be constructed? More precisely, suppose that for any $M$ accepting $$ \text{coBHP}=\{\langle ...
9
votes
2answers
1k views

Is the solution bounded Diophantine problem NP-complete?

Let a problem instance be given as $(\phi(x_1,x_2,\dots, x_J),M)$ where $\phi$ is a diophantine equation, $J\leq 9$, and $M$ is a natural number. The decision problem is whether or not a given ...
9
votes
4answers
2k views

Why relativization can't solve NP !=P?

If this problem is really stupid, please close it. But I really wanna get some answer for it. And I learnt computational complexity by reading books only. When I learnt to the topic of relativization ...
1
vote
0answers
1k views

Quantum computation implications of (P vs NP) [duplicate]

Possible Duplicate: What impact would P!=NP have on the characterization of BQP? Before I begin, I had a similar post closed for mentioning the recently released (to be verified) proof that ...
11
votes
2answers
2k views

What impact would P!=NP have on the characterization of BQP?

Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever ...
7
votes
3answers
1k views

Decidable but nonrecursive sets

Until recently, I believed that recursive=decidable, subscribing to this Wikipedia quote: "In computability theory, a set is decidable, computable, or recursive if there is an algorithm that ...
14
votes
2answers
1k views

Structure theorems for Turing-decidable languages?

Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such ...
2
votes
7answers
910 views

Completeness, easiest, hardest problems

One says that a language $L$ is complete for a complexity class $\mathcal{C}$ if $L$ is in $\mathcal{C}$ and every language in $\mathcal{C}$ is reducible to $L$. Thus, in a sense, $L$ is the hardest ...
23
votes
2answers
1k views

Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program?

Question. Given a Turing-machine program $e$, which is guaranteed to run in polynomial time, can we computably find such a polynomial? In other words, is there a computable function $e\mapsto p_e$, ...
16
votes
5answers
3k views

The problem of finding the first digit in Graham's number

Motivation In this BBC video about infinity they mention Graham's number. In the second part, Graham mentions that "maybe no one will ever know what [the first] digit is". This made me think: Could ...
19
votes
3answers
2k views

Is the theory of categories decidable?

There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...
1
vote
4answers
1k views

A subset of all languages which is uncountable?

Maybe I'm being dense here, but can someone give me a subset of the set of all languages which is uncountable and the subset is easy to describe? (Some natural subset -- not like "take the set of all ...