1
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0answers
93 views

reference on aperiodicity and cluster [closed]

From this image: I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)
2
votes
3answers
740 views

An established proof in Wang Tile which I doubt

When I was reading the paper: Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305. from http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf I could not ...
2
votes
0answers
120 views

characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set. Now, I wish to characterize all the periodic tilings of this set (better if they are ...
0
votes
0answers
125 views

Basis of periodic tiling of Wang tile

Given a set of Wang tile, Given 3 periodic tiling: A, B, C We define 3 vector F[A], F[B], F[C] each vector correspond to the appearing frequency of each type of tiles in the tiling. Now, we ...
1
vote
1answer
90 views

simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...
1
vote
1answer
157 views

relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color. However, could we convert edge type of Wang Tile ...
1
vote
0answers
104 views

Graph theoretical representation of Wang Tile

We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution. However, is there a well established counter-part ...
3
votes
1answer
365 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
2answers
188 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
6answers
686 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 ...
6
votes
0answers
170 views

When is a reduction not a reduction?

Every mathematician understands the concept of reducing a complicated problem to a simpler problem. "Without loss of generality, we may assume…" However, I've noticed that some kinds of ...
2
votes
0answers
92 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 ...
10
votes
1answer
281 views

Harvey Friedman's strict reverse mathematics vs. Cook-Nguyen's V$^0$

Harvey Friedman posted several manuscripts [1] proposing a program for "strict" reverse mathematics, in the sense that the base theory should be mathematically natural and coding-free. In them he ...
24
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
239 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 ...
17
votes
2answers
810 views

Deep theorems and long proofs

I ran across this discussion by Daniel Shanks, "Is the quadratic reciprocity law a deep theorem?." Solved and Unsolved Problems in Number Theory. Vol. 297. AMS, 2001. p.64ff. which made me ...
3
votes
2answers
262 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 ...
5
votes
0answers
158 views

Feasible Type Theories

I am looking for references about efficient type theories, efficiency in the sense of computational complexity, and type theory in the sense of Martin-Lof's type theories. Has there been any studies ...
23
votes
1answer
698 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 ...
7
votes
1answer
427 views

Normality of Chaitin's constant

Can anyone provide an overview of the proof that Chaitin's constant is normal, or better yet, the guiding intuition? Even if we replace the existential quantifiers in the assertion of non-normality ...
1
vote
1answer
154 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 ...
16
votes
1answer
571 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 ...
9
votes
6answers
1k views

Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity. The wikipedia article on constructive proof begins, "a constructive ...
13
votes
1answer
613 views

Forcing over set theory versus forcing over arithmetic

I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...
6
votes
3answers
603 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
261 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 ...
14
votes
4answers
3k views

What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper: Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by ...
4
votes
1answer
447 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
486 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
582 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 ...
3
votes
1answer
219 views

Is $MIN^P$ search problem (partial order) reducible to $MIN^L$ (linear order) search problem?

Search problem $MIN^P$ is, given a polynomial-time computable predicate that is a partial order, to find its minimum (any will do). Search problem $MIN^L$ is, given a polynomial-time computable ...
3
votes
2answers
517 views

Measure of progress towards a proof

Can one define some measure of progress towards a proof of a statement? I'm not sure if it's even possible for general first order logic statements so let's restrict ourselves to propositional ...
7
votes
2answers
863 views

Distribution of the computable numbers on the real number line

If we order all the positive computable real numbers $r_1,r_2,r_3...$ by their Kolmogorov complexity in some language $L$, then make a histogram plot of the $r_i$ on the real line, and we scale it ...
8
votes
3answers
688 views

Definition of relativization of complexity class

Is there any general definition, for a class $C$ of languages, what is the relativized class $C^A$ for an oracle $A$? Usually, these classes and their relativizations seem to be defined in an ad-hoc ...
2
votes
1answer
457 views

How much of P versus NP's difficulty stems from having to rule out the existence of Turing machines that “accidentally” solve, say, 3-SAT efficiently?

It seems like there is a sense in which a Turing machine that demonstrates P=NP could be said to "accidentally" exist. I'm wondering the extent to which the possibility of such machines is the main ...
3
votes
1answer
464 views

Proof systems and their hierarchy

Why ZFC is placed in top of the proof system hierarchy? How it can p-simulate other systems?
28
votes
2answers
1k views

The NP version of Matiyasevich's theorem

By Matiyasevich, for every recursively enumerable set $A$ of natural numbers there exists a polynomial $f(x_1,...,x_n)$ with integer coefficients such that for every $p\ge 0$, $f(x_1,...,x_n)=p$ has ...
2
votes
0answers
194 views

Oracle separating FIP for bounded-depth Frege from FIP for Frege (and hardness conditions on DDH)

Is there an oracle such that in the relativized world, bd-Frege (bounded depth Frege propositional proof system) has FIP (feasible interpolation property) but Frege does not have FIP? Such an ...
2
votes
2answers
544 views

Natural numbers of great kolmogorov complexity

Before I ask my question, let me give you a mini-preamble: in 2006, during an animated discussion on feasibility, ultrafinitism, and what else on FOM, I introduced (informally, and to speak the tuth, ...
2
votes
2answers
1k views

Do you believe P=NP? [closed]

Do you believe P=NP? I've seen some mathematicians say that if P=NP their work would be worthless and restricted to enunciating theorems. They seem to believe that there exist an almost philosophical ...
8
votes
5answers
416 views

Syntactically capturing complexity classes

Primitive recursive functions are syntactically constructible in the sense that from a set of "axioms" we can build every function in the set $PR$. This basicly means that we can build a machine that ...
5
votes
4answers
1k views

Zero-knowledge proof that 0 = 1

Suppose one day I came up with a proof that 0 = 1 in some formal system such as PA or ZFC that cannot prove its own consistency (unless it is inconsistent). Would it be possible to have a ...
25
votes
8answers
2k views

Is P=NP relevant to finding proofs of everyday mathematical propositions?

Disclaimer: I don't know a whole lot about complexity theory beyond, say, a good undergrad class. With increasing frequency I seem to be encountering claims by complexity theorists that, in the ...
13
votes
2answers
826 views

Is there a name for sets for which it is easier to test membership than to find members---and vice versa?

This is a question my son Bob asked me. For some sets it is relatively easy to test for membership but a lot more difficult to find members, and for others the reverse is true. Here is an elementary ...
11
votes
2answers
725 views

Which recursively-defined predicates can be expressed in Presburger Arithmetic?

In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for ...
6
votes
2answers
1k views

Horn clauses and satisfiability

It is well known that satisfiability of Horn formulae can be checked in polynomial time using unit propagation. But suppose we relax the condition for horn clauses from at most one un-negated ...
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 ...
3
votes
0answers
275 views

To what extent MSO = WS1S, when adding relations?

Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma=\{a_1, \ldots, a_n\}$, I define two structures: $${\mathbb{N}}(w) = \langle {\mathbb{N}}, <, Q_{a_1}, \ldots, Q_{a_n} ...
3
votes
4answers
448 views

Deriving the complete set of “non-redundant” true statements in disjunctive form in propositional logic

Given a finite set of statements known to be true, I need to derive all the "non-redundant" statements in disjunctive form using only literals that can be derived from this set of statements, i e all ...