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3
votes
1answer
70 views

A problem related to routing in a graph

I have come across a new problem - I want to know whether this problem is similar to some existing problem or not. The new problem is this. There is a tourist who has a having the following ...
8
votes
1answer
240 views
+50

Fast checking that overdetermined polynomial system does not have a solution

As a result of some inductive procedure for each $n$ I have a system of about $n^2$ polynomial equations with $n$ variables with integer coefficients, which can be precisely computed. As the system is ...
3
votes
0answers
85 views

Correspondence between numerical semigroups and polynomials?

A numerical semigroup $A$ is defined as a subsemigroup of the semigroup $(\mathbb{N},+)$ of the positive integers such that the set $\mathbb{N}\setminus A$ is finite. Equivalently (for a subsemigroup) ...
1
vote
0answers
37 views

Connection between inf-entropy rate and min-entropy

I am reading the paper "Generating random bits from an arbitrary source: fundamental limits" by Vembu and Verdu. This paper is written in the language of information theory, however, I need to ...
2
votes
0answers
23 views

largest size for a randomness extractor

I am not so expert in theoretical computer science, so sorry if the question is trivial, i just could not find it in literature. Suppose we have a source $X$ with min-entropy $\ell$, the randomness ...
3
votes
1answer
100 views

Which automated theorem provers can address the combinatorics of periods in strings?

Five years ago, I made a conjecture on the number of correlation classes that are exhibited by pairs of words in an alphabet of a given size. I later speculated that the conjecture could be tackled ...
5
votes
1answer
258 views

Number of partitions whose blocks form arithmetic progressions

As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B_n$ (the $n$th Bell number) many partitions, where clearly $B_n<2^{2^n}$ and it is actually known that $B_n<n^n$ for large $n$. ...
1
vote
1answer
106 views

How many edges can you put in a graph such that every edge belongs to a minimal $k$-cycle?

I am trying to solve: Given $n, k$, find maximum $m$ such that there exists a graph on $n$ nodes, $m$ edges such that every edge is part of a minimal $k$-cycle. I only care about the asymptotic ...
4
votes
1answer
115 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 ...
2
votes
0answers
44 views

Private Randomness extractor

Suppose we are given two random variables $X$ and $Y$ with fixed marginal and joint distribution. What is the maximum randomness that we can extract from $Y$ that is independent from $X$, that is, if ...
31
votes
1answer
1k views

Wanted: a “Coq for the working mathematician”

Sorry for a possibly off-topic question -- there are four StackExchange subs each of which could be construed as the proper place for this question, and I've just picked the one I'm most familiar ...
3
votes
1answer
307 views

Fundamental Problems in Mathematics that, without Computer Sciences, would not be resolved? [closed]

Could you please give examples of fundamental questions in mathematics (let us say, pure mathematics) which were resolved fundamentally by the use of computers? More precisely, are there examples that ...
6
votes
1answer
115 views

Separating infinite words sharing factors by automata

Two infinite words $\xi, \eta \in X^{\omega}$ are separated by an (Büchi-)automaton if it accepts one but not the other. Denote by $F_n(\xi)$ the factors of length $n$ of an infinite word $\xi$ and ...
44
votes
1answer
842 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 ...
5
votes
1answer
69 views

Generalising the adherence operator and its closure properties with regard to regular (rational) languages

Let $X$ be an alphabet and denote by $X^{\omega}$ the set of all infinite sequences (i.e. words) in $X$. A subset $L \subseteq X^{\omega}$ is called $\omega$-regular if it is acceptable by some ...
5
votes
1answer
281 views

Subsets of all Diophantine's sets

I have asked this question on math.stackexchange already: http://math.stackexchange.com/questions/627461/subsets-of-all-diophantines-sets Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable ...
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
177 views

Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances

Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances. The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?
5
votes
2answers
379 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 ...
6
votes
0answers
716 views

How many 2L-bit numbers are the product of two L-bit numbers?

If I multiply two integers $x, y $ in $[0,2^L)$, I get an integer in $[0,2^{2L})$. Clearly, this map from $[0,2^L) \times [0,2^L) \to [0,2^{2L})$ is not bijective. I am interested in the size of ...
13
votes
7answers
1k views

Two questions from combinatorics on words

Question 1. Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true ...
5
votes
1answer
108 views

What prefix and factors determine a ultimately periodic word uniquely

Let $\xi$ be an ultimately periodic sequence, i.e. there exists finite sequences $p, q \in X^*$ such that $\xi = pq^{\omega}$. Does there exists a $n > 0$ such that the prefix of length $n$ and all ...
0
votes
0answers
64 views

Proof of conjecture that permutation-free automata restrict the possible states visitable from a stringset sharing prefixes and infixes

An automaton $\mathcal A = (X, Q, \delta, q_0)$ is called permutation-free iff no word $w \in X^*$ induces a nontrivial permutation of a subset of the states of $\mathcal A$. More formally for any $R ...
9
votes
2answers
1k views

How difficult will it be for me to switch fields (details below) after my Ph.D. in pure mathematics?

I'm a first year postdoctoral researcher, working in pure areas of Riemann surfaces and differential geometry, after just finishing my Ph.D. in 2013. Recently I've also started taking interest in ...
8
votes
1answer
456 views

How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? ...
6
votes
3answers
401 views

Are paths in HoTT perhaps just “cost-free” paths?

Homotopy type theory (HoTT) doesn't seem to say anything about "mutations" of values in type $T$, an important concept in computer science. Mutations occur when you "change a value" of some variable ...
20
votes
4answers
1k views

Securing privacy of “who communicates with whom” under Orwell-like conditions

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data ...
3
votes
1answer
96 views

Subsets of $\omega$-regular lanuages accepted by automata with special acceptance condition

Let $\mathcal A = (X, Q, \delta, q_0, F)$ be a deterministic finite automata with the following acceptance condition on infinite words: The automata accepts $\xi \in X^{\omega}$ with respect to $F$ ...
8
votes
1answer
276 views

Categorical Semantics for Second-Order Logics

I am currently doing some work using a categorical semantics of first-order logic. The specific semantics I am using is due to Andrew Pitts, as described in: Categorical Logic, Andrew M. Pitts, ...
6
votes
6answers
317 views

Physical Disturbances to Computations [closed]

In this paper, page 7 (160 of the Journal), Fig 3, there is a particularly amusing (not to the authors!) caption: "... On April 1 of year 2 in the $S_0$ experiment, the computer was hit by a cosmic ...
0
votes
1answer
208 views

Proof of the lower bounds of time of algorithm working [closed]

I have asked this question on math.stackexchange already: http://math.stackexchange.com/questions/515920/lower-bounds-on-the-running-time There are some problems, when there is non-trivial lower ...
6
votes
2answers
407 views

A simple language and systematic computations

The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact ...
2
votes
0answers
68 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 ...
1
vote
0answers
35 views

Finite window transformations--input+output (pure algebra)

This q. presents a complete approach to my previous q.: Indecomposability of image transformations ... Let $\ A\ B\ $ be finite sets of cardinality $\ > 1$.   Let $\ D:=A\times B$,   ...
2
votes
0answers
87 views

Indecomposability of image transformations (pure algebra). Open questions

W-transformations -- definitions We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
19
votes
4answers
761 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 ...
3
votes
1answer
265 views

The relationship between P vs NP problem and “Kolmogorov complexity with time”

Let $P$ - polynomial($P(x) \ge x$), $n \in \mathbb{N}$, $l < log(n)$. Problem1: "Is there program with length $\le l$ that print $n$ by using $\le P(log(n))$ time?" Is it Problem1 $\in ...
1
vote
0answers
66 views

Testing functional equivalence

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
0
votes
0answers
125 views

k-means type clustering of binary data, under capacity constraints per cluster. Proof of NP-hardness?

Suppose you are given a set of $I$ binary vectors in ${\mathbb R}^N$, a number of clusters $k$, and positive integers $\{ c_i \}_{i=1}^k$ where $\sum_{i=1}^k c_i = I$. I am interested in finding a ...
9
votes
2answers
430 views

Efficiently determine if convex hull contains the unit ball

Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time? The convex hull itself might have ...
7
votes
1answer
212 views

Main problems on lattice-basis reduction algorithms (such as LLL)?

What are the main open problems on lattice-basis reduction algorithms (such as LLL)? I am looking for problems satisfying the following two conditions: (a) their solution would likely be of some ...
10
votes
2answers
332 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 ...
8
votes
2answers
339 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
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 ...
4
votes
0answers
152 views

Rough structure of the double coset space/Graph bijections up to automorphisms

I am dealing with bijective maps $\pi:\Gamma_1\to \Gamma_2$ between two graphs with the same number of vertices $N=O(10)$. The graphs have a significant automorphism group (these are disconnected ...
3
votes
0answers
76 views

Question about constructing an admissible ideal of a quiver of an algebra with the aid of a computer

Let $k$ be an algebraically closed field and $A$ a finite-dimensional, basic, connected $k$-algebra. Then $A$ is Morita-equivalent to a quotient of a path algebra $kQ/I$ and $I$ is an admissible ...
7
votes
1answer
244 views

Constructing Metrics for specific Topological Spaces, and Refinements of the Cantor-Space in particular

I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very ...
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 $\mathcal A(i)$ ...