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2
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0answers
22 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 ...
-1
votes
0answers
70 views
what is the Erdos Discrepancy Problem connection(s) to (T)CS? [migrated]
recently there are some new results on computer-based empirical/experimental study of the Erdos Discrepancy Problem (via SAT solvers, cited below). this problem has been cited & studied by several ...
3
votes
1answer
93 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
254 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
101 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
113 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 ...
32
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
296 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
99 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
823 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
66 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
278 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
194 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
173 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
350 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
709 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
104 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
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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 ...
8
votes
2answers
980 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
433 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
396 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
271 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
315 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
183 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
406 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
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0answers
67 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
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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
755 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
253 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
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0answers
64 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
120 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
410 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
205 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
329 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
130 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
337 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
94 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
73 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
234 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
286 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)$ ...
11
votes
4answers
418 views
Mathematics of privacy?
I wonder to which extent the current public debate on privacy issues (not only by state sniffing, but e.g. by microtargetting ads too an issue) offers interesting questions in mathematics?
Can we ...
14
votes
2answers
851 views
Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc?
Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ...
0
votes
1answer
141 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 ...
