All mathematical questions about computer science, including theoretical computer science, formal methods, verification, and artificial intelligence. For questions about Turing computability, please use the (computability) tag instead. For numerical analysis, use the (numerical-methods) tag. For ...

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12
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569 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
11
votes
0answers
209 views

Reference on standard types

This question is about what I presume is a basic construction in type theory. The finite types are defined as follows: 0 is a finite type; if $\sigma, \tau$ are finite types, then so is ...
11
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0answers
175 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
7
votes
0answers
125 views

How to attack universal hash function based on finite-field arithmetic?

As per the Recursive n-gram hashing is pairwise independent, at best paper, I want to use the algorithm described in chapter 6 and 7 (page 7 - 10). The hash works as follows: Define a random ...
7
votes
0answers
190 views

How to maximize the number of operations in process

In my research project I have encountered the following problem, concerning a tuple of words in the formal language $L=\{0,1\}^*$, with $\epsilon$ denoting the empty word. If we are given an ordered ...
7
votes
0answers
314 views

Homomorphic Compression

Can there be an algorithm such that, given plaintext data P,Q, and compression function e, Such that if we treat P and Q as a number (a series of bits): $$\begin{eqnarray*}e(P + Q)& =& e(P) ...
6
votes
0answers
110 views

This proof in my textbook involving the pumping lemma appears incorrect - is it?

It states Let $B$ be the language $\{0^n1^n2^n | n \geq 0\}$. We use the pumping lemma to prove that $B$ is not regular. The proof is by contradiction. Assume to the contrary that $B$ is regular. ...
6
votes
0answers
190 views

How to list the prime factorised natural numbers?

Today I set out to invent a two character numeral system designed to make factorization trivial. Indeed, it lets one factor non-trivial numbers with over thousand digits within 30 seconds per hand - ...
5
votes
0answers
82 views

partial derivative of a facet normal wrt to one of its vertex

I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16). Basically they define three ...
5
votes
0answers
80 views

How to find all integral elements over a subring using Macaulay2

I have the following question about Macaulay2: how to find all integral elements over a subring? I mean suppose $A$ is a subring of $B$. How can I find the set $L=\{x\in B:x \text{ is integral over ...
5
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0answers
62 views

Where can I learn more about the “else” operation / “else monoid”?

(The set of natural numbers $\mathbb{N}$ starts at $0$ for me.) Let $X$ denote a set, and define $X_\bot = X \uplus \{\bot\}.$ Let $\mathbf{else}$ denote the binary operation on $X_\bot$ defined as ...
5
votes
0answers
78 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
5
votes
0answers
119 views

Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I'm programming a maths environment. I'm computer science, so please excuse any probable ignorance and lack of precision in my question. It seems $i$ and complex numbers were ...
5
votes
0answers
346 views

Constructor And\Or-graph on function transition of the alternating automata

In a And\Or-graph induced by the transition function, each node of $G$ corresponds to a state $q$ belonging to a set $Q$ of the state of the Automaton, for $q$ with $\delta(q,a)=q_1*q_2$, the node is ...
4
votes
0answers
73 views

Best sources on complete transforms (classic orthonormal transforms) and overcomplete transforms in signal processing

In the introduction section of a thesis I read a little about classic orthonormal transforms such as Fourier, discrete cosine and wavelet transforms and their application in signal processing. Then ...
4
votes
0answers
56 views

Combining vector coordinates in $\mathbb{R^3}$ from two perspectives

DISCLAIMER: I would like to apologize beforehand for asking this stupid question. Somehow I have a feeling that I should know this, even with basic math. Some of the examples I will give are probably ...
4
votes
0answers
116 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
4
votes
0answers
162 views

Computational hard math problem

Given a square filled randomly with the numbers $1$ to $N$, for instance $$\begin{array}{cccc} 16 &12 & 9 & 1\\ 11 & 3 & 4 & 7\\ 2& 8 & 5&14\\ 6& 10& ...
4
votes
0answers
516 views

algorithm for solving diagonal quadratic equations over real or complex numbers

I found the following statement in the paper http://www.math.uni-bonn.de/~saxena/papers/cubic-forms.pdf (page 22, in the middle): For $\mathbb F\in\{\mathbb R, \mathbb C\}$ and $b, a_i\in\mathbb ...
4
votes
0answers
138 views

Hardness of perimeter minimization?

Given $xy=C$ where $x, y$ are integer variables and $C$ is integer constant. What is the most efficient algorithm that finds $x,y$ such that $x+y$ is minimum? Providing references is highly ...
3
votes
0answers
84 views

Reverse Automorphic Numbers

I recently stumbled across Automorphic Numbers (definition and examples). In simple words, a number $n$ is said to be automorphic if last $d(n)$ digits of $n^2$ are $n$ itself (where $d(n)$ is ...
3
votes
0answers
59 views

Is there a better representation than p-adics for exact computer arithmetic?

I stumbled across Quote notation and went hog wild. But when I stumbled on a technical detail I received a very discouraging comment: I think those authors may have been a bit short-sighted, ...
3
votes
0answers
179 views

Finding a minimal perfect hash function for small sets quickly

I'm trying to solve the computer science problem "Minimal perfect hash function" (MPHF). I have an algorithm that can generate a MPHF for very large sets in $O(n)$ that only needs 1.54 bits/key, very ...
3
votes
0answers
39 views

Finding if f(n) = O(g(n) and vice versa…

Define $f(n) = \{ 3n-1$ if n is divisible by 3.. $n^2$ otherwise and $g(n) = \{ n^2$ if n is divisble by 3.. $2n$ otherwise Is $f(n) = O(g(n))$, is $g(n) = O(f(n))$? Here is my attempt at solving ...
3
votes
0answers
28 views

What does “the activation of a basis” mean?

In the paper Rajat Raina, Alexis Battle, Honglak Lee, Benjamin Packer, Andrew Y. Ng, Self-taught learning: transfer learning from unlabeled data, ICML '07 Proceedings of the 24th international ...
3
votes
0answers
36 views

NP-hardness of solving congruence equations in several variables

You are given the following equation modulo $N$ (where the $\beta_i$'s are given integers modulo $N$, and the $x_i$'s are unknown integers modulo $N$): $$\beta_1x_1 = \beta_2 x_2 = \ldots = \beta_l ...
3
votes
0answers
60 views

Complexity of finding set of sets with maximum cardinality and constrained coverage.

Given a set of sets $S = \{S_1, S_2, \dots, S_n$}, let $S^{'} \subset S$ be the largest subset of S that obeys $\left| \bigcup_{S_i \in S^{'}}{S_i} \right| \leq k$. What is the complexity of finding ...
3
votes
0answers
113 views

Tree decomposition by hand for understanding

I am implementing "algorithm 2" from the paper "Treewidth computations I. Upper bounds" by Bodlander and Koster[1,page5] and I am not sure if I understand it or not. As I understand, the algoritm ...
3
votes
0answers
58 views

Why is this not a poset after adding zero?

The problem    Consider the following set for divisibility. {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 96}. If 0 is added, the divisibility relation set will no longer be a poset. Please ...
3
votes
0answers
203 views

Execution time of function

At the algorithm SELECT, there is the following step: ...
3
votes
0answers
243 views

Converting a pushdown automaton (that accepts by final state) to a context-free grammar

Given the following PDA: $$ P = (\{q, p\}, \{0, 1\}, \{Z_0, X\}, \delta, q, Z_0, \{p\}) $$ where the transition function $\delta$ is given by: $$ \delta(q, 0, Z_0) = \{(q, XZ_0)\} \\ \delta(q, 0, ...
3
votes
0answers
115 views

A Tricky Quetiones on Creative Algorithm in Graph

an agent is works between n producer and m consumers. i'th producer, generate $s_i$ candy and j'th consumer, consumes $b_j$ candy, in this year. for each candy that sales, agent get 1 dollar payoff. ...
3
votes
0answers
96 views

Computability and continuous real functions

I have found somewhere the following statement: "Every computable real function has to be continuous," but I'm not able to prove it and the "proofs" that I found in some blog posts don't seem ...
3
votes
0answers
92 views

How to find the shortest path of a graph in a turing machine

I'm reading about Turing machine and I saw some examples as: Let $M_{1}$ a Turing Machine and the language $B = \{w\#w \vert w \in \{0,1\}^{*}\}$, We want $M_{1}$ to accept if its input is a member of ...
3
votes
0answers
89 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
3
votes
0answers
138 views

Related paradigms in Computer Science and Mathematics

I've read in a number of places discussions on whether Mathematics is a branch of Computer Science and vice-versa (see here for example) Having a background in both, I know that the Computer Sciences ...
3
votes
0answers
100 views

Minimizing set intersection/block design

I asked this question on the CS theory stack exchange, but didn't get an answer. Was wondering whether anyone here might have some insight. Thanks in advance for any help. Given 3 parameters $s, r$ ...
3
votes
0answers
96 views

Which takes more energy: Shuffling a sorted deck or sorting a shuffled one?

You have an array of length $n$ containing $n$ distinct elements. You have access to a comparator on the elements (a black-box function that takes $a$ and $b$ and returns true if $a < b$, false ...
3
votes
0answers
92 views

Boltzmann machines - motivation for the energy function

I've been studying Boltzmann machines lately and was wondering if anyone could give me a "high-level" explanation or motivation for the energy function used: $$E = -\sum_{i<j} w_{ij} \, s_i \, s_j ...
3
votes
0answers
235 views

Binomial Coefficients optimization

Given n and R, I have to find the minimum value of k such that: $${(2^n)-1 \choose k}\bmod(2^n)==R$$ Where $k = \{0, 1, 2, \dots, 2^n-1\}$ Here ${n \choose k}$ is the binomial coefficient ...
3
votes
0answers
55 views

Covering $n$ points with fewest disks with fixed radius $\epsilon$

The title says it all. I have a set of $n$ points in $\mathbb{R^{2}}$ and I am looking for an algorithm that tells me the fewest numbers of disks of radius $\epsilon$ that cover the set of $n$ ...
3
votes
0answers
224 views

Amortized Analysis for (2,5)-Tree

I need some help with the following problem Definition: A (2,5)-tree is an external search tree, where all leaves have the same depth. Each inner node in a (2,5)-tree has at least 2, and at most 5 ...
2
votes
0answers
51 views

Optimization of English Braille: Using the fewest dots

Background: The English Braille system is laid out in such a way so that the letters can be referenced by their position in the alphabet. Of the six dots available for each character, the top four ...
2
votes
0answers
41 views

How to choose set in Group Isomorphism Algorithm( Quasipolynomial time)

In the paper titled "On the $n^{log_2(n)}$ Isomorphism Technique" by Gary L. Miller, it is written A group is a binary operation * , satisfying 1) and 2) . 1) a)$ \exists! x(a*b = x)$ ...
2
votes
0answers
22 views

Math of Repetition

Let R = A (A | B) (A | B | C) (A | B | C | D) ... be an R-Form(A sort of repetition matching grammar). Where A,B,C,D,... are symbols that allow equality comparison. Any sequence of A,B,C,D,... can ...
2
votes
0answers
34 views

Are binary bit-strings the most efficient representation of integers?

There is no format more popular in the world than the representation of Integers: 32-bit and 64-bit strings are used by basically every single computer in existence and there's no practical reason to ...
2
votes
0answers
73 views

How to generalize “Seven trees in one” to labelled/colored trees?

In the famous paper Seven trees in one, Andreas Blass showed that there is "a particularly elementary bijection between the set $T$ of finite binary trees and the set $T^7$ of seven-tuples of such ...
2
votes
0answers
31 views

Probabilistic methods and equations over $m$-dimensional space

Given a set $A$ of $n$ different points in the space $(\mathbb{Z}_p)^m$ (assume $p$ is prime), and given $\delta>0$. show the following property holds for a big enough $n$ and $p$ (you can demand ...
2
votes
0answers
31 views

Equivalence relation on set $X$

Me and my friends were complaining about one of the exercises in discrete math. If there are two equivalence relations $R_1$ and $R_2$ on set $X$. Is $R_{1}\setminus R_{2}$ still an equivalence ...
2
votes
0answers
12 views

Naive Euclidean algorithm - average complexity?

Suppose I compute the GCD in a rather simple-minded recursive way: $$ \gcd(a, b) = \begin{cases} \gcd(a, b-a) & \text{ if }{a < b}, \\ \gcd(a-b, b) & \text{ if }{a > b}, \\ a & ...