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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270 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 ...
9
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175 views

Algorithm for obtaining the surface of a mirror

My colleague and I have been trying to implement an algorithm described in the paper "Recovering local shape of a mirror surface from reflection of a regular grid", primary author of which being ...
9
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177 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 ...
7
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194 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
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155 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 $b_1=(u,v_1)$ ...
5
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0answers
146 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
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0answers
99 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 ...
4
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37 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 ...
4
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102 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
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106 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
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331 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
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129 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
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75 views

For those wanting to study theoretical computer science, is Math a degree that worth more than Computer Science?

At my University, everyone first enrol in a Science and Technology, a very general course, and then choose the specialisation (like Computer Science, Math, some Engineering, etc). As I was finishing ...
3
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0answers
57 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
64 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. ...
3
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80 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
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0answers
375 views

How to reverse this bitwise AND-XOR encoding algorithm?

I have been given an "encoding" algorithm that does bitwise XOR and bitwise AND. Originally it's a C code that operates on integers with bit-shifts, but I have translated it into a simpler pseudocode ...
3
votes
0answers
69 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
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0answers
90 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
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0answers
77 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
140 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
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0answers
47 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
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0answers
184 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 ...
3
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0answers
317 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 set Q of the state of the Automaton, for q with $\delta(q,a)=q1*q2$, the node is a $*-node$ ...
2
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0answers
19 views

Decidability of given languages

Given are the following languages: $L_1 = \{0\}\\ L_2 = \{w \in \{0,1\}^{*} | L(M_w) = \{0\}\}\\ L_3 = \{w \in \{0,1\}^{*} | M_w \text{ stops at all entries }\} \\ L_4 = \{w \in \{0,1\}^{*} | ...
2
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0answers
53 views

Is a “network topology'” a topological space?

Is there any connection between the computer science phrase "network topology" and the mathematical notion of a topological space (or, is there any other way to connect "network topologies" with ...
2
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0answers
50 views

What computations would advance math knowledge a lot?

Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations ...
2
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0answers
37 views

Delete nodes that satisfy a property

I want to write a function that takes as argument a pointer A to the root of a binary tree that simulates a (not necessarily binary) ordered tree. We consider that each node of the tree saves apart ...
2
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0answers
87 views

Traveling salesman neighborhood

I am solving some TSP problems and i got this one and i am not pretty sure about my answer. By seeing TSP as a formal combinatorial problem, i have that the Feasible solutions $F$ is the set defined ...
2
votes
0answers
92 views

How can we find the elements?

I want to describe an algorithm with time complexity $O(m)$ that, given a set $M$ with $m$ numbers and a positive integer $p \leq m$, returns the $p$ closest numbers to the median element of the set ...
2
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0answers
52 views

How to check homeomorphic embedding relation programmatically?

This is a follow up to this question and Deedlit's answer. I'm looking for a precise definition of the "hem?" (tree A homeomorphically embeddable in tree B?) relation, preferably in terms of a ...
2
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0answers
34 views

Size of increments in commutative ring to reach given number

I have the ring $\Bbb Z_q = \{0,1,\ldots,q-1\}$, where $q$ is a prime. Starting from $0$, I want to make exactly $n$ equally sized increments and reach $a\in \Bbb Z_q$, with $n<q-1$. For example if ...
2
votes
0answers
23 views

Deriving a recurrence equation and apply Master's Thrm to it

So for a function such as: function Hi(n) if n > 1 then for j ← 1 to n do print(”Hi”) Hi(n/2) Hi(n/2) Hi(n/2) It's very easy to eyeball this and get a ...
2
votes
0answers
44 views

Recursive definition of a language

Define recursively the language L of all finite strings over the alphabet Σ={a b} satisfying both criteria: All words in L contain the substring aa an odd number of times. All words in L are such ...
2
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0answers
81 views

Humankind knows the prime factorization of the first how many consecutive integers?

I am only looking for an approximation. I'm guessing the answer must be somewhere between $10^{20}$ and $10^{50}$. . Edit: Okay so my first initial estimation was pretty poor... I should have ...
2
votes
0answers
46 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 ...
2
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0answers
60 views

Algorithm to reduce expressions to canonical form

I'm writing a small computer algebra system that only knows rational numbers and all expressions that you can get from them by using basic arithmetic operations and powers. So the expressions are ...
2
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0answers
33 views

Prove the big theta

I need to find a $n_0$ and $k$ for Big Oh and an $n_0$ and $k$ for Big Omega, to find a big theta bound for: $5n^2 - 9n = \theta(n^2)$ Can anyone help me and show me how to find these for this ...
2
votes
0answers
38 views

Computer program to simplify formulas

What is the computer program that attempts to simplify sums of binomial coefficients, factorials, etc.? Possibly Zeilberger wrote it, but I'm unsure. If so, possibly it was talked about in his A=B ...
2
votes
0answers
71 views

Local informatics Olympiad and Algorithm

I see one of recent local informatics Olympiad question. i have a trouble to solve it. any idea? hint? or solutions? thanks to all creative man. We have two function $P_1, P_2$ and input an array $n$ ...
2
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0answers
32 views

Generating interesting random TMs

To get a more intuitive understanding of the halting problem I want to generate some random TMs and see how they behave, what some heuristics can tell about them, etc. The problem is that, if I ...
2
votes
0answers
31 views

Bijection of simple set

Let $X$ is simple set (http://en.wikipedia.org/wiki/Simple_set) $Z \subset X$ is infinite recursive set. $Y = X$ \ $Z$. How to prove that there is a computable bijection $f$ that $x \in X ...
2
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0answers
48 views

Turning a cup into a donut: parametrising the continuous deformation

A cup and a donut are topologically equivalent because one can be continuously deformed into another. What if you had a particular cup and a particular donut and wanted to write down (or simulate on a ...
2
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0answers
35 views

Chaitin's constant for lambda calculus and combinatory logic

I have found some approximations of Chaitin's Constant for turing machines but I have not found approximations for others. I'd like to have a rough estimate or upper bound on it for lambda calculus ...
2
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0answers
59 views

How to represent an algebraic number in algorithm

I'm studying the problem of computing $n$th digit of given (irrational) algebraic number. Is there some canonical way to represent an algebraic number as a string I could use in computing? Basically ...
2
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0answers
32 views

How do you find a minimum of a function with these tools?

Let's say I can define a group $G$ acting on a set of combinatorial objects $X$ and I have a function $f: X \to \Bbb{N}$ that I want to find a minimum of in $X$. Is there a polynomial time ...
2
votes
0answers
78 views

Todd-Coxeter algorithm: coincidences

I'm trying to understand the Todd-Coxeter algorithm with the help of a multiplication and relator table, but there is one thing about coincidences that is not really clear. For some small groups (for ...
2
votes
0answers
22 views

A confusion about RP class of problems

I have some notes which introduces the quantifier $\exists^+x$ and interprets it as "the overwhelming majority of $x$". Then, it defines RP (Randomized Polynomial) as: $$ L\in RP\Leftrightarrow ...
2
votes
0answers
66 views

Implications of NP=coNP for PSPACE

If NP = coNP, then the Polynomial Hierarchy collapses to its first level (NP). Intuitively, it seems to me that PSPACE should collapse down to NP as well. As a loose heuristic argument, take the ...
2
votes
0answers
80 views

Cut-off Subtraction in Coq

I am new to the world of computer assistant proof programs in general, and Coq in particular. As a result, I have sought to prove some elementary results about integers as a way to … At the moment, I ...