Questions tagged [computer-science]
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641 questions
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Reference request: Time and proofs of shared pasts
Is there research about structures for notions of time with distributed systems of information, as with blockchains?
I am thinking of tuples $(I, T, P, A, \prec, s, \eta, u)$ where
$I$, $T$ and $P$ ...
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About the complexity of some operation involving integers
There are two integers: $A, B$. Given the below four allowed operations (and only them):
$A+1$, $A-1$, $\sqrt{A}$, $A^2$
Also, it is only allowed to take the square root of $A$ when this square root ...
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Can information be extracted more precisely using more random trials?
Write $n$ iid draws of $(x,y)$ as $(x^n, y^n)$. Fix $R\in (0,H(x))$. What is the min of $n^{-1}H(y^n|f(x^n))$ over maps $f$ with range $\lbrace 1,\dots,\exp nR\}$, taking $n\to \infty$?
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Can computers find zeros of order $2$?
We assume we are given an entire function $f: \mathbb C \to \mathbb C$ with $f(0)=1$ and $f'(0)=0$ and $f$ is real on the real axis.
We assume (as a fact about $f$, that we want to demonstrate ...
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Can we explicitly compute this "shift"-quantity over Boolean functions $u:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$?
This question is a follow-up of this question.
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements, and suppose that $n$ is odd.
Question: Can we compute the exact minimum $$A:=
\min_{u:\mathbb{...
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Smallest relation in complement of partial order that prohibits its extension
Let $P$ be a partial order on a finite set $S$ (assume that every element is related to at least one other element besides itself…this raises a few quick questions: is this implied by the definition ...
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Finding the union of N random circles arbitrarily (or conspiratorially) placed on a two-dimensional surface
Please consider a two-dimensional surface populated with a set of Cartesian coordinates $(x_i, y_i)$ for $N$ circles with individual radii $r_i$, where $r_{\min} < r_i < r_{\max}$.
Here, the ...
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Non-isomorphic graphs with identical iterated degree matrix
If $G = (V, E)$ is a simple, undirected graph and $T \subseteq V$, let $$N(T) = \{v \in V: \{v, t\}\in E \text{ for some }t\in T\}.$$
Given $v\in V$ we let $N_0(v) = \{v\}$ and $N_{k+1}(v) = N_k(v) \...
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SAT and Arithmetic Geometry
This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...
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Enumerative algorithm through inclusion-exclusion
Hello everybody !
I wondered, without really knowing where to search, whether there was a "smart" way to enumerate/iterate over all the elements of a set which can be counted by inclusion-exclusion. ...
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Polynomial size embeddings of toric varieties from polytopes?
Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...
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Union of admissible words are subshift of finite type
Assume that $Q=(q_{ij})$ is a $k\times k$ with $q_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma_{Q}\rightarrow \Sigma_{Q}$, where
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Can we cover the unit square by these rectangles?
The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \...
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Nearly De Bruijn sequences constructed from De Bruijn sequences
Let $w$ be a De Bruijn $01$-sequence of the type $B(2,n)$; i.e., a cyclic $01$-sequence that contains every $n$-digit $01$-sequence exactly once. Let $x$ be a $01$-sequence of length $n$. When and ...
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How is Ricci flow related to computer graphics?
I recently came across the book Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications by Wei Zeng and Xianfeng David Gu. Because, I just saw the book on the ...
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$\omega$ incompleteness of $\lambda$ calculus
In Plotkin's 'The $\lambda$-Calculus is $\omega$-Incomplete' (The Journal of Symbolic Logic Vol. 39, No. 2 (Jun., 1974), pp. 313-317), an example is given of two (untyped) $\lambda$-terms $M$ and $N$ ...
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3D interpolation function
I've got a 3D figure created using around 30k points and has different regions colored in an specific way according to some unrelated variables that come from a project I'm creating. Taking in ...
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Are there functions $\mathbb{F}_2^n \to \mathbb{F}_2$ satisfying these special relations?
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements, and let
$u:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$. Suppose that $n$ is odd.
Is it possible that
$$
\sum_{x \in \mathbb{F}_2^n}(-1)^{u(x)+u(...
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What is the mathematical name for Haskell's Alternative Functor
Haskell's Applicative Functor is called Lax Monoidal Functor in mathematics.
What is Haskell's Alternative Functor called in mathematics?
Recall that Haskell's Alternative Functor is defined as ...
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Is there a complete countable axiomatization of conditional independence? (Graphoids)
Note: A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly ...
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Hamilton cycles in $\{0,1\}^n$ with fixed Hamming distance
Let $n>1$ be an integer. For $a,b\in \{0,1\}^n$ let $d_h(a, b)$ denote the Hamming distance of $a$ and $b$. For $k\in \{1,\ldots,n-1\}$ let $H(n,k)$ be the graph on $\{0,1\}^n$ given by the edge ...
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Cycling through $\{0,1\}^n$ by shifting and applying a $n$-ary function
This question is motivated by Linear Feedback Shift Registers, which cycle through $\{0,1\}^n \setminus \{(0,\ldots,0)\}$ by shifting and applying a small set of XOR operations.
Let $n>1$ be an ...
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Cycling through $\{0,1\}^{(2^n)}$ such that all Hamming distance appear equally frequently
Let $n\in\mathbb{N}$ be a positive integer. Let $\{0,1\}^{(2^n)}$ be the set of $0,1$-sequences of length $2^n$. For $a,b\in \{0,1\}^{(2^n)}$ let $d_h(a,b)$ be the Hamming distance between $a$ and $b$....
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Is normalcy preserved under the swapping operation?
Let $\mathbb{N}$ denote the set of non-negative integers. We say that a sequence $f:\mathbb{N}\to \{0,1\}$ is normal if every finite $\{0,1\}$-sequence appears in $f$.
Let the swapping operation $\...
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Binary operation approximating "addition" on $2^\omega$
Motivation. In computer science, addition of integers $a+b$ can be approximated by a very fast operation: $(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$, where $\oplus$ denotes bitwise XOR, $\...
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Decision problems for which it is unknown whether they are decidable
In computability theory, what are examples of decision problems of which it is not known whether they are decidable?
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Understanding statement about bounds of vector in the context of a RSDF ≤ₘ WOPT proof
I'm trying to follow the proof of Lemma 4 of "Strong NP-Hardness of the Quantum Separability Problem", by S. Gharibian, 2010 [1], which, roughly, states that there is a many-one reduction ...
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Boolean function : approximation by a linear function
Let $f$ be a balanced Boolean function.
Are there $g$ linear functions, with $$\frac1{2^n}\mathrm{card} \big(\big\{\mathrm{sign} (g (x)) = 2f (x) -1, x \in \{0,1\}^n\big\}\big) > 0.55\quad ?$$
$g ...
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When does Le Cam's method give tight lower bounds for distribution testing?
In the context of statistical estimation or distribution testing, Le Cam's method is a way to prove lower bounds on the minimax sample complexity ([1,2,3,4], further details below). My question is: ...
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Games and the right mathematical framework for GANs
Generative Adversarial Networks were introduced in http://papers.nips.cc/paper/5423-generative-adversarial-nets and has more than 20000 citations.
It is an important topic within deep learning.
Are ...
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Cramer–Castillon problem like
Special case of Golden ratio as a property of conic section (is it known?) as follows:
Let $ABC$ be arbitrary triangle and $DEF$ is the its tangential triangle. Let $CF$ meets $AB$ at $G$ and $BE$ ...
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Can finite sets be non-c.e. depending on how they are presented?
I ask the question because of the following statement found in Mark Burgin's paper, "Algorithmic complexity of recursive and inductive algorithms", Theoretical Computer Science 317 (2004) 31-...
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How to draw Archimedean-Galileo spiral?
It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and ...
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Is the following two-dimensional graph likely to be globally rigid?
Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...
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Finding a $k$-subset which maximizes a matrix sum
Let $M\in \mathbb{R}^{N\times N}$ be a given matrix and $k\ge 2$ be a given integer. Then my question is the following optimization problem:
Is there a polynomial-time solution to the following ...
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Is Toom's rule robust under local but non-on-site noise?
Toom's rule is a 2-dimensional cellular automaton which is known to have two distinct stationary measures in the thermodynamic limit, even after small perturbations to a probabilistic cellular ...
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Are there (probablistic) uniform 1D cellular automata which can fault-tolerantly store one bit?
In two dimensions, "Toom's rule" is known to be a cellular automaton which can fault-tolerantly store one bit of information. This means that, if we start with the all-0 configuration on an $...
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Graph Theory Conjectures [closed]
What are some important conjectures in graph theory that have been checked by computer up to order 11?
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How do I fit flow values to connections in a known network?
This is not my area and I'm new to its terminology, and am posting my problem in the hope that someone will be able to direct me to where it has been solved, or who has written about it.
I have a flow ...
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CCCs, computational calculi and point-surjectivity
The models of some computational calculi are in a correspondence with Cartesian Closed Categories with an object $U$ that has some relationship to its exponential object $U^U$ e.g. a retraction ...
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Complete problems for randomized complexity classes
It is believed that $BPP$ has no complete problems. Even for $BPP^O$ for a suitable oracle $O$ it is believed not to have complete problems, unless P=BPP. I wonder if the class MA (the randomized ...
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Is factorial computation known to be in a class smaller than $FEXP$?
Functional version of the counting hierarchy is $FCH$. It is an open problem whether there a sequence of $poly(log(n))$ number of $+,\times$ operations utilizing the assistance of $O(1)$ number of ...
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Efficient Algorithm to Find Subset of Vectors Over $\mathbb{F}_q$ Living in Low Dimensional Subspace
Let $q$ be a fixed prime, $P, Q$ be polynomials with $\mathrm{deg}(Q) < \mathrm{deg}(P)$ and $h = O(\log n)$.
Let $S$ be a subset of $\mathbb{F}_q^n$ of size $P(n)$ such that there exists a subset ...
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How does the greedy algorithm for CSES problem collecting numbers work? [closed]
The collecting numbers problem in the CSES problem set has a greedy solution where we compare the position of a number x with the position of x-1. If pos(x) < pos(x-1) then we increment rounds ...
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Most efficient reductions between NP-complete problems
Assume I need to solve an NP-complete problem, for which problem-specific methods (e.g. efficient heuristics or exponential algorithms faster than naive one) are not well developed. If the size of ...
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Search algorithms with mappings/functions/sets as variables
I apologize in advance if this sounds vague but I am trying to find directions as to what to look for.
All the sets in this problem are finite.
Suppose we have two functions $f_1\colon X_1\times Y_1\...
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Are there some algorithms which have high consistency strength?
Are there some algorithms, their time complexity is relatively good, for example polynomial time.
And the correctness of them has high consistency strength.
And these algorithms shouldn't able to ...
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Estimating the fractal dimension of a point cloud
I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most ...
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Does the morphism of composition have some universal property?
Let $A$, $B$ and $C$ be three objects in the category Set. For simplicity, assume that their underlying sets contain a finite number of elements, a, b and c respectively. Using the usual Haskell ...
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Programming Languages Based on Category Theory
Since some computer scientists use category theory, I was wondering if there are any programming languages that use it extensively.
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