Questions tagged [computer-science]
For question borderline with, or having application to, computer science. Consider also posting http://cs.stackexchange.com/ or http://cstheory.stackexchange.com/ instead of here, if appropriate.
656 questions
3
votes
2
answers
183
views
Why is graph non-isomorphism in coC_=P?
I recently read in the paper "Quantum NP is hard for PH" by S. Fenner et. al. that "graph non-isomorphism is known to be in coC_=P", but they did not attach a reference. I have ...
- 241
1
vote
0
answers
132
views
What is the minimal strength a logic should have to be able to express, say, first order logic?
I am trying to figure out what would qualify as something like the mother of all logics. The motivation of this question comes from studying some model theory. There I came across this interesting ...
- 1,693
-1
votes
0
answers
81
views
Do we have a hierarchy/generalisations of effectively axiomatised theories, just like we have a hierarchy of Turing machines?
The notion of an effectively axiomatised theory is based on computing abilities of a Turing machine. For instance, the wffs and proofs are required to be decidable, the theorems are required to be ...
- 107
46
votes
16
answers
9k
views
Examples for the use of AI and especially LLMs in notable mathematical developments
The purpose of this question is to collect examples where large language models (LLMs) like ChatGPT have led to notable mathematical developments.
The emphasis in this question is on LLMs, but ...
6
votes
2
answers
798
views
Which conjectures have been solved (or partially solved) using entropy methods?
I am studying how entropy-based arguments (in the sense of Shannon, Boltzmann, or Perelman's geometric entropy) can be used to prove or guide the resolution of major mathematical conjectures.
Some ...
3
votes
1
answer
200
views
Supernormal sequences
We call a binary sequence $s:\mathbb{N}\to\{0,1\}$ supernormal if
for every injective, increasing and computable function $\iota:\mathbb{N}\to\mathbb{N}$, the binary sequence $s\circ\iota:\mathbb{N}\...
- 54.4k
11
votes
1
answer
464
views
Is there any algorithm which can find a common divisor of two polynomials modulo $p^k$?
Let us consider two monic polynomials $f(X), g(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]$. Now, we call $h(X)$ is a divisor of $f(X)$, if there exists a $l(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]...
- 311
0
votes
0
answers
172
views
Algorithms for searching equal sums of like powers (taxicab problem)
I am developing a brute-force algorithm to search for solutions to the generalized taxicab equation, $a^k + b^k = c^k + d^k$ between a lower and upper bound. My current approach is a priority queue-...
8
votes
0
answers
677
views
Automating the resolution of IMO 2025 Problem 1
The problem 1 of the 2025 IMO is the following:
A line in the plane is called sunny if it is not parallel to any of the x-axis, the y-axis, and the line $x + y = 0$.
Let $n ⩾ 3$ be a given integer.
...
- 371
1
vote
0
answers
111
views
Is the NH hash family $\varepsilon$-AXU?
As context, I'll start with summarizing and simplifying the section of "UMAC: Fast and Secure Message Authentication", by Black et al.(https://www.cs.ucdavis.edu/~rogaway/papers/umac-full....
- 111
2
votes
1
answer
178
views
Gröbner Basis That do not Introduce New Indeterminates
Let $\vec{X}$ be a finite set of indeterminates, and $\sqsubseteq$ be a monomial ordering.
A Gröbner Basis $B$ of an ideal $I$ of polynomials can be characterized as a finite set of polynomials ...
20
votes
4
answers
2k
views
Is there a concept of Turing Machine over a group, not just over the integers as a model of the tape?
For a Turing machine with an infinite tape, the tape may be modeled by the group of integers. So a rule $(q,a,b,m,r)$ says "in state $g$ if see $a$ on tape replace it by $b$ and go left if $m=-1$ ...
- 521
15
votes
2
answers
1k
views
What is the implicit pseudorandomness conjecture behind the use of e.g. numpy.random() for probabilistic applications?
Suppose I want to investigate some complicated probabilistic phenomenon numerically, e.g. the eigenvalues of random matrices. One thing I might do is (ask some software to) generate a bunch of random ...
- 124k
6
votes
1
answer
290
views
Deciding positivity for polynomials summed across a regular language
Motivation: Long story short, this problem arose after a number of reductions and simplifications from a question in symbolic dynamics motivated by the paper Symbolic discrepancy and self-similar ...
- 1,483
7
votes
1
answer
252
views
Polynomially normal binary sequences
Motivation. When we try to construct a (pseudo-)random sequence $s:\newcommand{\N}{\mathbb{N}}\N\to\{0,1\}$ we often want $s$ itself, and some of its subsequences, to be normal.
Question. Is there a ...
- 54.4k
1
vote
1
answer
114
views
Surjective hash functions $h:\{0,1\}^* \to \{0,1\}^{2n}$ with avalanche effect
Motivation. In computer science, hash functions are maps that convert binary strings of arbitrary length to a fixed-length binary string. In symbols, we have a map $h:\{0,1\}^* \to \{0,1\}^n$ for some ...
- 54.4k
1
vote
1
answer
138
views
Graph classes which have small edge k-cuts
I am interested in graph classes that have the following property: There exists a function $f(k)$ such that for every graph $G$ in the class, for every choice of $k$ vertices $v_1, \ldots, v_k$ in the ...
- 528
5
votes
2
answers
2k
views
Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
- 54.4k
5
votes
1
answer
544
views
Is the set of generalized Fermat triples computable?
Is $\;\big\{(a,b,c)\in\mathbb{N}^3: \big(\exists m,n,\ell \in (\mathbb{N}\setminus\{0,1,2\})\big): a^m + b^n= c^\ell\big\}\;$ computable?
- 54.4k
6
votes
0
answers
149
views
Computer program for free restricted Lie polynomial
I am conducting numerical experiments involving the Gröbner–Shirshov Basis for restricted Lie algebras. At each step of the computation, I need to work with restricted Lie polynomials. Specifically, I ...
- 1,143
3
votes
1
answer
154
views
References: rigorous algorithms for elementary computations in base-b with complexity estimates
Definitions/Notation: Fix positive integers $b$ and $M$. Consider the set of real numbers which can be exactly expressed with $2M+1$ coefficients in base $b$, defined by
$$\mathcal{X}(b,M):=\{x\in \...
- 4,842
3
votes
0
answers
149
views
References on P vs NP under various axiomatic systems
I am teaching algorithms and theory of computation this semester and had the opportunity to dig a bit into the details of one way functions and the P vs NP problem.
This problem has resisted attacks ...
- 31
1
vote
0
answers
58
views
Computing all roots of a function with square-root terms
Given $3n$ positive numbers $a_1, \ldots, a_n$, $b_1, \ldots, b_n$, and $x_1, \ldots, x_n$, we are given a function
$$f(x) = \sum_{i = 1}^n \frac{a_i}{\sqrt{(x - x_i)^2 + b_i}}.$$
Can we find all the ...
- 11
1
vote
1
answer
406
views
Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
- 11
3
votes
0
answers
208
views
Lower Bound of Solutions to P=NP?
Do we at least know that simulating polynomial time non-deterministic Turing machines requires more than a linear slowdown? That is, do we know there is some non-deterministic Turing machine with ...
- 4,039
9
votes
1
answer
2k
views
2
votes
0
answers
105
views
Is this variant of post correspondence problem undecidable?
The post correspondence problem, as defined by wikipedia, is undecidable. The problem is defined as follows.
Let $A$ be an alphabet with at least two symbols. The input of the problem consists of ...
- 21
1
vote
0
answers
86
views
On binary constraints defined by vanishing of bivariate polynomials modulo $n$ [duplicate]
In an answer here
Dima Pasechnik showed that constraints of the form $x_i x_j + a_{ij}x_i + b_{ij}x_j + c_{ij}$ are efficiently solvable modulo $2$ using Groebner basis.
In comments he suggested that ...
- 25.8k
3
votes
1
answer
500
views
About Shor's quantum algorithm
I know very little about quantum computing, and I've been trying to understand Shor's algorithm for the factorization of an integer $N$. I'm following Computational Complexity — a modern approach by ...
- 2,359
1
vote
0
answers
49
views
Multipoint evaluation in Lagrange basis on subset smaller than degree
Setup. Let $\mathbb{F}$ be a finite field with a multiplicative subgroup $E = \{e_1, \dots, e_k\}$ of order $k$. Given a list $y = y_1, \dots, y_k\in \mathbb{F}$ let $p$ be the unique polynomial of ...
2
votes
0
answers
176
views
What do we know about efficiently finding a solution to a system of multivariate polynomials over finite fields?
Consider the following (NP-complete) problem:
Given a system of polynomials $f_1, f_2, \ldots, f_m \in \mathbb{F}_q[x_1, x_2, \ldots, x_n]$ of total degree at most $d$, find an $\mathbb{F}_q$-rational ...
- 121
0
votes
0
answers
209
views
How can I transform every graph into one with constant out-degree?
I am working on my master thesis and try to implement a new shortest path algorithm from the following paper: https://arxiv.org/abs/2203.03456
In some of the functions (for example ScaleDown), ...
1
vote
1
answer
147
views
Probabilistic 2D cellular automata with memory lifetime increasing like $e^{L^2}$
Consider 2-state probabilistic cellular automata on an $L\times L$ torus square lattice which has the all-$0$ and all-$1$ configurations as fixed points, thinking of something similar to Toom's rule ...
- 3,105
4
votes
1
answer
436
views
rank of an integer valued matrix
I make some numerical experiments, involving rank of integer valued matrices of the size about $14\times 24$. As the matrix is integer valued, theoretically there should be no room for errors. However ...
- 678
2
votes
0
answers
212
views
NP-hardness of a string transformation problem with k templates
Given strings $x$ and $y$, a template length $l$, and a maximum number of different templates $k$, the task is to determine if it's possible to convert $x$ into $y$ using no more than $k$ different ...
- 21
2
votes
0
answers
73
views
graphs which have polynomial bounded number of cycles
How does the graph class defined as those graphs which have polynomial (or quasi polynomial) bounded number of cycles look? (in number of vertices)
I suspect it will rather non-interesting as ...
- 129
8
votes
1
answer
3k
views
Polynomial-time quantum algorithms for lattice problems (GapSVP, SIVP, LWE)
The author of a recent preprint claims to have found polynomial-time quantum algorithms for solving the following lattice problems: the Decisional Shortest Vector Problem (GapSVP), the Shortest ...
- 434
1
vote
2
answers
299
views
Topology of directed graph $G$ with non-singular adjacency matrix
Given a directed graph $G$ with non-singular adjacency matrix,
Q. Is there a directed
subgraph $H$ in $G$ that can be represented as the union of disjoint cycles such that $H$ contains all nodes of $...
- 4,150
1
vote
1
answer
231
views
Permutation graph with insert-and-shift
Motivation. I am working with a database software that allows
you to sort the fields of any given table in the following
peculiar way. Suppose your fields are numbered $1,\ldots, 18$.
Next to every ...
- 54.4k
0
votes
2
answers
188
views
Isometric path cover number of the 2 dimensional grid graph
I am looking for a proof of the fact that at least $2n/3$ isometric paths (i.e. shortest paths between the end points) are required to cover the vertices of the $n\times n$ grid graph (i.e. Cartesian ...
- 1,435
14
votes
2
answers
3k
views
Soft question: Deep learning and higher categories
Recently, I have stumbled upon certain articles and lecture videos that use category theory to explain certain aspects of machine learning or deep learning (e.g. Cats for AI and the paper An enriched ...
- 367
1
vote
0
answers
83
views
A small lemma on cache resets (Bloom filters in particular)
Assume a fixed set of message $D$ and an associated distribution for selecting each message $d_i$ such that the total probability $\sum_{i \in D} d_i = 1$. We create a cache with $M$ bits and $k$ ...
1
vote
0
answers
114
views
Interpreting multiple property tests at different values of $\epsilon,\delta$ [closed]
I am doing some work in the area of Property Testing, as in Goldreich, Goldwasser, and Ron (2008) or the textbook Introduction to Property Testing (Goldreich). In this framework, I run a test to see ...
- 171
3
votes
0
answers
131
views
What does the computation of irrationality and transcendentality via a fancy implementation of analytic Markov's property look like?
Proofs that various real numbers are not rational or not algebraic tend to be constructively valid as is. Examples include the proofs that $\sqrt 2$ and $\log_2(3)$ are not rational and that $e$ is ...
- 7,158
1
vote
0
answers
95
views
Set functions satisfying if $f(X) \le f(Y)$ and $Z \cap (X \cup Y) = \emptyset$, then $f(X \cup Z) \le f(Y \cup Z)$
I am investigating set functions $f : 2^\Omega \to \mathbb{N}$ satisfying the following two properties:
(monotone) For all $X, Y \subset \Omega$, if $X \subseteq Y$, then $f(X) \le f(Y)$.
(property ...
- 151
1
vote
0
answers
153
views
When is a container a monad?
The category of polynomial functors on Set is equivalent to the category of containers.
We have a prescription for when a container is a comonad.
There are a few other questions that come to mind. ...
- 1,289
-4
votes
1
answer
138
views
Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
38
votes
3
answers
7k
views
Using Busy Beavers to prove conjectures
I've been pondering some stuff on Shtetl Optimized where Yedidia and Aaronson construct Turing machines that will only halt if (e.g.) the Riemann Hypothesis is false, or Goldbach's conjecture is false....
- 513
3
votes
0
answers
151
views
Computational complexity of exact computation of the doubling dimension
Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
3
votes
1
answer
368
views
Root finding algorithm for an analytic function
Given an analytic function $f(x)$. What is the best algorithm to find roots on the interval $[a,b]$ inside the radius of convergence> What is its complexity with respect to the length of input of ...
- 81