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Computer Science Stack Exchange

Questions tagged [algebra]

50 questions
25
votes
2 answers

Is there a non-trivial type which is equal to its own derivative?

An article called The Derivative of a Regular Type is its Type of One-Hole Contexts shows that the "zipper" of a type—its one hole contexts—follow the differentiation rules in type algebra. We have: \begin{align} \partial_x x &\mapsto 1…
Matthew Piziak
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8
votes
1 answer

Convert HSV to RGB colors

HSV colors are composed of a triple of numbers: hue $\in [0, 360)$ (in degrees), saturation $\in [0, 1]$ and value or brightness $\in [0, 1]$. RGB colors instead are more well-known and are also composed of a triple of numbers all of them in the…
user20691
8
votes
2 answers

Solving systems of linear equations over semirings

So I have come across an issue where it would be very nice to solve systems of linear equations over semirings but I have no clue how to do that. Over a field I would use Gaussian elimination but I'm at a loss of how to find solution spaces to even…
Jake
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6
votes
1 answer

What is the fastest algorithm to check whether input table is a group?

Given an input table with binary operation $\circ$. I want to check whether given input table represents a group or not. I need to verify the four axioms of a group. The identity I can find it by just scanning the first row of the given table.…
sssa
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5
votes
1 answer

Complexity of covering subset of the monoid $(\{0,1\}^n, \text{OR})$

(At the very bottom of this, I will shortly describe the motivation for this question.) Assume we have a commutative monoid $(G,\circ)$, i.e. a set $G$ with a commutative binary operation $\circ$ that satisfies associativity and the existence of a…
G. Bach
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5
votes
1 answer

Why is Dyck-2 so important for the Chomsky-Schützenberger theorem?

I have read a lot of times, that models that can parse Dyck-2 are of great importance. It appears that Dyck-2 is interchangeably used like Dyck-N. Afaik the Chomsky-Schützenberger representation theorem states that you can convert and context-free…
Crea Teeth
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4
votes
1 answer

What is the difference between Boolean Algebra and Boolean Lattice?

What is the difference between Boolean Algebra and Boolean Lattice? I have already searched on Google but could not find a reasonable answer?
Kishan Kumar
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4
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Can every sentence of first-order logic be converted into an equisatisfiable equation in Boolean algebra?

There may be some theoretical literature, unknown to me, that addresses this question. If possible, I would like a practical approach to this problem. My attempt involves the use of an equational specification language called CafeOBJ (see…
Patrick Browne
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4
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Testing whether polynomial is in algebra of other polynomials

A collection $\Sigma$ of polynomials is an algebra if: (1) $\lambda f + \eta g \in \Sigma$ for any $f,g \in \Sigma, \lambda,\eta \in \mathbb{R}$ and (2) $f,g \in \Sigma$ implies $fg \in \Sigma$. We say that $P$ is in the algebra of…
user114441
4
votes
1 answer

Decidability of factoring algebraic equations

Given an arbitrary algebraic equation, say for example the likelihood of the bernoulli distribution: $$\prod_{i}^{n}\theta^{x_i}(1-\theta)^{1-x_i}$$ And some arbitrary factorization constraints, say: $$f(x)g(\theta)e^{\phi(\theta)^{T}u(x)}$$ Can I…
JackSprat
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3
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A book introducing proof theory needed (many-sorted FOL, classical non-Gentzen calculus, satisfiability in partial algebras, induction)

We define a signature as a triple $$\Sigma\ =\ (S,F,\mathrm{type})$$ where $S$ is a set of sorts, $F$ a set of $n$-ary function symbols $f$ of the type $\mathrm{type}(f)$ $=$ $(M_1,\dotsc,M_n\dashrightarrow M_{n+1})$, where $n\ge 0$ and all $M_i$…
user90064
3
votes
1 answer

Algorithm to find the size of a quotient of a free group

Are there any algorithms to find the size of an algebraic quotient of a free group? It would take the generators as input and output the size. For example, an input could be something like {a,b: a^8=b^2=1, ab=ba^3} (a quasidihedral group), and the…
Alex Li
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2
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2 answers

How to Apply Elementary Axioms from Kleene Star to an Inequality

Axioms For * \begin{align} 1 + aa^* &\leq a^* \\ 1 + a^*a &\leq a^* \\ b + ax &\leq x \to a^*b \leq x \\ b + xa &\leq x \to ba^* \leq x \\ \end{align} Elementary Results \begin{align} a \leq b &\to a + c \leq b + c \\ a \leq b &\to ac \leq bc\,…
grant2088
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2
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Optimal vector decomposition

I have a vector $v \in \mathbb{N}^k$ and a set of vectors $R \subset \mathbb{N}^k$, with $k \ll \left\vert R \right\vert $. I would like to find a way to obtain all the possible bases of $\mathbb{N}^k$ taking $k$ elements from $R$. The ultimate goal…
Michele Ippolito
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2
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1 answer

Producing an algebric equation from a graph

I'm writing a computer game and one of my game's objects must follow a movement path that is very similar to the following graph. The bold lines are the Y and X axis. In order to be able to code this movement thought I need the algebric equation…
exophrenik
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