Questions tagged [algebra]
The algebra tag has no usage guidance.
105 questions
81
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Uses of algebraic structures in theoretical computer science
I'm a software practitioner and I'm writing a survey on algebraic structures for personal research and am trying to produce examples of how these structures are used in theoretical computer science (...
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votes
12
answers
3k
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Gröbner bases in TCS?
Does anyone know of interesting applications of Gröbner bases to theoretical computer science?
Gröbner bases are used to solve multi-variate polynomial equations, an NP-hard problem in general. I was ...
- 11k
33
votes
12
answers
7k
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Algebra oriented branch of theoretical computer science
I have a very strong base in algebra, namely
commutative algebra,
homological algebra,
field theory,
category theory,
and I am currently learning algebraic geometry.
I am a math major with an ...
- 481
28
votes
6
answers
3k
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Alternative proofs of Schwartz–Zippel lemma
I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz.
Are there any other ...
- 3,684
27
votes
2
answers
4k
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What is the logarithm or root operation in type-space?
I was recently reading The Two Dualities of Computation: Negative and Fractional Types. The paper expands on sum-types and product-types, giving semantics to the types ...
- 373
23
votes
4
answers
9k
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Abstract algebra for Theoretical Computer Scientists
I have a reasonable undergrad math education but have never been 100% comfortable with abstract algebra (the mathematics of groups, rings, fields etc. ). I think this was partly as I needed to see ...
- 353
20
votes
4
answers
3k
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Data Structure isomorphisms
Disclaimer: I am not a CS theorist.
Coming from abstract algebra, I'm used to dealing with things that are equal up to a isomorphism - but I'm having a trouble translating this concept to data ...
- 203
20
votes
3
answers
889
views
Computing sum of sparse polynomials squared in O(n log n) time?
Suppose we have polynomials $p_1,...,p_m$ of degree at most $n$, $n>m$, such that the total number of nonzero coefficients is $n$ (i.e., the polynomials are sparse). I am interested in an efficient ...
- 972
18
votes
1
answer
4k
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The quad-edge data structure (Delaunay/Voronoi)
2 questions for the computational geometers or algebraists:
I am just beginning to dive into computational geometry and I am loving it =)
I am attempting to read the famous article by Guibas and ...
- 283
18
votes
2
answers
2k
views
Is there a theory that combines category theory/abstract algebra and computational complexity?
Category theory and abstract algebra deal with the way functions can be combined with other functions. Complexity theory deals with how hard a function is to compute. It's weird to me that I haven't ...
- 1,073
17
votes
3
answers
682
views
Formal representation of rings in computations
While reading a paper about using algebraic methods to detect some induced subgraphs, it appears that edge ideal is an important tool connecting commutative algebra and graph theory. Since I'm not ...
- 7,668
16
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3
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Hardness Guarantees for AES
Many public-key cryptosystems have some kind of provable security. For example, the Rabin cryptosystem is provably as hard as factoring.
I wonder whether such kind of provable security exists for ...
- 16.6k
16
votes
0
answers
372
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Looking for an operator on polynomials
I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc).
I wonder ...
- 11k
15
votes
3
answers
2k
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On the realisation of monoids as syntactic monoids of languages
Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as
$$
u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L
$$
and the quotient ...
- 2,057
14
votes
2
answers
751
views
Are There Highly Symmetric NP- or P-complete Languages?
Does there exist $L$, an NP- or P-complete language which has some family of symmetry groups $G_n$ (or groupoid, but then the algorithmic questions become more open) acting (in polynomial time) on ...
- 1,582
14
votes
2
answers
513
views
What's the bias of random polynomials with low degree over GF(2)?
I have a question concerning low-degree polynomials and probability:
What is the (assyptotic behavior of the) probability that a
random* polynomial, $p$, over GF(2), with degree $\le d$ and n ...
- 363
13
votes
4
answers
2k
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(N)DFA with same initial/accepting state(s)
What is known about the class of languages recognized by finite automata having the same initial and accepting state? This is a proper subset of the regular languages (since every such language ...
- 4,831
12
votes
2
answers
1k
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Applications of algebraic geometry in type theory/programming language theory
Lately, I have become interested in algebraic geometry and have started reading on it. I still know very little about this field, but I do want to know if it has any connection with my main field, ...
- 1,195
12
votes
2
answers
1k
views
List of number theoretic or algebraic problems in various complexity classes
I am looking for a list about the known or unknown complexity of various number theoretic /algebraic problems. For example,
GCD in $NC^1$ is open,
factoring in $P$ is open,
computing sheaf ...
- 13.3k
12
votes
1
answer
2k
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Complexity of convolution in the max/plus ring
We can do convolution in $O(n\log n)$ for plus/multiply polynomials with FFT. However, the approach doesn't seem very generalisable to rings in general.
Has there been any progress over the naive $O(...
- 958
11
votes
2
answers
361
views
Are there any 'graphical' algebras that can describe the 'shape' of graphs?
One of the main problems in graph enumeration is determining the 'shape' of a graph, e.g. the isomorphism class of any particular graph. I am fully aware that every graph can be represented as a ...
- 323
11
votes
0
answers
200
views
Are there cascade decompositions of machines that are more general than finite automata?
The idea of decomposing automata and their associated semi-groups into irreducible sub-components is due to Krohn & Rhodes and has been explored relatively thoroughly. Krohn & Rhodes gave an ...
- 361
10
votes
2
answers
684
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Complexity of computing the order of a permutation group
Given two permutations $g$ and $h$ over $n$ elements (i.e., members of $S_n$), what is the complexity of computing the order of the subgroup generated by $g,h$? Or just of deciding whether the ...
- 10.6k
10
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2
answers
1k
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What kind of theoretical object corresponds to a C++ concept?
I am lacking a background in theoretical computer science but I would have liked to understand to what kind of theoretical objects C++ concepts corresponds to. Basically, C++ concepts allow to define ...
- 307
10
votes
1
answer
571
views
Chomsky Schützenberger enumeration theorem
In many textbooks the Chomsky-Schützenberger enumeration theorem is stated as that the characteristic formal power series of a language is $\mathbb N$-algebraic, if the grammar is unambigious. In some ...
9
votes
2
answers
849
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Formal representation of an abstraction hierarchy
Introduction
I'm writing my PhD thesis on Abstract Delta Modeling (ADM), an abstract algebraic description of modifications (known as deltas) able to act on products (as in 'software products'). This ...
- 231
9
votes
1
answer
459
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Turing Machines as Coalgebras
I'm looking to write a survey on the method of representing the dynamics of state-based computation within the framework of coalgebras. So far I've managed to find papers on coalgebra representations ...
- 163
9
votes
1
answer
319
views
Generalisation of the statement that a monoid recognizes language iff syntactic monoid divides monoid
Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $...
- 2,057
9
votes
3
answers
607
views
Find the remainder of a large fixed polynomial when divided by a small unknown polynomial
Assume we operate in a finite field. We are given a large fixed polynomial p(x) (of, say, degree 1000) over this field. This polynomial is known beforehand and we are allowed to do computation using a ...
- 91
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votes
0
answers
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What does a private coin $\mathsf{IP}$ protocol for Hilbert's Nullstellensatz look like?
$\mathsf{GNI}$ Private Coin
In [GMW85], the authors provided the famous interactive proof $\mathsf{IP}$ of Graph Non Isomorphism $\mathsf{GNI}$.
The $\mathsf{GNI}$ protocol entails a verifier ...
- 1,145
8
votes
1
answer
459
views
What is an unambiguous language in the sense of Schützenberger?
I'm reading Thomas Wilke's survey on the connections between Temporal Logic and finite automata, finite semigroups and first-order logic.
In Theorem 6 (by Kamp), the fragment $\mathrm{TL}[\mathsf{F},\...
- 1,406
8
votes
2
answers
412
views
Sufficient conditions to guarantee unique fixpoint (not unique least/greatest fixpoint) for monotone functions on complete lattice
Tarski's fixpoint theorem states, that the fixpoints of a monotone operator on a complete lattice is a complete lattice. By consequence, we have a unique greatest fixpoint and unique least fixpoint ...
- 265
8
votes
1
answer
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Factoring low-degree polynomials
What is the fastest algorithm known for factoring polynomials with $n$ variables and total degree $\leq d$? Here, $n$ is growing and $d$ is fixed. Most work seem to consider the case when $d$ is ...
- 7,030
8
votes
2
answers
925
views
What category are Tagless Final Algebras final In?
The Haskell and Scala community have been very enamored recently with what they call tagless final 'pattern' of programming. These are referenced as dual to initial free algebras, so I was wondering ...
- 595
8
votes
1
answer
272
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Algebraic (or numeric) invariants of complexity classes
I hope this question isn't too naive for this site.
In mathematics (topology, geometry, algebra) it is common for one to distinguish between two objects by coming up with an algebraic or numerical ...
- 1,328
8
votes
2
answers
540
views
Bivariate low-degree polynomial testing of Polishchuk-Spielman
In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
- 81
8
votes
1
answer
211
views
Complete axiomatization of relation algebras without ${}^-$ and $\top$
I'm working on a more thorough algebraic treatment of delta modeling. Briefly, deltas are syntactic entities that can modify products (as in 'software products'). They actually represent relations on ...
- 231
7
votes
3
answers
507
views
Field extensions in CS
A field is a set with two binary operations called addition and multiplication satisfying various axioms. Wikipedia article: Field_(mathematics)
A field extension is when you add a new element and ...
7
votes
2
answers
322
views
Smallest representatives of a quotient by an equivalence relation
Background
Let $\mathcal{A}=(Q,\Sigma,\delta,q_0,F)$ be a minimal DFA for a regular language $L$ such that $|Q|=n$, and let $\equiv_L$ be the relation given by $$x\equiv_Ly\text{ iff for all $u$: }xu\...
- 1,406
7
votes
1
answer
658
views
On the relation for the Myhill-Nerode theorem/syntactic monoid of a language
In order to characterize regular languages one finds the following definition useful:
Let $\Sigma$ be an alphabet and $L\subseteq\Sigma^*$. Say that $x,y\in\Sigma^*$ are $\equiv_L$-related, and ...
- 1,406
7
votes
1
answer
417
views
Kleene Algebra for star-free regular expressions
TLDR: Is there a notion of Kleene Algebra for star-free regular expressions?
Kleene Algebras are algebraic structures that are equivalent to regular expressions. A Kleene Algebra is an idempotent ...
- 203
7
votes
1
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Technical lemma about curves used in original proof of PCP theorem
I am reading the proof from here and found a technical lemma that seems to be incorrect (its proof is short and very vague). I know this is rather specific and the context is problematic, but I couldn'...
- 153
7
votes
0
answers
68
views
Complexity of deciding whether subspaces of Z_2^n cover every point 3*x times
When studying the complexity of checking identities in certain finite algebras, I came across the following decision problem:
Input: A positive integer $n \in N$ and a set of affine subspaces $H_1,...
- 71
6
votes
2
answers
673
views
Proof of a unique homomorphism from an initial object
What is the proof that there is only one homomorphism from an initial object to another object?
- 293
6
votes
1
answer
173
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Where does randomness help when deciding algebraic geometry over $\mathbb{C}$?
If we have a single straight line program expressing a multivariate polynomial equation with integer coefficients, the Schwartz-Zippel lemma gives a simple randomized algorithm for deciding whether ...
- 3,313
6
votes
5
answers
285
views
What is/are the lower bounds for finding a something akin to complex residue?
Given a function $\sum_{i=-N}^N{c_i x^i}$:
$f(x) \equiv \sum_{i=-N}^N{c_i x^i}$ where $c_i$ is an integer; $0 \le c_i \le a$ for some $a$.
The constant $c_0$ is desired, and we start with only $f(x)...
6
votes
1
answer
418
views
Algebra and algebraic data types
Which of the well-known structures of modern algebra (monoids, groups, rings etc) can be expressed as algebraic data types (ADTs)?
Presumably a free monoid can be considered to be isomorphic to the ...
- 785
6
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0
answers
240
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Satisfiability and a Galois Theory Analog
Let $v(a, b)$ be a binary predicate, and define $\phi$ as follows:
$$\phi: v(a_1, b_1) \land v(a_1, b_2) \land (a_1, b_3)$$
where our universe consists of two sorts $A: \{a_1, a_2, a_3\}$ and $B: \{...
- 101
5
votes
1
answer
194
views
Why is it difficult to develop a subexponential functoral approach to discrete log
Call a discrete logarithm algorithm functoral if the commutative algebraic object that the algorithm acts on can be changed with another commutative object in the same category and the algorithm does ...
- 13.3k
5
votes
1
answer
238
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Terminology about computation and Finite algebra
I am looking for the name of something that may have one.
A finite algebra $\mathcal{A} = (E, \{f_1, f_2, \ldots, f_k\})$ is a non-empty set $E$ together with some functions $f_i$ from $E^{r_i} \to E$...
- 1,002