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40

A type $C$ has a logarithm to base $X$ of $P$ exactly when $C \cong P\to X$. That is, $C$ can be seen as a container of $X$ elements in positions given by $P$. Indeed, it's a matter of asking to what power $P$ we must raise $X$ to obtain $C$. It makes sense to work with $\mathop{log}F$ where $F$ is a functor, whenever the logarithm exists, meaning $\mathop{...


27

There have been recent developments in dependent type theory which relate type systems to homotopy types. This is now a relatively small field, but there is a lot of exciting work being done right now, and potentially a lot of low hanging fruit, most notably in porting results from algebraic topology and homological algebra and formalizing the notion of ...


24

Algebraic geometry is used heavily in algebraic complexity theory and in particular in geometric complexity theory. Representation theory is also crucial for the latter, but it's even more useful when combined with algebraic geometry and homological algebra.


18

Algebraic geometry Noether's Normalization Lemma (NNL) for explicit varieties is currently only known to be in $\mathsf{EXPSPACE}$ (like general NNL), but is conjectured to be in $\mathsf{P}$ (and is in $\mathsf{P}$ assuming that PIT can be black-box derandomized). Update 4/18/18: It was recently shown that for the variety $\overline{\mathsf{VP}}$ it is in $...


17

You could try the notes from Madhu Sudan's course: Algebra and Computation


15

Your knowledge of field theory would be useful in cryptography, while category theory is heavily used in the research on programming languages and typing systems, both of which are closely related to the foundations of mathematics.


14

Finite automata in which the initial state is also the unique accepting state have the form $r^∗$, where $r$ is some regular expression. However, as J.-E. Pin points out below, the converse is not true: there are languages of the form $r^*$ which are not accepted by a DFA with a unique accepting state. Intuitively, given a sequence of states $q_0, \ldots, ...


13

Rings, modules, and algebraic varieties are used in error correction and, more generally, coding theory. Specifically, there is an abstract error correcting scheme (algebraic-geometry codes) which generalizes Reed-Solomon codes and Chinese Remainder codes. The scheme is basically to take your messages to come from a ring R and encode it by taking its ...


12

[Computational complexity and category theory] seem like such natural pairs. Given the prominence of computational complexity as a research field, if they were such natural bedfellows, maybe somebody would have brought out the connection already? Wild speculation. Let me entertain the reader with thoughts about why a categorical rendering of ...


12

The order of permutation groups can be computed in polynomial-time. In fact, I believe even in $\mathsf{NC}$ and also nearly linear Las Vegas time. See, e.g., the book by Seress. For reference, subgroups of $S_n$ (and algorithms related thereto) are typically called "permutation groups" rather than merely "subgroups (of $S_n$)". So you can google "...


12

For NP, this seems hard to construct. In particular, if you can also sample (nearly) uniform elements from your group - which is true for many natural ways of constructing groups - then if an NP-complete language has a poly-time group action with few orbits, PH collapses. For, with this additional assumption about sampleability, the standard $\mathsf{coAM}$ ...


11

One possibily path into abstract algebra could be to look at it from point of view of cryptography, which is about algorithms on finite field. Fields are rings, and fields are also two groups coupled by simple laws. Field theory uses vector spaces in prominent position (Galois theory), so this angle should cover a lot of abstract algebra. The book A ...


11

There is a more general question on mathoverflow. I asked a similar question on CS.SE. jbapple provided a good answer. To quote "Necklaces, Convolutions, and X+Y", By Bremner et al. shows a $O\left(\frac{n^2(\lg \lg n)^3}{\lg^2 n}\right)$ algorithm for this problem on the real RAM and a $O(n \sqrt{n})$ algorithm in the nonuniform linear decision ...


11

Field theory and algrebraic geometry would be useful in topics related to error correcting codes, both in the classical setting as well as in studying locally decodable codes and list decoding. I believe this goes back to work on the Reed-Solomon and Reed-Muller codes, which was then generalized to algebraic geometric codes. See for example, this book ...


11

It seems there is a paper answering this exact question, and even in the more general case of $\omega$-regular languages, but I cannot find an open-access version. If somebody finds a link without paywall it would be great. I requested the full-text on ResearchGate. Title: Which Finite Monoids are Syntactic Monoids of Rational omega-Languages. Authors: ...


11

In a more elementary way than Denis's answer, the following is extracted from Pippenger's "Theories of Computability", p.87, and immediate to check. Definition: Let $M$ be a monoid, and $Y \subseteq M$. Define the congruence relation $\equiv_Y$ over $M$ by $x \equiv_Y y$ iff $\big[\forall w, z \in M$, $wxz \in Y \Leftrightarrow wyz \in Y\big]$. Definition:...


11

The terminology rigid seems to be relatively new compared to the term disjunctive used in the late 70's (and probably before, I didn't check for earlier references). A subset $P$ of a monoid $M$ is disjunctive if and only if the syntactic congruence of $P$ in $M$ is the equality relation. Thus a monoid is the syntactic monoid of a language if and only if it ...


11

From a programming language theory perspective, as opposed to the computability perspective other answers and comments have offered, C++ templates combined with concepts correspond to bounded polymorphism or constrained genericity. Concepts themselves correspond to the constraints or bounds placed on a type. A template is type-level function, parameterised ...


10

An important subclass of this family is a sub-class of 0-reversible languages. A language is 0-reversible if the reversal of the minimal DFA for the language is also deterministic. The reversing operation is defined as swapping initial and final states, and inverting the edge relation of the DFA. This means that a 0-reversible language can have only one ...


10

To my knowledge (which is definitely incomplete), there has been relatively little work on this, presumably because it requires assimilating two relatively intricate bodies of knowledge. However, little does not mean nonexistent. Thierry Coquand and his collaborators have written quite a few papers on the connections between commutative algebra and ...


9

You are working on your PhD. Saying "I am not well versed in $X$" is not an excuse. And if you're good, then saying "my advisor does not know $X$" is not an excuse either. You are using monoids where you should be using categories. Your monoid operations presupposes that you can combine any $\delta$'s together. But does this really make sense, for example, ...


9

As a complement to Joshua Grochow's answer: Computing the order of a permutation group given generators is in P by Schreier–Sims algorithm, see also p. 8-9 of these lectures notes by Luks. Just as membership in permutation groups, the problem was believed to be P-complete by many researchers, but it was finally shown to be in NC by Babai, Luks & Seress. ...


8

This question is solved for deterministic automata and for unambiguous automata in the book [1] [1] J. Berstel, D. Perrin, C, Reutenauer, Codes and automata, Vol. 129 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2009. In the case of deterministic automata, the characterization is given in Proposition 3.2.5. Recall that a ...


8

I believe it would be beneficial for you to look up the theory of abstract interpretation, which provides very thorough answers to similar questions in the slightly different area of lattice-based program analysis. It appears to me that you are using a framework based on algebras. I'm using the word algebra here in the sense of universal algebra, where I ...


8

Let $\mathbb K$ be a field of characteristic $0$ or at least $d(d-1)+1$, and $p\in\mathbb K[x_1,\dotsc,x_n]$ be a polynomial of total degree at most $d$. If $d$ is fixed and $n$ is growing, one has the following complexity bounds for the reduction of the factorization of $p$ to the factorization of a degree-$d$ univariate polynomial: (The notation $\tilde{\...


8

The reason that sub exponential algorithms for the discrete log in $\mathbb{Z}/p\mathbb{Z}$ do not carry over to groups based on elliptic curves is because $\mathbb{Z}/p\mathbb{Z}$ has extra structure that doesn't seem to be present in elliptic curve groups. In particular, $\mathbb{Z}/p\mathbb{Z}$ has both addition and multiplication, whereas the elliptic ...


7

Algebra (and algebraic geometry) has had a pretty big role to play in cryptography, with elliptic curve groups, (number-theoretic) lattices, and of course $\mathbb{Z}_p$ being the basis for nearly all modern cryptographic work.


7

For the question of when does $p=0 \Rightarrow q=0$, randomness can indeed help, as follows. First, factor $p$ (uses randomness when $p$ is given as a straight-line program; doesn't need randomness if $p$ is given as a coefficient vector). Let $\tilde{p}$ denote the square-free version of $p$ - that is, if $p=p_1^{k_1} p_2^{k_2} \dotsb p_\ell^{k_\ell}$ then $...


7

Short answer The general answer is that computability is easy for locally compact spaces, doable for separable spaces, and hard for non-separable spaces. The short anwer for a particular example, namely $\ell^2$ is: in order to get computability on $\ell^2$ working, represent an element $x$ of $\ell^2$ by a program which accepts $n$ and gives (the list of ...


7

There are some problems in computational learning theory, machine learning and computer vision that can be solved using commutative algebra and algebraic geometry. For instance, convergence of the Belief Propagation algorithm, a message passing algorithm for Bayesian inference, can be formulated in terms of characterizing the affine variety of system of ...


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