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3
votes
2answers
146 views

efficiently checking that a field extension is Galois

Let $K \subset L$ be an algebraic extension of fields finitely presented over a prime field or over an algebraically closed field. Is there an efficient procedure to check that $L/K$ is Galois? To ...
0
votes
1answer
147 views

Refutation of $A \land \lnot\lnot\lnot A$ by resolution [closed]

$ A \land \lnot\lnot\lnot A $ this is a very simple example. Resolution is refutation complete. So it should be able to refute this formula. However, I don't see how would it do that without using ...
3
votes
0answers
57 views

Weak randomness relative to finite-state machines

Is there a nice example of a sequence that looks random to any predictor whose predictions use a finite-state machine? More precisely, consider a finite-state machine $M$ with input alphabet {0,1} ...
4
votes
5answers
285 views

procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...
0
votes
3answers
353 views

System of quadratic equations with 18 unknown

So I want to solve for a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r which satisfy the following system of equations: ( I only need positive integer (or 0) solution) a g + c h + b i + g j + i ...
38
votes
11answers
2k views

Important open problems that have already been reduced to a finite but infeasible amount of computation

Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer." Some questions ...
15
votes
3answers
1k views

What to do when your research runs into a computationally challenging problem?

Occasionally, but more frequently lately, I would like to perform some hard computations. As an example, yesterday the following question came up: What is the projective dimension of the edge ...
3
votes
1answer
424 views

Efficiently computing with pullbacks and pushouts

Often when computing in category theory, one has to show that some square is cartesian. Depending on the number of maps involved, and their arrangement, it's somewhat difficult to write down exactly ...