Problems in NC
As with P, by a slight abuse of language, one might classify function problems and search problems as being in NC. NC is known to include many problems, including
- Integer addition, multiplication and division;
- Matrix multiplication, determinant, inverse, rank;
- Polynomial GCD, by a reduction to linear algebra using Sylvester matrix (it is open whether integer GCD is in NC);
- Finding a maximal matching.
Often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence. One might contrast ripple carry adder with a carry-lookahead adder.
Read more about this topic: NC (complexity)
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“The truth of the thoughts that are here set forth seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, the final solution of the problems. And if I am not mistaken in this belief, then the second thing in which the value of this work consists is that it shows how little is achieved when these problems are solved.”
—Ludwig Wittgenstein (1889–1951)