Problems
| Is P = BPP ? |
Besides the problems in P, which are obviously in BPP, many problems were known to be in BPP but not known to be in P. The number of such problems is decreasing, and it is conjectured that P = BPP.
For a long time, one of the most famous problems that was known to be in BPP but not known to be in P was the problem of determining whether a given number is prime. However, in the 2002 paper PRIMES is in P, Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena found a deterministic polynomial-time algorithm for this problem, thus showing that it is in P.
An important example of a problem in BPP (in fact in co-RP) still not known to be in P is polynomial identity testing, the problem of determining whether a polynomial is identically equal to the zero polynomial. In other words, is there an assignment of variables such that when the polynomial is evaluated the result is nonzero? It suffices to choose each variable's value uniformly at random from a finite subset of at least d values to achieve bounded error probability, where d is the total degree of the polynomial.
Read more about this topic: BPP (complexity)
Famous quotes containing the word problems:
“I rarely speak about God. To God, yes. I protest against Him. I shout at Him. But to open a discourse about the qualities of God, about the problems that God imposes, theodicy, no. And yet He is there, in silence, in filigree.”
—Elie Wiesel (b. 1928)
“To be a philosopher is not merely to have subtle thoughts, nor even to found a school, but so to love wisdom as to live according to its dictates a life of simplicity, independence, magnanimity, and trust. It is to solve some of the problems of life, not only theoretically, but practically.”
—Henry David Thoreau (1817–1862)
“It’s so easy during those first few months to think that the problems will never end. You feel as if your son will never sleep through the night, will always spit up food after eating, and will never learn to smile—even though you don’t know any adults or even older children who still act this way.”
—Lawrence Kutner (20th century)