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)
“In many ways, life becomes simpler [for young adults]. . . . We are expected to solve only a finite number of problems within a limited range of possible solutions. . . . It’s a mental vacation compared with figuring out who we are, what we believe, what we’re going to do with our talents, how we’re going to solve the social problems of the globe . . .and what the perfect way to raise our children will be.”
—Roger Gould (20th century)
“I believe that if we are to survive as a planet, we must teach this next generation to handle their own conflicts assertively and nonviolently. If in their early years our children learn to listen to all sides of the story, use their heads and then their mouths, and come up with a plan and share, then, when they become our leaders, and some of them will, they will have the tools to handle global problems and conflict.”
—Barbara Coloroso (20th century)