Definition
A language L is in PP if and only if there exists a probabilistic Turing machine M, such that
- M runs for polynomial time on all inputs
- For all x in L, M outputs 1 with probability strictly greater than 1/2
- For all x not in L, M outputs 1 with probability less than or equal to 1/2.
Alternatively, PP can be defined using only deterministic Turing machines. A language L is in PP if and only if there exists a polynomial p and deterministic Turing machine M, such that
- M runs for polynomial time on all inputs
- For all x in L, the fraction of strings y of length p(|x|) which satisfy M(x,y) = 1 is strictly greater than 1/2
- For all x not in L, the fraction of strings y of length p(|x|) which satisfy M(x,y) = 1 is less than or equal to 1/2.
In both definitions, "less than or equal" can be changed to "less than" (see below), and the threshold 1/2 can be relaced by any fixed rational number in (0,1), without changing the class.
Read more about this topic: PP (complexity)
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than thisdevoted and obedient. This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (18201910)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (18421910)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)