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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“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 (1842–1910)