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)
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