Definition
A language L is in BPP 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 greater than or equal to 2/3
- For all x not in L, M outputs 1 with probability less than or equal to 1/3
Unlike the complexity class ZPP, the machine M is required to run for polynomial time on all inputs, regardless of the outcome of the random coin flips.
Alternatively, BPP can be defined using only deterministic Turing machines. A language L is in BPP 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 greater than or equal to 2/3
- For all x in not in L, the fraction of strings y of length p(|x|) which satisfy M(x,y) = 1 is less than or equal to 1/3
In this definition, the string y corresponds to the output of the random coin flips that the probabilistic Turing machine would have made. For some applications this definition is preferable since it does not mention probabilistic Turing machines.
Read more about this topic: BPP (complexity)
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