Editing RP (complexity) (section)

== Formal definition ==
A language ''L'' is in '''RP''' 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 1/2
* For all ''x'' not in ''L'', ''M'' outputs 0

Alternatively, '''RP''' can be defined using only deterministic Turing machines. A language ''L'' is in '''RP''' if and only if there exists a polynomial ''p'' and deterministic Turing machine ''M'', such that
* ''M'' runs for polynomial time p on all inputs
* For all ''x'' in ''L'', the fraction of strings ''y'' of length ''p''(|''x''|) which satisfy {{tmath|1=M(x,y) = 1}} is greater than or equal to 1/2
* For all ''x'' not in ''L'', and all strings ''y'' of length ''p''(|''x''|), {{tmath|1=M(x,y) = 0}}
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.