Editing RP (complexity) (section)

== Connection to P and NP ==
{{unsolved|computer science|{{tmath|1= \mathsf P \overset{?}{=} \mathsf{RP} }}}}
'''[[P (complexity)|P]]''' is a subset of '''RP''', which is a subset of '''[[NP (complexity)|NP]]'''.  Similarly, '''P''' is a subset of '''co-RP''' which is a subset of '''[[co-NP]]'''.  It is not known whether these inclusions are strict. However, if the commonly believed conjecture '''P''' = '''BPP''' is true, then '''RP''', '''co-RP''', and '''P''' collapse (are all equal). Assuming in addition that [[P = NP problem|'''P''' ≠ '''NP''']], this then implies that '''RP''' is strictly contained in '''NP'''. It is not known whether '''RP''' = '''co-RP''', or whether '''RP''' is a subset of the intersection of '''NP''' and '''co-NP''', though this would be implied by '''P''' = '''BPP'''.

A natural example of a problem in '''co-RP''' currently not known to be in '''P''' is [[Polynomial identity testing|Polynomial Identity Testing]], the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, {{nowrap|''x''·''x'' − ''y''·''y'' − (''x'' + ''y'')·(''x'' − ''y'')}} is the zero-polynomial while
{{nowrap|''x''·''x'' + ''y''·''y''}} is not.

An alternative characterization of '''RP''' that is sometimes easier to use is the set of problems recognizable by [[nondeterministic Turing machine]]s where the machine accepts if and only if at least some constant fraction of the computation paths, independent of the input size, accept. '''NP''' on the other hand, needs only one accepting path, which could constitute an exponentially small fraction of the paths. This characterization makes the fact that '''RP''' is a subset of '''NP''' obvious.