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Did You Know?

Informally, lowness means that problems in B are not only solvable by machines which can solve problems in A, but are "easy to solve." An A machine can simulate many oracle queries to B without exceeding its resource bounds.Results and relationships that establish one class as low for another are often called lowness results.Several natural complexity classes are known to be low for themselves.

This allows us to reason about it in the same manner we normally would.

Similarly, the argument for BPP almost goes through for BQP, but we have to additionally show that quantum queries can be performed in coherent superposition.Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411-1473, 1997.

PSPACE is also low for itself, and this can be established by exactly the same argument.

1993.Lowness is particularly valuable in relativization arguments, where it can be used to establish that the power of a class does not change in the "relativized universe" where a particular oracle machine is available for free.

Similarly, L is low for itself because it can simulate log space oracle queries in log space, reusing the same space for each query.BPP is also low for itself and the same arguments almost work for BPP, but one has to account for errors, making it slightly harder to show that BPP is low for itself.