# AM/MA and NP in analogy to P and BPP

Arora and Barak show that $\mathsf{AM}$ can be expressed as $\mathsf{BP}\cdot \mathsf{NP}$ i.e the set of languages that have randomized reductions to 3SAT. $\mathsf{MA}$ is also a natural randomized generalization of $\mathsf{NP}$ in that you replace the deterministic verifier by a randomized one.

Is there a sense in which one of these is a closer fit in the "P is to BPP as NP is to ?" relation ?

• Just to give credit where it's due, Zachos was the first to express AM as BP$\cdot$NP. – Lance Fortnow Apr 11 '12 at 12:51
• Yes, I was referring to the textbook without being careful. Thanks ! – Suresh Venkat Apr 11 '12 at 17:54

This is of course a very subjective matter, but here is something that might be interpreted as saying that $\mathbf{MA}$ is a closer fit: The same assumptions that imply that $\mathbf{P} = \mathbf{BPP}$ also imply that $\mathbf{NP} = \mathbf{MA}$, but those assumptions are not known to imply $\mathbf{NP} = \mathbf{AM}$. In addition, the assumption that $\mathrm{promise}\mathbf{P} = \mathrm{promise}\mathbf{BPP}$ implies that $\mathrm{promise}\mathbf{NP} = \mathrm{promise}\mathbf{MA}$, but is not known to imply $\mathrm{promise}\mathbf{NP} = \mathrm{promise}\mathbf{AM}$.
However, there is an alternative view saying that $\mathbf{MA}$ is the non-deterministic variant of $\mathbf{BPP}$ while $\mathbf{AM}$ is the probabilistic variant of $\mathbf{NP}$. The foregoing facts can also be interpreted as evidence for this view.