If NP has Polynomial-Size Circuits, then MA=AM

It is shown that the assumption of NP having polynomial size circuits implies apart from a collapse of the polynomial time hierarchy as shown by Karp and Lip ton that the classes AM and MA of Babai s Arthur Merlin hierarchy coincide This means that also a certain inner collapse of the remaining classes of the polynomial time hierarchy occurs It is well known KL that the assumption of NP having polynomial size circuits in symbols NP P poly implies that the polynomial time hierarchy collapses to level two in symbols PH P P The textbooks BDG KST BC Pa can be consulted for the basic notations and results Furthermore this collapse level was shown to be optimal up to relativization in He There it is shown that under a suitable oracle the collapse cannot go down to the next lower level of the polynomial time hierarchy P P NP What we show here is under the same assumption an additional inner collapse namely of the two classes AM and MA which are not known to be equal to each other and which are not known to be equal to P Figure shows the known inclusion structure of the classes in the polynomial time hierarchy whereas Figure shows these inclusions under the assumption NP P poly The proof is not di cult and just a combination of known techniques but the result as such has not been observed before and we think it has some signi cance In both gures the relative position of the classes NP and BPP is also out lined By La Si used in a relativized version BPP is included in the class P P NP P P By the fact that PH P P holds under the assumption NP P poly the class BPP is a subset of P P in Figure It is still open whether the classes NP and BPP are also a ected by the collapse P co NP NP BPP MA co MA co AM AM P BPP P Q Q Q Q Qk Q Q Q Q Qk H H H H H HY Q Q Q Q Q Q Q Q Q Q Q k