We consider Arthur-Merlin proof systems where (a) Arthur is a probabilistic quasi-polynomial time Turing machine
and (AMQ)(b) Arthur is a probabilistic exponential time Turing machine (AME). We prove two new results related to these classes.

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In this paper, we study two questions related to
the problem of testing whether a function is close to a homomorphism.
For two finite groups $G,H$ (not necessarily Abelian),
an arbitrary map $f:G \rightarrow H$, and a parameter $0 < \epsilon <1$,
say that $f$ is $\epsilon$-close to a homomorphism ... more >>>

An instance of a constraint satisfaction problem is $l$-consistent
if any $l$ constraints of it can be simultaneously satisfied.
For a set $\Pi$ of constraint types, $\rho_l(\Pi)$ denotes the largest ratio of constraints which can be satisfied in any $l$-consistent instance composed by constraints from the set $\Pi$. In the ... more >>>