Interactive proof system
In the case where a valid proof certificate exists, the prover is always able to make the verifier accept by giving it that certificate. In the case where there is no valid proof certificate, however, the input is not in the language, and no prover, however malicious it is, can convince the verifier otherwise, because any proof certificate will be rejected.
The class MA in particular is a simple generalization of the NP interaction above in which the verifier is probabilistic instead of deterministic. Also, instead of requiring that the verifier always accept valid certificates and reject invalid certificates, it is more lenient:
In a public coin protocol, the random choices made by the verifier are made public. They remain private in a private coin protocol.
The essential problem with public coins is that if the prover wishes to maliciously convince the verifier to accept a string which is not in the language, it seems like the verifier might be able to thwart its plans if it can hide its internal state from it. This was a primary motivation in defining the IP proof systems.