closure properties of IP(2pfa) and AM(2pfa) IP(2pfa) and AM(2pfa) are the classes of languages recognized with bounded error by private and public coin versions, respectively, of interactive proof systems with verifiers that are probabilistic finite automata with a two-way input head.

Are any closure properties of these classes known?
You can take a look at the following paper:

Dwork, C. and Stockmeyer, L. 1992. Finite state verifiers I: the power of interaction. J. ACM 39, 4 (Oct. 1992), 800-828. DOI= http://doi.acm.org/10.1145/146585.146599

or the related paper:

Dwork, C. and Stockmeyer, L. 1992. Finite state verifiers II: zero knowledge. J. ACM 39, 4 (Oct. 1992), 829-858. DOI= http://doi.acm.org/10.1145/146585.146601