What are semantic classes that have a syntactic equivalent?
FWIW, the ostensibly semantic class APP defined in [1] was shown to be syntactic in [2].

[1] Valentine Kabanets, Charles Rackoff, Stephen A. Cook, Efficiently
  approximable real-valued functions, ECCC Report TR00-034, 2000.

[2] Emil Jeřábek, Approximate counting in bounded arithmetic, Journal of Symbolic Logic 72 (2007), no. 3, pp. 959–993. doi, preprint