# What are semantic classes that have a syntactic equivalent?

This question is related to Benefits for syntactic and semantic classes. As mentioned there, $\mathsf{PSPACE} = \mathsf{IP}$, which can be interpreted as the semantic class $\mathsf{IP}$ obtaining a syntactic definition. What are other non-trivial examples for "syntacticizing" a class?

• Does MIP=NEXP count? Or is it too closely related to IP=PSPACE to count as another example? Also QIP=PSPACE? – Joshua Grochow Mar 9 '18 at 16:39
• Another example is ReachUL. See this paper. – William Hoza Mar 11 '18 at 5:43
• @William I suggest you turn this into an answer with a link that doesn't require a U Texas account... – domotorp Mar 11 '18 at 6:43
• @domotorp Oops. Done. – William Hoza Mar 13 '18 at 5:14

A few others; related to $\mathsf{IP} = \mathsf{PSPACE}$, but there are enough of them that weren't mentioned in the OQ that I figured it's worth putting them here:
Another example is $\mathbf{ReachUL}$, the class of languages decidable by nondeterministic log-space Turing machines such that for any input and any configuration, there is at most one sequence of nondeterministic choices leading to that configuration. See "An unambiguous class possessing a complete set" by Lange.
One of my favorites is $IND[t(n)] = FO[t(n)]$, where $IND[t(n)]$ is the class of problems decidable with an inductive definition which closes in less than or equal to $t(n)$ iterations and $FO[t(n)]$ are the problems decidable using a first order logic sentence with less than or equal to $t(n)$ iterated quantifier blocks.