Monadic First Order Logic is FOL with no function symbols, and predicate symbols restricted to arity 1. For this question, let's say that the = symbol is also forbidden. I want to know the complexity of determining if a monadic FOL sentence is valid, ideally in the form of a completeness result (say, under some notion of reduction that the class NE is closed under).
According to this presentation, it was proved a century ago that if S is a satisfiable monadic FOL sentence, then S has a model of size at most $2^kr$, where $r$ is the number of variables and $k$ is the number of predicate symbols. From that it follows that the problem is in $NTIME(2^{n^2})$ and $SPACE(2^n)$: First, guess a (respectively, try every) structure of size $\leq 2^k r \leq 2^n n$. Then evaluate the sentence on that structure in the most straightforward deterministic way, by enumerating all the elements of the structure at every quantifier, and evaluating the quantifier-free parts of the sentence in polynomial time. That takes time $(2^n n)^r poly(n)$, which is in $O(2^{n^2})$. On the other side, it is easy to show that the problem is hard for $PSPACE$.
