We show that any formula not involving Until is equivalent to one without nested fixed point operators.
We then prove that USF has expressive power matching that of the monadic second-order logic S1S. The proof shows that any USF-formula is equivalent to one with at most two nested fixed point operators - i.e., no branch of its formation tree has more than two fixed point operators.
We then axiomatise USF and prove that it is decidable, with PSPACE-complete satisfiability problem.
Finally, we discuss an application of these results to the executable temporal logic system `MetateM'.