Partial fixed-point logic ($\mathrm{PFP}$)

Extend first-order logic by iterating definable relation operators from the empty relation. If iteration reaches a fixed point, use that relation; if it enters a nontrivial cycle, the partial fixed point is the empty relation. For a fixed arity on a finite universe, a relation requires polynomially many bits and there are exponentially many states, so iteration is evaluable in PSPACE. On finite ordered structures this logic captures PSPACE.