Descriptive complexity of deterministic polylogarithmic time and space

We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and space (PolylogSpace). introduce a novel two-sorted logic that separates the elements input domain from bit positions needed to address these elements. prove inflationary partial fixed point variants this capture PolylogTime PolylogSpace, respectively. In course proving our indeed captures on finite ordered structures, we variant random-access Turing machines can access relations functions structure directly. investigate whether an explicit predicate for ordering is logic. Finally, present open problem finding exact characterization order-invariant queries PolylogTime.