Descriptive complexity of deterministic polylogarithmic time and space

F. Ferrarotti, S. González, J. Turull Torres, J. Van den Bussche, J. Virtema. Descriptive complexity of deterministic polylogarithmic time and space. Journal of Computer and System Sciences, volume 119, pages 145-163, DOI, 8, 2021.

  • Flavio Ferrarotti
  • Senén González
  • José María Turull Torres
  • Jan Van den Bussche
  • Jonni Virtema
JournalJournal of Computer and System Sciences

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