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.

我们提出了在确定性对数时间（PolylogTime）和对数空间（PolylogSpace）中可解决的问题的逻辑表征。我们介绍了一种新颖的两类逻辑，该逻辑将输入域的元素与解决这些元素所需的位位置分开。我们证明了该逻辑的通货膨胀和部分不动点变体分别捕获了PolylogTime和PolylogSpace。在证明我们的逻辑确实捕获了有限有序结构上的PolylogTime的过程中，我们引入了一种随机访问的图灵机的变体，它可以直接访问结构的关系和功能。我们调查PolylogTime逻辑中是否需要一个明确的谓词来对域进行排序。最后，