DSPACE [n^k]=VAR[k+1]

The author proves that the set of properties checkable by a Turing machine in DSPACE[n^k] is exactly equal to the set of properties describable by a uniform sequence of first-order sentences using at most k+1 distinct variables. He proves that this is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE=VAR[O[1]]. The author suggests some directions for exploiting this result to derive tradeoffs between the number of variables and the quantifier-depth in descriptive complexity. This has applications to parallel complexity