DSPACE (n/sup k/)=VAR(k+1)

The author proves that the set of properties checkable by a Turing machine in DSPACE(n/sup 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.<
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