Number of Variables Is Equivalent to Space

We prove that the set of properties describable by a uniform sequence of rst-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACEn k ] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative deenition for a nite set of relations of arity k. This is a reenement of the theorem PSPACE = VARO1]] 7]. We suggest some directions for exploiting this result to derive trade-oos between the number of variables and the quantiier depth in descriptive complexity.