Number of Variables Is Equivalent To Space

Abstract

We prove that the set of properties describable by a uniform sequence of firstorder sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE [n k ] (where n is the size of the universe). This set 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]] [7]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity. 1 Introduction In Descriptive Complexity one analyzes the complexity of a language in terms of the complexity of describing the language. It is known that the quantifier depth and number of variables needed to express the membership property of a language is closely related to the Research supported in part by NSF grant CCR-9505446. y Research supported in part by NSERC, and p..