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Number of variables is equivalent to space

Published online by Cambridge University Press:  12 March 2014

Neil Immerman ,
Jonathan F. Buss  and
David A. Mix Barrington
Neil Immerman
Affiliation:
Department of Computer Science, University of Massachusetts, Amherst, Massachusetts 01003, USA, E-mail: immerman@cs.umass.edu
Jonathan F. Buss
Affiliation:
Department of Computer Science, University of Waterloo, Waterloo, Ontario N2L3G1, Canada, E-mail: jfbuss@math.uwaterloo.ca
David A. Mix Barrington
Affiliation:
Department of Computer Science, University of Massachusetts, Amherst, Massachusetts 01003, USA, E-mail: barring@cs.umass.edu
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Abstract

We prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[nk] (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]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1217 - 1230
Copyright
Copyright © Association for Symbolic Logic 2001

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References

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