Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-26T12:38:14.186Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Adler, M. and Immerman, N., An n!, lower bound on formula size, LICS, 2001.CrossRefGoogle Scholar
[2]Barrington, D. Mix, Immerman, N., and Straubing, H., On uniformity within NC1, Journal of Computer and System Sciences, vol. 41 (1990), no. 3, pp. 274–306.CrossRefGoogle Scholar
[3]Cai, J., Fürer, M., and Immerman, N., An optimal lower bound on the number of variables for graph identification, 30th IEEE FOCS symposium, 1989, pp. 612–617.Google Scholar
[4]Chandra, A. and Harel, D., Structure and complexity of relational queries, Journal of Computer and System Sciences, vol. 25 (1982), pp. 99–128.CrossRefGoogle Scholar
[5]Fagin, R., Finite-model theory—a personal perspective, Third international conference on database theory, 1990.Google Scholar
[6]Gurevich, Y., Logic and the challenge of computer science, Current trends in theoretical computer science (Börger, Egon, editor), Computer Science Press, 1988, pp. 1–57.Google Scholar
[7]Immerman, N., Number of quantifiers is better than number of tape cells, Journal of Computer and System Sciences, vol. 22 (1981), no. 3, pp. 65–72.CrossRefGoogle Scholar
[8]Immerman, N., Upper and lower bounds for first order expressibility, Journal of Computer and System Sciences, vol. 25 (1982), no. 1, pp. 76–98.CrossRefGoogle Scholar
[9]Immerman, N., Relational queries computable in polynomial time, Information and Control, vol. 68 (1986), pp. 86–104, a preliminary version of this paper appeared in 14th ACM STOC Symposium, (1982), 147-152.CrossRefGoogle Scholar
[10]Immerman, N., Languages that capture complexity classes, SIAM Journal on Computing, vol. 16 (1987), no. 4, pp. 760–778.CrossRefGoogle Scholar
[11]Immerman, N., Expressibility and parallel complexity, SIAM Journal on Computing, vol. 18 (1989), pp. 625–638.CrossRefGoogle Scholar
[12]Immerman, N., DSPACE[nk] = VAR[k + 1], Sixth IEEE structure in complexity theory symposium, 07 1991, pp. 334–340.Google Scholar
[13]Immerman, N., Descriptive complexity, Springer Graduate Texts in Computer Science, New York, 1998.Google Scholar
[14]Karchmer, M. and Wigderson, A., Monotone circuits for connectivity require super-logarithmic depth, SIAM Journal on Discrete Mathematics, vol. 3 (1990), no. 2, pp. 255–265.CrossRefGoogle Scholar
[15]Moschovakis, Y., Elementary induction on abstract structures, North Holland, 1974.Google Scholar
[16]Tarksi, A., A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, vol. 55 (1955), pp. 285–309.Google Scholar
[17]Vardi, M., Complexity of relational query languages, 14th ACM symposium on theory of computing, 1982, pp. 137–146.Google Scholar