# Completeness and weak completeness under polynomial-size circuits

## Abstract

This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform many-one reductions that are computed by polynomial-size circuits (P/Poly-many-one reductions). Every weakly P/Poly-many-one-complete problem is shown to have a dense, exponential, nonuniform complexity core. An exponential lower bound on the space-bounded Kolmogorov complexities of weakly P/Poly-Turing-complete problems is established. More importantly, the P/Poly-many-one-complete problems are shown to be *unusually simple* elements of ESPACE, in the sense that they obey nontrivial *upper* bounds on nonuniform complexity (size of nonuniform complexity cores and space-bounded Kolmogorov complexity) that are violated by almost every element of ESPACE. More generally, a Small Span Theorem for P/Poly-many-one reducibility in ESPACE is proven and used to show that every P/Poly-many-one degree -including the complete degree — has measure 0 in ESPACE. (In contrast, almost every element of ESPACE is weakly P-many-one complete.) All upper and lower bounds are shown to be tight.

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### References

- 1.E. Allender and R. Rubinstein. P-printable sets.
*SIAM Journal on Computing*, 17:1193–1202, 1988.Google Scholar - 2.E. W. Allender. Some consequences of the existence of pseudorandom generators.
*Journal of Computer and System Sciences*, 39:101–124, 1989.Google Scholar - 3.E. W. Allender and O. Watanabe. Kolmogorov complexity and degrees of tally sets.
*Information and Computation*, 86:160–178, 1990.Google Scholar - 4.K. Ambos-Spies, S. A. Terwijn, and Zheng Xizhong. Resource bounded randomness and weakly complete problems. In
*Proceedings of the Fifth Annual International Symposium on Algorithms and Computation*, pages 369–377. Springer-Verlag, 1994.Google Scholar - 5.J. L. Balcázar and R. V. Book. Sets with small generalized Kolmogorov complexity.
*Acta Informatica*, 23:679–688, 1986.Google Scholar - 6.J. L. Balcázar, J. Díaz, and J. Gabarró.
*Structural Complexity I*. Springer-Verlag, Berlin, 1988.Google Scholar - 7.J. L. Balcázar and U. Schöning. Bi-immune sets for complexity classes.
*Mathematical Systems Theory*, 18:1–10, 1985.Google Scholar - 8.R. Book and D.-Z. Du. The existence and density of generalized complexity cores.
*Journal of the ACM*, 34:718–730, 1987.Google Scholar - 9.R. Book, D.-Z Du, and D. Russo. On polynomial and generalized complexity cores. In
*Proceedings of the Third Structure in Complexity Theory Conference*, pages 236–250, 1988.Google Scholar - 10.G. J. Chaitin. On the length of programs for computing finite binary sequences.
*Journal of the Association for Computing Machinery*, 13:547–569, 1966.Google Scholar - 11.D.-Z. Du.
*Generalized complexity cores and levelability of intractable sets*. Ph.D. thesis, University of California, Santa Barbara, 1985.Google Scholar - 12.D.-Z. Du and R. Book. On inefficient special cases of
*NP*-complete problems.*Theoretical Computer Science*, 63:239–252, 1989.Google Scholar - 13.S. Even, A. Selman, and Y. Yacobi. Hard core theorems for complexity classes.
*Journal of the ACM*, 35:205–217, 1985.Google Scholar - 14.J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In
*Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science*, pages 439–445, 1983.Google Scholar - 15.J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities.
*Theoretical Computer Science*, 34:17–32, 1984.Google Scholar - 16.D. T. Huynh. Resource-bounded Kolmogorov complexity of hard languages,
*Structure in Complexity Theory*, pages 184–195. Springer-Verlag, Berlin, 1986.Google Scholar - 17.D. T. Huynh. On solving hard problems by polynomial-size circuits.
*Information Processing Letters*, 24:171–176, 1987.Google Scholar - 18.D. W. Juedes.
*The Complexity and Distribution of Computationally Useful Problems*. Ph.D. thesis, Iowa State University, 1994.Google Scholar - 19.D. W. Juedes. Weakly complete problems are not rare. submitted, 1994.Google Scholar
- 20.D. W. Juedes and J. H. Lutz. The complexity and distribution of hard problems.
*SIAM Journal on Computing*, 24, 1995. to appear.Google Scholar - 21.D. W. Juedes and J. H. Lutz. Weak completeness in E and E
_{2}.*Theoretical Computer Science*, 1995. to appear.Google Scholar - 22.R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets.
*Information and Control*, 55:40–56, 1982.Google Scholar - 23.R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors,
*Complexity of Computer Computations*, pages 85–104. Plenum Press, New York, 1972.Google Scholar - 24.R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In
*Proceedings of the 12th ACM Symposium on Theory of Computing*, pages 302–309, 1980.Google Scholar - 25.K. Ko. On the notion of infinite pseudorandom sequences.
*Theoretical Computer Science*, 48:9–33, 1986.Google Scholar - 26.A. N. Kolmogorov. Three approaches to the quantitative definition of ‘information'.
*Problems of Information Transmission*, 1:1–7, 1965.Google Scholar - 27.L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories.
*Information and Control*, 61:15–37, 1984.Google Scholar - 28.L. Longpré.
*Resource Bounded Kolmogorov Complexity, a Link Between Computational Complexity and Information Theory*. Ph.D. thesis, Cornell University, 1986. Technical Report TR-86-776.Google Scholar - 29.J. H. Lutz. Resource-bounded measure. in preparation.Google Scholar
- 30.J. H. Lutz. Weakly hard problems.
*SIAM Journal on Computing*, to appear. See also*Proceedings of the Ninth Structure in Complexity Theory Conference*, 1994, pp. 146–161. IEEE Computer Society Press.Google Scholar - 31.J. H. Lutz. Category and measure in complexity classes.
*SIAM Journal on Computing*, 19:1100–1131, 1990.Google Scholar - 32.J. H. Lutz. Almost everywhere high nonuniform complexity.
*Journal of Computer and System Sciences*, 44:220–258, 1992.Google Scholar - 33.J. H. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small.
*Theoretical Computer Science*, to appear. See also*Proceedings of the Eleventh Symposium on Theoretical Aspects of Computer Science*, Springer-Verlag, 1994, pp. 415–426.Google Scholar - 34.J. H. Lutz and E. Mayordomo. Measure, stochasticity, and the density of hard languages.
*SIAM Journal on Computing*, 23:762–779, 1994.Google Scholar - 35.N. Lynch. On reducibility to complex or sparse sets.
*Journal of the ACM*, 22:341–345, 1975.Google Scholar - 36.P. Martin-Löf. Complexity oscillations in infinite binary sequences.
*Zeitschrift für Wahrscheinlichkeitstheory und Verwandte Gebiete*, 19:225–230, 1971.Google Scholar - 37.E. Mayordomo.
*Contributions to the study of resource-bounded measure*. Ph.D. thesis, Universitat Politècnica de Catalunya, 1994.Google Scholar - 38.P. Orponen. A classification of complexity core lattices.
*Theoretical Computer Science*, 70:121–130, 1986.Google Scholar - 39.P. Orponen and U. Schöning. The density and complexity of polynomial cores for intractable sets.
*Information and Control*, 70:54–68, 1986.Google Scholar - 40.D. A. Russo and P. Orponen. On P-subset structures.
*Mathematical Systems Theory*, 20:129–136, 1987.Google Scholar - 41.M. Sipser. A complexity-theoretic approach to randomness. In
*Proceedings of the 15th ACM Symposium on Theory of Computing*, pages 330–335, 1983.Google Scholar - 42.S. Skyum and L. G. Valiant. A complexity theory based on boolean algebra.
*Journal of the ACM*, 32:484–502, 1985.Google Scholar - 43.R. J. Solomonoff. A formal theory of inductive inference.
*Information and Control*, 7:1–22, 224–254, 1964.Google Scholar