Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T20:25:58.060Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost weakly 2-generic sets

Published online by Cambridge University Press:  12 March 2014

Stephen A. Fenner
Stephen A. Fenner*
Affiliation:
Computer Science Department, University of Southern Maine, Portland, Maine 04103, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

There is a family of questions in relativized complexity theory—weak analogs of the Friedberg Jump-Inversion Theorem—that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets, i.e., sets which meet every dense set of strings that is re. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2-generic set G such that NPG ∩ co-NPG ⊈ PG⊕x. (On the other hand, if G is 1-generic, then NPG ∩ co-NPG ⊆ PG⊕SAT, where SAT is the NP-complete satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets differ radically from those of 1-generic sets: every degree above 0′ contains an aw2-generic set: no aw2-generic set exists below any incomplete r.e. set: there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0′ is the jump of an aw2-generic degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 868 - 887
Copyright
Copyright © Association for Symbolic Logic 1994

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.)

References

REFERENCES

[1]Allender, E. W., The complexity of sparse sets in P. Structure in complexity theory, Lecture Notes in Computer Science, vol. 223. Springer-Verlag. Berlin and New York. 1986, pp. 1–11.Google Scholar
[2]Ambos-Spies, K., Fleischhack, H., and Huwig, H., P-generic sets, Proceedings of the 11th international colloquium on automata, languages, and programming (Paredaens, , editor). Lectures Notes in Computer Science, vol. 172, Springer-Verlag. Berlin. 1984, pp. 58–68.Google Scholar
[3]Ambos-Spies, K., Fleischhack, H., and Huwig, H., Diagonalizations over polynomial time computable sets, Theoretical Computer Science, vol. 51, (1987), pp. 177–204.CrossRefGoogle Scholar
[4]Balcázar, J. L., Díaz, J., and Gabarró, J., Structural complexity I. EATCS Monographs on Theoretical Computer Science, vol. 11, Springer-Verlag, Berlin and New York. 1988.CrossRefGoogle Scholar
[5]Blum, M., A machine-independent theory of the complexity of recursive functions, Journal of the Association for Computing Machinery, vol. 14 (1967), pp. 322–336.CrossRefGoogle Scholar
[6]Blum, M. and Impagliazzo, R., Generic oracles and oracle classes, Proceedings of the 28th annual IEEE symposium on foundations of computer science, IEEE Computing Society Press, Washington. D.C., 1987, pp. 118–126.Google Scholar
[7]Chong, C. T. and Downey, R. G., Minimal degrees recursive in l-generic degrees, Annals of Pure and Applied Logic, vol. 48 (1990), pp. 215–225.CrossRefGoogle Scholar
[8]Fenner, S., Notions of resource-bounded category and genericity, Technical Report 90-32, Department of Computer Science, University of Chicago, Chicago, Illinois, 1990.Google Scholar
[9]Fenner, S., Notions of resource-bounded category and genericity, Proceedings of the 6th annual IEEE structure in complexity theory conference, IEEE Computer Science Press, Washington, D.C., 1991, pp. 196–212. journal version in preparation.Google Scholar
[10]Fenner, S., Tight lower bounds on genericity required to prevent one-way functions, Technical Report 91-04, Department of Computer Science, University of Chicago, Chicago, Illinois. 1991.Google Scholar
[11]Fenner, S., Fortnow, L., and Kurtz, S., Gap-definable counting classes, Journal of Computer and System Sciences, vol. 48 (1994), pp. 116–148.CrossRefGoogle Scholar
[12]Fenner, S., Fortnow, L., Kurtz, S., and Li, L., An oracle builder's toolkit, Proceedings of the 8th IEEE structure in complexity theory conference, IEEE Computer Society Press, Washington, D.C., 1993, pp. 120–131.Google Scholar
[13]Foster, J. A., Forcing and genericity on the polynomial hierarchy, Ph.D. thesis, Illinois Institute of Technology, Chicago. Illinois. 1990.Google Scholar
[14]Friedberg, R. M., A criterion for completeness of degrees of unsolvability, this Journal, vol. 22 (1957), pp. 159–160.Google Scholar
[15]Grollmann, J. and Selman, A., Complexity measures for public-key cryptosystems, SIAM Journal on Computing, vol. 17 (1988), pp. 309–335.CrossRefGoogle Scholar
[16]Hinman, P. G., Some applications of forcing to hierarchy problems in arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 341–352.CrossRefGoogle Scholar
[17]Hopcroft, J. and Ullman, J., Introduction to automata theory, languages, and computation, Addison-Wesley, Reading, Massachusetts, 1979.Google Scholar
[18]Impagliazzo, R. and Naor, M., Decision trees and downward closures, Proceedings of the 3rd IEEE structure in complexity theory conference, IEEE Computer Science Press. Washington. D.C., 1988, pp. 29–38.Google Scholar
[19]Jockusch, C. G., Degree of generic sets, Recursion theory: Its generalizations and applications (Drake, F. R. and Wainer, S. S.. editors), Cambridge University Press. London and New York, 1980, pp. 110–139.Google Scholar
[20]Jockusch, C. G. and Posner, D. B., Double jumps of minimal degrees, this Journal, vol. 43 (1978), pp. 715–724.Google Scholar
[21]Kumabe, M., A 1-generic degree which bounds a minimal degree, this Journal, vol. 55 (1990), pp. 733–743.Google Scholar
[22]Kurtz, S. A., Notions of weak genericity, this Journal, vol. 48 (1983), pp. 764–770.Google Scholar
[23]Lutz, J. H., Resource-bounded Baire category and small circuits in exponential space, Proceedings of the 2nd annual IEEE structure in complexity theory conference, IEEE Computer Science Press. Washington. D.C., 1987. pp. 81–91.Google Scholar
[24]Lutz, J. H., Almost everywhere high nonuniform complexity, Proceedings of the 4th annual IEEE structure in complexity theory conference, IEEE Computer Science Press. Washington, D.C., 1989, pp. 37–53. An updated version appears as Iowa State University Computer Science Department Technical Report #91-18.Google Scholar
[25]Lutz, J. H., Category and measure in complexity classes, SIAM Journal on Computing, vol. 19 (1990), pp. 1100–1131.CrossRefGoogle Scholar
[26]Mayordomo, E., Almost every set in exponential time is P-bi-immune, unpublished manuscript. (1991).CrossRefGoogle Scholar
[27]Ogiwara, M. and Hemachandra, L. A., A complexity theory of feasible closure properties, Proceedings of the 6th annual IEEE structure in complexity theory conference, IEEE Computer Science Press, Washington. D.C., 1991, pp. 16–29.Google Scholar
[28]Poizat, B., Q = NQ?, this Journal, vol. 51 (1986), pp. 22–32.Google Scholar
[29]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, 1967: reprinted by MIT Press, Cambridge, Massachusetts, 1987.Google Scholar
[30]Shore, R. A., private communication, 1991.Google Scholar
[31]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin and New York, 1987.CrossRefGoogle Scholar