Structure in Complexity Theory pp 1-11 | Cite as

# The complexity of sparse sets in P

## Abstract

P-printable sets, defined in [HY-84], arise naturally in the study of P-uniform circuit complexity, generalized Kolmogorov complexity, and data compression, as well as in many other areas. We present new characterizations of the P-printable sets and present necessary and sufficient conditions for the existence of sparse sets in P which are not P-printable. The complexity of sparse sets in P is shown to be central to certain questions about circuit complexity classes and about one-way functions. Among the main results are:

(1) There is a sparse set in P which is not P-printable iff

There is a sparse set in DLOG which is not P-printable iff

There is a sparse set in FNP — P

(where FNP is a class related to U: FNP=the class of sets accepted by nondeterministic polynomial-time Turing machines which accept inputs of size n with n^{O(1)} accepting computations; FNP stands for NP with “few” accepting computations).

(2) A set S is P-printable iff

S is sparse and S is accepted by a one-way logspace-bounded AuxPDA.

(3) NC=PUNC iff

All P-printable sets in P are in NC iff

All Tally languages in P are in NC.

## Keywords

Polynomial Time Turing Machine Ranking Function Circuit Complexity Input Head## Preview

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## References

- [Al-85]E. W. Allender,
*Invertible functions*, Doctoral Dissertation, Georgia Institute of Technology.Google Scholar - [Al-86]E. W. Allender,
*Characterizations of PUNC and precomputation*, to be presented at the 13th International Colloquium on Automata, Languages and Programming, and will appear in Lecture Notes in Computer Science.Google Scholar - [Al-86a]E. W. Allender,
*Isomorphisms and 1L reductions*, These Proceedings.Google Scholar - [BB-86]J. L. Balcazar and R. V. Book,
*Sets with small generalized Kolmogorov complexity*, Technical Report MSRI 00918-86, Mathematical Sciences Research Institute, Berkeley.Google Scholar - [BDG-85]J. L. Balcazar, J. Diaz, J. Gabarro,
*On some “non-uniform” complexity measures*, 5th Conference on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 199, pp. 18–27.Google Scholar - [BCH-84]P. W. Beame, S. A. Cook, and H. J. Hoover,
*Log depth circuits for division and related problems*, Proc. 25th IEEE Symposium on Foundations of Computer Science, pp. 1–11.Google Scholar - [Be-77]L. Berman,
*Polynomial reducibilities and complete sets*, Doctoral Dissertation, Cornell University.Google Scholar - [Bo-74]R. V. Book,
*Tally languages and complexity classes*, Information and Control 26, 186–193.Google Scholar - [Br-77a]F.-J. Brandenburg,
*On one-way auxiliary pushdown automata*, Proc. 3rd GI Conference, Lecture Notes in Computer Science 48, pp. 133–144.Google Scholar - [Br-77b]F.-J. Brandenburg,
*The contextsensitivity of contextsensitive grammars and languages*, Proc. 4th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 52, pp. 272–281.Google Scholar - [Ch-77]M. P. Chytil,
*Comparison of the active visiting and the crossing complexities*, Proc. 6th Conference on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 53, pp. 272–281.Google Scholar - [Co-71]S. A. Cook,
*Characterizations of pushdown machines in terms of time-bounded computers*, J. ACM 19, 175–183.Google Scholar - [vzG-84]J. von zur Gathen,
*Parallel powering*, Proc. 25th Annual ACM Symposium on Theory of Computing, pp. 31–36.Google Scholar - [GS-85]A. V. Goldberg and M. Sipser,
*Compression and ranking*, Proc. 17th Annual ACM Symposium on Theory of Computing, pp. 440–448.Google Scholar - [Gr-84]J. Grollmann,
*Complexity measures for public-key cryptosystems*, Doctoral Dissertation, Dortmund.Google Scholar - [GS-84]J. Grollmann and A. Selman,
*Complexity measures for public-key cryptosystems*, Proc. 25th IEEE Symposium on Foundations of Computer Science, pp. 495–503.Google Scholar - [Ha-83]J. Hartmanis,
*Generalized Kolmogorov complexity and the structure of feasible computations*, Proc. 24th IEEE Symposium on Foundations of Computer Science, pp. 439–445.Google Scholar - [Ha-83a]J. Hartmanis,
*On sparse sets in NP = P*, Information Processing Letters 16, 55–60.Google Scholar - [HSI-83]J. Hartmanis, V. Sewelson, and N. Immerman,
*Sparse sets in NP-P: EXPTIME versus NEXPTIME*, Proc. 15th Annual ACM Symposium on Theory of Computing, pp. 382–391.Google Scholar - [HY-84]J. Hartmanis and Y. Yesha,
*Computation times of NP sets of different densities*, Theoretical Computer Science 34, 17–32.Google Scholar - [HU-79]J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Mass.Google Scholar
- [Hu-85]D. T. Huynh,
*Non-uniform complexity and the randomness of certain complete languages*, Technical Report TR 85-34, Computer Science Department, Iowa State University.Google Scholar - [Hu-86]D. T. Huynh,
*Resource-bounded Kolmogorov complexity of hard languages*, These Proceedings.Google Scholar - [Ko-83]K.-I. Ko,
*On the definition of some complexity classes of real numbers*, Mathematical Systems Theory 16, 95–109.Google Scholar - [Ko-84]K.-I. Ko,
*A definition of infinite pseudorandom sequences*, manuscript, University of Houston.Google Scholar - [Pi-81]N. Pippenger,
*Pebbling with an auxiliary pushdown*, J. Computer and System Sciences 23, 151–165.Google Scholar - [Ru-86]R. Rubinstein,
*Generalized Kolmogorov complexity, tally sets, sparseness, etc. — a few notes*, manuscript, Iowa State University.Google Scholar - [Ru-81]W. L. Ruzzo,
*On uniform circuit complexity*, J. Computer and System Sciences 21, 365–383.Google Scholar - [Ru-85]W. L. Ruzzo, personal communication.Google Scholar
- [Se-83]V. Sewelson,
*A study of the structure of NP*, Doctoral Dissertation, Cornell University.Google Scholar - [Va-76]L. Valiant,
*Relative complexity of checking and evaluating*, Information Processing Letters 5, 20–23.Google Scholar - [We-80]G. Wechsung,
*A note on the return complexity*, Elektronische Informationsverarbeitung und Kybernetik 16, 139–146.Google Scholar - [WB-79]G. Wechsung and A. Brandstadt,
*A relation between space, return and dual return complexities*, Theoretical Computer Science 9, 127–140.Google Scholar