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The complexity of sparse sets in P

Preliminary report
  • Eric W. Allender
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 223)

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 nO(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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Eric W. Allender
    • 1
  1. 1.Department of Computer ScienceRutgers UniversityNew BrunswickUSA

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