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Structure in Complexity Theory pp 1–11Cite as

The complexity of sparse sets in P

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Part of the book series: Lecture Notes in Computer Science ((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.

Portions of this research were carried out while the author was supported by NSF grant MCS 81-03608.

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Authors and Affiliations

  1. Department of Computer Science, Rutgers University, 08903, New Brunswick, N.J., USA

    Eric W. Allender

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  1. Eric W. Allender
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Alan L. Selman

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© 1986 Springer-Verlag Berlin Heidelberg

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Allender, E.W. (1986). The complexity of sparse sets in P. In: Selman, A.L. (eds) Structure in Complexity Theory. Lecture Notes in Computer Science, vol 223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16486-3_85

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  • DOI: https://doi.org/10.1007/3-540-16486-3_85

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  • Print ISBN: 978-3-540-16486-9

  • Online ISBN: 978-3-540-39825-7

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