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Dimension, Halfspaces, and the Density of Hard Sets

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Computing and Combinatorics (COCOON 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4598))

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Abstract

We use the connection between resource-bounded dimension and the online mistake-bound model of learning to show that the following classes have polynomial-time dimension zero.

  1. 1

    The class of problems which reduce to nondense sets via a majority reduction.

  2. 1

    The class of problems which reduce to nondense sets via an iterated reduction that composes a bounded-query truth-table reduction with a conjunctive reduction.

As corollary, all sets which are hard for exponential time under these reductions are exponentially dense. The first item subsumes two previous results and the second item answers a question of Lutz and Mayordomo. Our proofs use Littlestone’s Winnow2 algorithm for learning r-of-k threshold functions and Maass and Turán’s algorithm for learning halfspaces.

This research was supported in part by NSF grant 0515313.

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Authors and Affiliations

  1. Department of Computer Science, University of Wyoming,  

    Ryan C. Harkins & John M. Hitchcock

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  1. Ryan C. Harkins
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  2. John M. Hitchcock
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Guohui Lin

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© 2007 Springer-Verlag Berlin Heidelberg

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Harkins, R.C., Hitchcock, J.M. (2007). Dimension, Halfspaces, and the Density of Hard Sets. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_15

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  • DOI: https://doi.org/10.1007/978-3-540-73545-8_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73544-1

  • Online ISBN: 978-3-540-73545-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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