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Dimension, Halfspaces, and the Density of Hard Sets

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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. 2.

    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.

Intuitively, polynomial-time dimension is a means of quantifying the size and complexity of classes within the exponential time complexity class E. The class P has dimension 0, E itself has dimension 1, and any class with dimension less than 1 cannot contain E. As a corollary, it follows that all sets which are hard for E under these types of 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.

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Authors and Affiliations

  1. Department of Computer Science, University of Wyoming, Laramie, WY, USA

    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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Correspondence to John M. Hitchcock.

Additional information

This research was supported in part by NSF grants 0515313 and 0652601.

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Harkins, R.C., Hitchcock, J.M. Dimension, Halfspaces, and the Density of Hard Sets. Theory Comput Syst 49, 601–614 (2011). https://doi.org/10.1007/s00224-010-9288-1

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