Reductions to sets of low information content

Arvind, Vikraman; Han, Yenjo; Hemachandra, Lane A.; Köbler, Johannes; Lozano, Antoni; Mundhenk, Martin; Ogiwara, Mitsunori; Schöning, Uwe; Silvestri, Riccardo; Thierauf, Thomas

doi:10.1007/3-540-55719-9_72

Cited by 27 publications

(16 citation statements)

References 35 publications

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“…The left set method turned out to be a well suited tooi to prove collapse results concerning sparse sets. Using this method similar results were obtained for polynomial-time conjunctive réductions [3,32]. Also the proof in [3] showing that no bounded Turing hard set for NP conjunctively reduces to a sparse set unless P = NP uses the left set technique.…”

supporting

confidence: 56%

“…Using this method similar results were obtained for polynomial-time conjunctive réductions [3,32]. Also the proof in [3] showing that no bounded Turing hard set for NP conjunctively reduces to a sparse set unless P = NP uses the left set technique. Furthermore, it makes use of the fact that the sets in R^T (R?c (SPARSE)) are monotonously reducible to a sparse set.…”

supporting

confidence: 56%

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Monotonous and randomized reductions to sparse sets

Arvind¹,

Köbler²,

Mundhenk³

1996

RAIRO-Theor. Inf. Appl.

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supporting

confidence: 56%

supporting

confidence: 56%

Monotonous and randomized reductions to sparse sets

Arvind¹,

Köbler²,

Mundhenk³

1996

RAIRO-Theor. Inf. Appl.

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“…The consequences in Theorem 6.1 are much stronger than what is known to follow from P = NP. If P = NP, then no ≤ p btt -hard or ≤ p c -hard set is sparse [26,3], but it is not known whether hard sets under disjunctive reductions or unbounded Turing reductions can be sparse.…”

Section: Hard Sets For Npmentioning

confidence: 99%

Online Learning and Resource‐Bounded Dimension: Winnow Yields New Lower Bounds for Hard Sets

Hitchcock¹

2007

SIAM J. Comput.

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We establish a relationship between the online mistake-bound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao (2000), and solves one of Lutz and Mayordomo's "Twelve Problems in Resource-Bounded Measure" (1999).

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“…Results (1) and (2) improved previous separations due to Watanabe [18]. The classes in (2) and (3) were reduced to disjunctions, which can be learned by Littlestone's Winnow algorithm [9]. We obtain further results in this direction using more sophisticated learning algorithms and concept classes that generalize disjunctions.…”

Section: Introductionsupporting

confidence: 67%

Dimension, Halfspaces, and the Density of Hard Sets

Harkins

Hitchcock

Lecture Notes in Computer Science

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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. The class of problems which reduce to nondense sets via a majority reduction.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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Reductions to sets of low information content

Cited by 27 publications

References 35 publications

Monotonous and randomized reductions to sparse sets

Monotonous and randomized reductions to sparse sets

Online Learning and Resource‐Bounded Dimension: Winnow Yields New Lower Bounds for Hard Sets

Dimension, Halfspaces, and the Density of Hard Sets

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