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On being incoherent without being very hard

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Abstract

Trakhtenbrot calls a setA autoreducible ifA is Turing-reducible toA via an oracle Turing machine that never queries the oracle about the input string. Yao considers sets that are autoreducible via probabilistic, polynomial-time oracle Turing machines, and he calls such setscoherent. We continue the study of autoreducible sets, including those that are autoreducible via a family of polynomial-sized circuits, which we callweakly coherent. Sets that are not weakly coherent are calledstrongly incoherent. We show

  • Ifs is superpolynomial and space-constructible, then there is a strongly incoherent set in DSPACE (s(n)).

  • If NEEEXP\( \nsubseteq \) BPEEEXP, then there is a set in NP that is incoherent.

  • IfA is complete for any of the classes ∑ p i , ∏ p i , or Δ p i ,i≥0, thenA is coherent. In particular, all NP-complete sets are coherent.

As corollaries, we obtain new lower bounds onprogram checkability andrandom-self-reducibility. These results answer open questions posed by Yao and by Blum and Kannan.

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Authors and Affiliations

  1. Department of Computer Science, Yale University, Yale Station, P. O. Box 2158, 06520-2158, New Haven, CT, USA

    Richard Beigel

  2. AT&T Bell Laboratories, Room 2C-473, 600 Mountain Avenue, 07974-0636, Murray Hill, NJ, USA

    Joan Feigenbaum

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  1. Richard Beigel
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Beigel, R., Feigenbaum, J. On being incoherent without being very hard. Comput Complexity 2, 1–17 (1992). https://doi.org/10.1007/BF01276436

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