On sparse hard sets for counting classes

Mitsunori Ogiwara, Antoni Lozano

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10 Scopus citations


In this paper, we study one-word-decreasing self-reducible sets which are introduced by Lozano and Torán (1991). These are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query and this is lexicographically smaller than the input. We show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact, we can prove that for any class K chosen from {PP, NP, C = P, MOD2P, MOD3 P,...} it holds that (1) if there is a sparse ≤Pbtt-hard set for K then K ⊆ P, and (2) if there is a sparse ≤SNbtt-hard set for K then K ⊆ NP {frown} co-NP. This generalizes the result of Ogiwara and Watanabe (1991) to the mentioned complexity classes.

Original languageEnglish (US)
Pages (from-to)255-275
Number of pages21
JournalTheoretical Computer Science
Issue number2
StatePublished - May 10 1993
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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