Space-efficient recognition of sparse self-reducible languages

Lane A. Hemaspaandra, Mitsunori Ogihara, Seinosuke Toda

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Mahaney and others have shown that sparse self-reducible sets have time-efficient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of complete sets for feasible classes until now. This paper shows that sparse self-reducible sets have space-efficient algorithms, and in many cases, even have time-space-efficient algorithms. We conclude that NL, NCk, ACk, LOG(DCFL), LOG(CFL), and P lack complete (or even Turing-hard) sets of low density unless implausible complexity class inclusions hold. In particular, if NL (respectively P, βk, or NP) has a polylog-sparse logspace-hard set, then NL{square image of or equal to}SC (respectively P{square image of or equal to}SC, βk, or PH{square image of or equal to}SC), and if P has subpolynomially sparse logspace-hard sets, then P≠PSPACE.

Original languageEnglish (US)
Pages (from-to)262-296
Number of pages35
JournalComputational Complexity
Issue number3
StatePublished - Sep 1994
Externally publishedYes


  • Subject classifications: 68Q15, 03D15

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics(all)
  • Computational Theory and Mathematics
  • Computational Mathematics


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