### Abstract

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, NC^{k}, AC^{k}, 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, β_{k}, 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, β_{k}, or PH{square image of or equal to}SC), and if P has subpolynomially sparse logspace-hard sets, then P≠PSPACE.

Original language | English (US) |
---|---|

Pages (from-to) | 262-296 |

Number of pages | 35 |

Journal | Computational Complexity |

Volume | 4 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1994 |

Externally published | Yes |

### Fingerprint

### Keywords

- Subject classifications: 68Q15, 03D15

### ASJC Scopus subject areas

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

### Cite this

*Computational Complexity*,

*4*(3), 262-296. https://doi.org/10.1007/BF01206639

**Space-efficient recognition of sparse self-reducible languages.** / Hemaspaandra, Lane A.; Ogihara, Mitsunori; Toda, Seinosuke.

Research output: Contribution to journal › Article

*Computational Complexity*, vol. 4, no. 3, pp. 262-296. https://doi.org/10.1007/BF01206639

}

TY - JOUR

T1 - Space-efficient recognition of sparse self-reducible languages

AU - Hemaspaandra, Lane A.

AU - Ogihara, Mitsunori

AU - Toda, Seinosuke

PY - 1994/9

Y1 - 1994/9

N2 - 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.

AB - 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.

KW - Subject classifications: 68Q15, 03D15

UR - http://www.scopus.com/inward/record.url?scp=33847520341&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847520341&partnerID=8YFLogxK

U2 - 10.1007/BF01206639

DO - 10.1007/BF01206639

M3 - Article

AN - SCOPUS:33847520341

VL - 4

SP - 262

EP - 296

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 3

ER -