### Abstract

Unger studied the balanced leaf languages defined via polylogarithmically sparse leaf pattern sets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to Θ_{2}^{p} and that Σ_{2}^{p}-complete sets are not polynomial-time bounded-truth-table reducible (respectively, polynomial-time Turing reducible) to any such balanced leaf language unless the polynomial hierarchy collapses to Δ_{2}^{p} (respectively, Σ_{4}^{p}). This paper studies the complexity of the class of such balanced leaf languages, which will be denoted by VSLL. In particular, the following tight upper and lower bounds of VSLL are shown: 1. coNP ⊆ VSLL ⊆ coNP/poly (the former inclusion is already shown by Unger). 2. coNP/1 ⊈ VSLL unless PH = Θ_{2}^{p}. 3. For all constant c > 0, VSLL ⊈ coNP/n^{c}. 4. P/(log log(n) + O(1)) ⊆ VSLL. 5. For all h(n) = log log(n) +ω(1), P/h ⊈ VSLL.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 375-386 |

Number of pages | 12 |

Volume | 4162 LNCS |

ISBN (Print) | 3540377913, 9783540377917 |

State | Published - 2006 |

Externally published | Yes |

Event | 31st International Symposium on Mathematical Foundations of Computer Science, MFCS 2006 - Stara Lesna, Slovakia Duration: Aug 28 2006 → Sep 1 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 4162 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 31st International Symposium on Mathematical Foundations of Computer Science, MFCS 2006 |
---|---|

Country | Slovakia |

City | Stara Lesna |

Period | 8/28/06 → 9/1/06 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 4162 LNCS, pp. 375-386). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4162 LNCS). Springer Verlag.

**Very sparse leaf languages.** / Fortnow, Lance; Ogihara, Mitsunori.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 4162 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4162 LNCS, Springer Verlag, pp. 375-386, 31st International Symposium on Mathematical Foundations of Computer Science, MFCS 2006, Stara Lesna, Slovakia, 8/28/06.

}

TY - GEN

T1 - Very sparse leaf languages

AU - Fortnow, Lance

AU - Ogihara, Mitsunori

PY - 2006

Y1 - 2006

N2 - Unger studied the balanced leaf languages defined via polylogarithmically sparse leaf pattern sets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to Θ2p and that Σ2p-complete sets are not polynomial-time bounded-truth-table reducible (respectively, polynomial-time Turing reducible) to any such balanced leaf language unless the polynomial hierarchy collapses to Δ2p (respectively, Σ4p). This paper studies the complexity of the class of such balanced leaf languages, which will be denoted by VSLL. In particular, the following tight upper and lower bounds of VSLL are shown: 1. coNP ⊆ VSLL ⊆ coNP/poly (the former inclusion is already shown by Unger). 2. coNP/1 ⊈ VSLL unless PH = Θ2p. 3. For all constant c > 0, VSLL ⊈ coNP/nc. 4. P/(log log(n) + O(1)) ⊆ VSLL. 5. For all h(n) = log log(n) +ω(1), P/h ⊈ VSLL.

AB - Unger studied the balanced leaf languages defined via polylogarithmically sparse leaf pattern sets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to Θ2p and that Σ2p-complete sets are not polynomial-time bounded-truth-table reducible (respectively, polynomial-time Turing reducible) to any such balanced leaf language unless the polynomial hierarchy collapses to Δ2p (respectively, Σ4p). This paper studies the complexity of the class of such balanced leaf languages, which will be denoted by VSLL. In particular, the following tight upper and lower bounds of VSLL are shown: 1. coNP ⊆ VSLL ⊆ coNP/poly (the former inclusion is already shown by Unger). 2. coNP/1 ⊈ VSLL unless PH = Θ2p. 3. For all constant c > 0, VSLL ⊈ coNP/nc. 4. P/(log log(n) + O(1)) ⊆ VSLL. 5. For all h(n) = log log(n) +ω(1), P/h ⊈ VSLL.

UR - http://www.scopus.com/inward/record.url?scp=33750045990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750045990&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33750045990

SN - 3540377913

SN - 9783540377917

VL - 4162 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 375

EP - 386

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -