### 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 | Mathematical Foundations of Computer Science 2006 - 31st International Symposium, MFCS 2006, Proceedings |

Publisher | Springer Verlag |

Pages | 375-386 |

Number of pages | 12 |

ISBN (Print) | 3540377913, 9783540377917 |

State | Published - Jan 1 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) |
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Volume | 4162 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

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

City | Stara Lesna |

Period | 8/28/06 → 9/1/06 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematical Foundations of Computer Science 2006 - 31st International Symposium, MFCS 2006, Proceedings*(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.