Very sparse leaf languages

Lance Fortnow; Mitsunori Ogihara

doi:10.1007/11821069_33

Very sparse leaf languages

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

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 Θ₂^p and that Σ₂^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 Δ₂^p (respectively, Σ₄^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 = Θ₂^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)
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
DOIs	https://doi.org/10.1007/11821069_33
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)	0302-9743
ISSN (Electronic)	1611-3349

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

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

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10.1007/11821069_33

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Fortnow, L., & Ogihara, M. (2006). Very sparse leaf languages. In 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. https://doi.org/10.1007/11821069_33

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

Mathematical Foundations of Computer Science 2006 - 31st International Symposium, MFCS 2006, Proceedings. Springer Verlag, 2006. p. 375-386 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4162 LNCS).

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

Fortnow, L & Ogihara, M 2006, Very sparse leaf languages. in Mathematical Foundations of Computer Science 2006 - 31st International Symposium, MFCS 2006, Proceedings. 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. https://doi.org/10.1007/11821069_33

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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.

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Very sparse leaf languages

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