Very Sparse Leaf Languages

Unger studied the balanced leaf languages defined via poly-logarithmically 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$^{p}_{\rm 2}$ and that Σ$^{p}_{\rm 2}$-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$^{p}_{\rm 2}$ (respectively, Σ$^{p}_{\rm 4}$). 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 $\not\subseteq$ VSLL unless PH = Θ$^{p}_{\rm 2}$. 3. For all constant c>0, VSLL $\not\subseteq$ coNP/nc. 4. P/(loglog(n)+O(1))⊆ VSLL. 5. For all h(n) = loglog(n) + ω(1), P$/h \not\subseteq$ VSLL.

[1]  Ran Canetti.  More on BPP and the Polynomial-Time Hierarchy , 1996, Inf. Process. Lett..

[2]  J. Cai,et al.  S^p _2 \subseteq ZPP^{NP} , 2001, FOCS 2001.

[3]  Pierluigi Crescenzi,et al.  A Uniform Approach to Define Complexity Classes , 1992, Theor. Comput. Sci..

[4]  Jin-Yi Cai,et al.  Competing provers yield improved Karp-Lipton collapse results , 2005, Inf. Comput..

[5]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[6]  Jin-Yi Cai,et al.  PSPACE Survives Constant-Width Bottlenecks , 1991, Int. J. Found. Comput. Sci..

[7]  Falk Unger,et al.  On Small Hard Leaf Languages , 2005, MFCS.

[8]  Osamu Watanabe,et al.  On polynomial time bounded truth-table reducibility of NP sets to sparse sets , 1990, STOC '90.

[9]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[10]  Alexander Russell,et al.  Symmetric alternation captures BPP , 1998, computational complexity.

[11]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[12]  Jim Kadin,et al.  P^(NP[O(log n)]) and Sparse Turing-Complete Sets for NP , 1989, J. Comput. Syst. Sci..

[13]  Silvio Micali,et al.  Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems , 1991, JACM.

[14]  Jim Kadin.  The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses , 1988, SIAM J. Comput..

[15]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[16]  Ian Parberry,et al.  On the Construction of Parallel Computers from Various Bases of Boolean Functions , 1986, Theor. Comput. Sci..

[17]  Michel Rigo,et al.  Abstract numeration systems and tilings , 2005 .