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 c0, 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.