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 Θp2 and that Σ p 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 ∆p2 (respectively, Σp4). 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. 2. coNP/1 6⊆ VSLL unless PH = Θp2. 3. For all constant c > 0, VSLL 6⊆ coNP/n. 4. P/(log log(n) +O(1)) ⊆ VSLL. 5. For all h(n) = log log(n) + ω(1), P/h 6⊆ VSLL.