Very sparse leaf languages

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 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. © Springer-Verlag Berlin Heidelberg 2006.