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 2 and that Σp2-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 ∆ p 2 (respectively, Σ p 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. 2. coNP/1 � ⊆ VSLL unless PH = Θ p 2. 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

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 Theta^p_2 and that Sigam^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 Delta^p_2 (respectively, Sigma^p_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 is included in VSLL and VSLL is included in coNP/poly (the former inclusion is already shown by Unger). 2. coNP/1 is not included in VSLL unless PH collapses to Theta^p_2. 3. For no constant c>0, VSLL is included coNP/n^c. 4. P/(loglog(n) + O(1)) is included in VSLL. 5. For no h(n) = loglog(n) + omega(1), P/h is included in VSLL

Very sparse leaf languages

Unger studied the balanced leaf languages defined via poly-logarithmically sparse leaf patternsets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to \Theta p2 and that \Sigma p2-complete setsare not polynomial-time bounded-truth-table reducible (respectively, polynomial-time Turing reducible) to any such balanced leaf language unless the polynomial hierarchy collapses to \Delta p2(respectively, \Sigma p 4).This paper studies the complexity of the class of such balanced leaf languages, which wil