On Small Hard Leaf Languages

This paper deals with balanced leaf language complexity classes, introduced independently in [1] and [14]. We propose the seed concept for leaf languages, which allows us to give “short” representations for leaf words. We then use seeds to show that leaf languages A with NP ⊆ BLeaf (A) cannot be polylog-sparse (i.e. censusA ∈ O(log )), unless PH collapses. We also generalize balanced ≤ m -reductions, which were introduced in [6], to other bit-reductions, for example (balanced) truth-tableand Turing-bit-reductions. Then, similarly to above, we prove that NP and Σ 2 cannot have polylog-sparse hard sets under those balanced truthtableand Turing-bit-reductions, if the polynomial-time hierarchy is infinite.