On Small Hard Leaf Languages

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Abstract

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 P (A) cannot be polylog-sparse (i.e. census A O(log O(1))), unless PH collapses.

We also generalize balanced ≤ \(^{P,{bit}}_{m}\) -reductions, which were introduced in [6], to other bit-reductions, for example (balanced) truth-table- and Turing-bit-reductions. Then, similarly to above, we prove that NP and Σ \(^{P}_{\rm 2}\) cannot have polylog-sparse hard sets under those balanced truth-table- and Turing-bit-reductions, if the polynomial-time hierarchy is infinite.