On Small Hard Leaf Languages

  • Falk Unger
Conference paper

DOI: 10.1007/11549345_67

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)
Cite this paper as:
Unger F. (2005) On Small Hard Leaf Languages. In: Jȩdrzejowicz J., Szepietowski A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg

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 ⊆ BLeafP(A) cannot be polylog-sparse (i.e. censusAO(logO(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.

Keywords

Computational Complexity Leaf Languages Seeds Sparseness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Falk Unger
    • 1
  1. 1.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands

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