Lowness and the complexity of sparse and tally descriptions

  • V. Arvind
  • J. Köbler
  • M. Mundhenk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)


We investigate the complexity of obtaining sparse descriptions for sets in various reduction classes to sparse sets. Let A be a set in a certain reduction class R r (SPARSE). Then we are interested in finding upper bounds for the complexity (relative to A) of sparse sets S such that A ∃ R r (S). By establishing such upper bounds we are able to derive the lowness of A. In particular, we show that if a set A is in the class R hd p (R c p (SPARSE)) then A is in R p c (R hd p (S)) for a sparse set S ∃ NP(A). As a consequence we can locate R hd p (R c p (SPARSE)) in the EL 3 Θ level of the extended low hierarchy. Since R hd p (R c p (SPARSE)) \(\supseteq\)R b p (R c p (SPARSE)) this solves the open problem of locating the closure of sparse sets under bounded truth-table reductions optimally in the extended low hierarchy. Furthermore, we show that for every AR d p (SPARSE) there exists a sparse set S ∃ NP(A ⊕ SAT)/Fθ 2 p (A) such that AR d p (S). Based on this we show that R 1−tt p (R d p (SPARSE)) is in EL 3 Θ .

Finally, we construct for every set AR c p (TALLY)∩R d p (TALLY) (or equivalently, A ∃ IC[log, poly], as shown in [AHH+92]) a tally set T ∃ P(A ⊕ SAT) such that AR c p (T) ∩ R d p (T). This implies that the class IC[log, poly] of sets with low instance complexity is contained in EL 1 Σ .


Polynomial Time Simplicity Result Reduction Class Polynomial Time Hierarchy SlAM Journal 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • V. Arvind
    • 1
  • J. Köbler
    • 2
  • M. Mundhenk
    • 2
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology, DelhiNew DelhiIndia
  2. 2.Abteilung für Theoretische InformatikUniversitÄt UlmUlmGermany

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