, Volume 81, Issue 1, pp 179–200 | Cite as

Sparse Selfreducible Sets and Nonuniform Lower Bounds

  • Harry Buhrman
  • Leen Torenvliet
  • Falk Unger
  • Nikolay VereshchaginEmail author


It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in \({\mathrm{EXP^{NP}}}\), or even in \({\mathrm{EXP}}\) that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that \({\mathrm{EXP^{NP}}}\) does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that \({\mathrm{NEXP}}\) does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of \({\mathrm{EXP}}\) is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for \({\mathrm{NEXP}}\).


Computational complexity Sparseness Selfreducibility 



We thank the anonymous referee for helpful suggestions. Funding was provided by Russian Foundation for Basic Research (Grant No. 16-01-00362), Russian Academic Excellence Project (Grant No. 5-100).


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Authors and Affiliations

  1. 1.CWI AmsterdamAmsterdamThe Netherlands
  2. 2.ILLCUniversiteit van AmsterdamAmsterdamThe Netherlands
  3. 3.Moscow State University, National Research University Higher School of EconomicsMoscowRussia

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