Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds

  • Harry Buhrman
  • Leen Torenvliet
  • Falk Unger
Conference paper

DOI: 10.1007/11672142_37

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3884)
Cite this paper as:
Buhrman H., Torenvliet L., Unger F. (2006) Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds. In: Durand B., Thomas W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg

Abstract

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 EXPNP, or even in 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 Toran [10] that EXPNP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also show that this result is optimal with respect to relativizing proofs, by exhibiting an oracle relative to which all of 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 NEXP.

Keywords

Computational Complexity Sparseness Selfreducibility 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Harry Buhrman
    • 1
    • 2
  • Leen Torenvliet
    • 2
  • Falk Unger
    • 1
  1. 1.CWI Amsterdam 
  2. 2.Universiteit van Amsterdam 

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