STACS 2006

Volume 3884 of the series Lecture Notes in Computer Science pp 455-468

Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds

  • Harry BuhrmanAffiliated withLancaster UniversityCWI AmsterdamUniversiteit van Amsterdam
  • , Leen TorenvlietAffiliated withLancaster UniversityUniversiteit van Amsterdam
  • , Falk UngerAffiliated withLancaster UniversityCWI Amsterdam

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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.


Computational Complexity Sparseness Selfreducibility