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 \({\mathrm{EXP^{NP}}}\), 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 \({\mathrm{NEXP}}\).
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Notes
Formally, the reader might notice, these questions are independent. They are related however as follows. If \({\mathrm{EXP}}^\mathrm{NP}\) or even \({\mathrm{NEXP}}\) has polynomial size circuits then \({\mathrm{P}}\ne {\mathrm{NP}}\) follows. Therefore, it seems that it should be easier to settle the former question, in the negative, than it does to settle the latter.
In several places in this paper we use “optimal” where this is not an exact statement. If we prove a problem to be in \({\mathrm{NEXP}}\) and show an oracle relative to which it is not in \({\mathrm{EXP}}\) then it could still be in many intermediate classes, and even a non-relativizing proof might still show it to be in \({\mathrm{EXP}}\). Though we always make the exact meaning of optimal precise in theorems following the statement, the reader should be cautioned.
Here \(S_x^{M}\) resp. T denote the nodes of the graphs \(S_x^{S,M}\), T.
A set S is called P-selective if there exists a polynomial time function f such that \(f(x,y)\in \{x,y\}\) and \([x\in S\vee y\in S]\Rightarrow f(x,y)\in S\).
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Acknowledgements
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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Buhrman, H., Torenvliet, L., Unger, F. et al. Sparse Selfreducible Sets and Nonuniform Lower Bounds. Algorithmica 81, 179–200 (2019). https://doi.org/10.1007/s00453-018-0439-0
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DOI: https://doi.org/10.1007/s00453-018-0439-0