Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis

@article{Cai1999SparseHS,
  title={Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis},
  author={Jin-Yi Cai and D. Sivakumar},
  journal={J. Comput. Syst. Sci.},
  year={1999},
  volume={58},
  pages={280-296}
}
Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non)existence of sparse sets complete for P under logspace many?one reductions. We show that if there exists a sparse hard set for P under logspace many?one reductions, then P=LOGSPACE. We further prove that if P has a sparse hard set under many?one reductions computable in NC1, then P collapses to NC1. 
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References

SHOWING 1-10 OF 44 REFERENCES
On the Existence of Hard Sparse Sets under Weak Reductions
TLDR
It is shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P=LOGSPACE, and the proof technique can be applied to resolve open questions about hard sparse sets for NP.
Sparse P-hard sets yield space-efficient algorithms
  • M. Ogihara
  • Computer Science
    Proceedings of IEEE 36th Annual Foundations of Computer Science
  • 1995
TLDR
It is shown that if P has sparse hard sets under logspace many-one reductions, then P/spl sube/DSPACE[log/sup 2/n].
Sparse Hard Sets for P Yield Space-Efficient Algorithms
  • M. Ogihara
  • Computer Science, Mathematics
    Chic. J. Theor. Comput. Sci.
  • 1996
TLDR
It is shown that if P has sparse hard sets under logspace many-one reductions, then P is a subset of DSPACE[log^2 n].
Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis
  • Stephen R. Mahaney
  • Mathematics, Computer Science
    21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
  • 1980
TLDR
The main result is that if there is a sparse NP-complete set under many-one reductions, then P = NP, and it is shown that if the set is under Turing reduction, then the polynomial time hierarchy collapses to Δ2P.
A Note on Sparse Complete Sets
  • S. Fortune
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1979
TLDR
It is shown that the existence of an NP- complete set whose complement is sparse implies P = NP, and that if there is a polynomial time reduction with sparse range to a PTAPE-complete set, then P=PTAPE.
Reducing P to a Sparse Set using a Constant Number of Queries Collapses P to L
  • D. Melkebeek
  • Computer Science, Mathematics
    Computational Complexity Conference
  • 1996
TLDR
It is proved that there is no sparse hard set for P under logspace computable bounded truth-table reductions unless P=L, and parameterizing the sparseness condition, the space bound and the number of queries of the reduction is extended to two-sided error randomized reductions in the multiple access model.
How reductions to sparse sets collapse the polynomial-time hierarchy: a primer: Part II restricted polynomial-time reductions
TLDR
It is proved that if SAT <1tt S for some sparse set S, then P = NP, a result which subsumed all earlier results on polynomial-time bounded-truth-table results on many-one reductions of SAT to sparse sets.
On polynomial time bounded truth-table reducibility of NP sets to sparse sets
TLDR
It is proved that if P not=NP, then there exits a set in NP that is polynomial-time bounded truth-table reducible to no sparse set and intractability of several number theoretic decision problems is investigated.
On Log-Tape Isomorphisms of Complete Sets
  • J. Hartmanis
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1978
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