# 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}
}
• Published 1 April 1999
• Mathematics
• J. Comput. Syst. Sci.
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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