# On the existence of hard sparse sets under weak reductions

## Abstract

Recently a 1978 conjecture by Hartmanis was resolved by Cai and Sivakumar, following progress made by Ogihara. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P=LOGSPACE. We extend these results to the case of sparse sets that are hard under more general reducibilities. Furthermore, the proof technique can be applied to resolve open questions about hard sparse sets for NP as well. Using algebraic and probabilistic techniques, we show the following results.

- (1)
If there exists a sparse set that is hard for P under bounded truthtable reductions, then P=NC

^{2}. - (2)
If there exists a sparse set that is hard for P under randomized logspace reductions with

*two-sided error*, then P=RLOGSPACE (with two-way access to the random tape). - (3)
If there exists an NP-hard sparse set under randomized polynomial-time reductions with two-sided error, then NP=RP.

- (4)
If there exists a disjunctive truth-table hard sparse set for NP, then NP=RP.

## Keywords

Boolean Circuit Random Tape Table Reduction SIGACT News Fast Parallel Algorithm## Preview

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