Abstract:
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 NC 1 , then P collapses to NC 1 .
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