Hardness Hypotheses, Derandomization, and Circuit Complexity

Abstract.We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: The measure hypothesis: NP does not have p-measure 0.The pseudo-NP hypothesis: there is an NP language that can be distinguished from any DTIME$$(2^{{n}^\epsilon})$$ language by an NP refuter.The NP-machine hypothesis: there is an NP machine accepting 0* for which no $$2^{{n}^\epsilon}$$ -time machine can find infinitely many accepting computations. We show that the NP-machine hypothesis is implied by each of the first two. Previously, no relationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuit-size lower bounds that are known to follow from the first two hypotheses also follow from the NP-machine hypothesis. In particular, the NP-machine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizes AM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses.

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