Optimal Proof Systems and Sparse Sets

We exhibit a relativized world where NP ∩ SPARSE has no complete sets. This gives the first relativized world where no optimal proof systems exist. We also examine under what reductions NP ∩ SPARSE can have complete sets. We show a close connection between these issues and reductions from sparse to tally sets. We also consider the question as to whether the NP ∩ SPARSE languages have a computable enumeration.

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