Resource bounded immunity and simplicity
Theoretical Computer Science
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A set is called NP-immune if it does not contain any infinite NP subsets, and NP-simple if it lies in NP, and its complement is NP-immune. Hartmanis, Li and Yehsa proved in 1986 that no m-hard (many-one hard) set for NP is NP-immune unless NP is contained in SUBEXP. On the other hand we can exhibit a relativized world in which there is an \NP-simple set that is complete under Turing reductions, even conjunctive reductions. We can therefore ask to what extent the original result generalizes to reductions which lie in strength in between many-one and Turing reductions. We show that no positive bounded truth-table complete set for NP is NP-simple, unless UP is contained in SUBEXP. Under the stronger assumption that the intersection of UP and co UP is not contained in SUBEXP it is even true that no bounded truth-table complete set for NP is NP-simple. Further results for different reductions are shown including a similar theorem about NEXP which does not require any assumptions. We derive the results by the use of inseparable sets. This technique turns out to be very powerful in the study of truth-table and even (honest) Turing reductions.