We show that there is a language that is Turing complete for NP but not many-one complete for NP, under a worst-case hardness hypothesis. Our hypothesis asserts the existence of a non-deterministic, double-exponential time machine that runs in time O(2^2^n^c) (for some c > 1) accepting Sigma^* whose accepting computations cannot be computed by bounded-error, probabilistic machines running in time O(2^2^{beta * 2^n^c) (for some beta > 0). This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis.
@InProceedings{mandal_et_al:LIPIcs.FSTTCS.2014.445, author = {Mandal, Debasis and Pavan, A. and Venugopalan, Rajeswari}, title = {{Separating Cook Completeness from Karp-Levin Completeness Under a Worst-Case Hardness Hypothesis}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {445--456}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.445}, URN = {urn:nbn:de:0030-drops-48621}, doi = {10.4230/LIPIcs.FSTTCS.2014.445}, annote = {Keywords: Cook reduction, Karp reduction, NP-completeness, Turing completeness, many-one completeness} }
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