On complete problems for NP∩CoNP
 Juris Hartmanis,
 Neil Immerman
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Abstract
It is not known whether complete languages exist for NP∩CoNP and Sipser has shown that there are relativizations so that NP∩CoNP has no ≤ _{ m } ^{ P } complete languages. In this paper we show that NP∩CoNP has ≤ _{ m } ^{ P } complete languages if and only if it has ≤ _{ T } ^{ P } complete languages. Furthermore, we show that if a complete language L _{0} exists for NP∩CoNP and NP∩CoNP≠NP then the reduction of L(N _{t}) ε NP∩CoNP cannot be effectively computed from N _{t}. We extend the relativization results by exhibiting an oracle E such that P ^{E}≠NP ^{E}∩CoNP ^{E}≠NP ^{E} and for which there exist complete languages in the intersection. For this oracle the reduction to a complete language can be effectively computed from complementary pairs of machines (N _{t}, N _{j}) such that L(N _{t})= $\overline {L(N_1 )} $ . On the other hand, there also exist oracles F such that P ^{F}≠NP ^{F}∩CoNP ^{F}≠NP ^{F} for which the intersection has complete languages, but the reductions to the complete language cannot be effectively computable from the complementary pairs of machines. In this case, the reductions can be computed from $$(N_t ,N_J , Proof that L(N_1 ) = \overline {L(N_1 )} ) .$$
 Baker, T., J. Gill and R. Solovay. “Relativization of the P=?NP Question”. SIAM Journal of Computing (1975), 431–442.
 Garey, M.R. and D.S. Johnson. “Computers and Intractability: A guide to the Theory of NPCompleteness”. W.H. Freeman and Co., San Francisco, 1979.
 Kowalczyk, W. “Some Connections Between Representability of Complexity Classes and the Power of Formal Systems of Reasoning”. Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 176. SpringerVerlag 1984, 364–369.
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 Sipser, M. “On Relativization and the Existence of Complete Sets”. Automata Languages and Programming Lecture Notes in Computer Science 140. SpringerVerlag 1982, 523–531.
 Title
 On complete problems for NP∩CoNP
 Book Title
 Automata, Languages and Programming
 Book Subtitle
 12th Colloquium Nafplion, Greece, July 15–19, 1985
 Pages
 pp 250259
 Copyright
 1985
 DOI
 10.1007/BFb0015750
 Print ISBN
 9783540156505
 Online ISBN
 9783540395577
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 194
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Juris Hartmanis ^{(1)}
 Neil Immerman ^{(2)}
 Author Affiliations

 1. Department of Computer Science, Cornell University, 14853, Ithaca, NY
 2. Department of Computer Science, Yale University, 06520, New Haven, CT
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