On complete problems for NPCoNP

  • Juris Hartmanis
  • Neil Immerman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 194)


It is not known whether complete languages exist for NPCoNP and Sipser has shown that there are relativizations so that NPCoNP has no ≤mP-complete languages. In this paper we show that NPCoNP has ≤mP-complete languages if and only if it has ≤TP-complete languages. Furthermore, we show that if a complete language L0 exists for NPCoNP and NPCoNPNP then the reduction of L(Nt) ε NPCoNP cannot be effectively computed from Nt. We extend the relativization results by exhibiting an oracle E such that PENPECoNPENPE 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 (Nt, Nj) such that L(Nt)=\(\overline {L(N_1 )} \). On the other hand, there also exist oracles F such that PFNPFCoNPFNPF 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
$$(N_t ,N_J , Proof that L(N_1 ) = \overline {L(N_1 )} ) .$$


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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Juris Hartmanis
    • 1
  • Neil Immerman
    • 2
  1. 1.Department of Computer ScienceCornell UniversityIthaca
  2. 2.Department of Computer ScienceYale UniversityNew Haven

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