# On complete problems for *NP*∩*CoNP*

Conference paper

- First Online:

DOI: 10.1007/BFb0015750

- Cite this paper as:
- Hartmanis J., Immerman N. (1985) On complete problems for
*NP*∩*CoNP*. In: Brauer W. (eds) Automata, Languages and Programming. ICALP 1985. Lecture Notes in Computer Science, vol 194. Springer, Berlin, Heidelberg

## 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$$(N_t ,N_J , Proof that L(N_1 ) = \overline {L(N_1 )} ) .$$

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## Copyright information

© Springer-Verlag 1985