Date: 07 Jun 2005

On complete problems for NPCoNP

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Abstract

It is not known whether complete languages exist for NPCoNP and Sipser has shown that there are relativizations so that NPCoNP has no ≤ m P -complete languages. In this paper we show that NPCoNP 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 NPCoNP and NPCoNPNP then the reduction of L(N t) ε NPCoNP cannot be effectively computed from N t. We extend the relativization results by exhibiting an oracle E such that P ENP ECoNP ENP 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 FNP FCoNP FNP 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 $$(N_t ,N_J , Proof that L(N_1 ) = \overline {L(N_1 )} ) .$$