On complete problems for NP∩CoNP

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 ≤ ^P _m -complete languages. In this paper we show that NP∩CoNP has ≤ ^P _m -complete languages if and only if it has ≤ ^P _T -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

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

Supported in part by National Science Foundation Grant DCR8301766.

Research supported by an NSF Postdoctoral Fellowship.