Relativized generic classes $\mathrm P$ and $\mathrm{NP}$

Classical theorem of Baker, Gill and Solovay states that there exist two oracles $A$ and $B$ such that $\mathrm P^A=\mathrm{NP}^A$, but $\mathrm P^B\neq\mathrm{NP}^B$. This result indicates that the classical tools of computability theory (such as diagonalization) are inapplicable to prove the inequality $\mathrm P\neq\mathrm{NP}$. Generic-case approach to algorithmic problems was suggested by Miasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. Many classical undecidable or hard algorithmic problems become feasible in the generic case. In this paper we introduce generic analogs $\mathrm{genP}$ and $\mathrm{genNP}$ of the classical computational complexity classes $\mathrm P$ and $\mathrm{NP}$. We prove a generic analog of the Baker–Gill–Solovay theorem: there exist two oracles $A$ and $B$ such that $\mathrm{genP}^A=\mathrm{genNP}^A$, but $\mathrm{genP}^B\neq\mathrm{genNP}^B$. Therefore the diagonalization arguments cannot be applied to prove the inequality $\mathrm P\neq\mathrm{NP}$ in the generic case too.