On the P vs NP problem over reals with integer oracle

Abstract

Relativized complexity classes are actively studied in the computation complexity theory since 1975. Classical result of Baker, Gill and Solovay states that there exist two oracles A and B such that P^A = NP^A, but P^B ≠ NP^B. This result indicates that the classical tools of theory of algorithms (such as diagonalization) are inapplicable to prove the inequality P ≠ NP. We consider a relativised Blum-Shub-Smale model of computations over the field of real numbers. We prove that relativized complexity classes P^Z and NP^Z are different in this model, where the oracle Z is the set of integers.