A polynomial jump operator

Abstract

For recursive sets A, define a complexity theoretic version of the ordinary recursion theoretic jump by setting A′ equal to the canonical NP^A-complete set. Thus A < ^P_T A′ iff P^A ≠ NP^A. The nth jump, A⁽ⁿ⁾, is defined by iteration. A jumps n times if A < ^P_T A′ < ^P_T… < ^P_T A⁽ⁿ⁾. It is straightforward that the jump operation is monotone. Post's theorem holds for the (relativized) polynomial hierarchy. We establish the following analogues of results in ordinary recursion theory: all relationships between pairs of polynomial Turing degrees and their jumps consistent with monotonicity can be realized by degrees which jump at least twice. For example, there are polynomially incomparable A and B with A′ ≡ ^P_T B′. Moreover, if for each recursive D the set of E such that D join E jumps at least n times is effectively comeager, then these relationships can be realized by degrees jumping at least n times. We also relativize some well-known results by showing that if A′ is polynomially many-one reducible to the join of A and a (co-)sparse set, then P^A = NP^A; and if A′ is polynomially Turing reducible to the join of A and an NP^A-(co-)sparse set, then the relativized polynomial hierarchy collapses to Δ^P,A₂.