Inverting the Turing Jump in Complexity Theory

This paper investigates the invertibility of certain analogs of the Turing jump operator in the polynomial-time Turing degrees. If C is some complexity class, the C-jump of a set A is the canonical C-complete set relative to A. It is shown that the PSPACE-jump and EXP-jump operators are not invertible, i.e., there is a PSPACE-hard (resp. EXP-hard) set that is not p-time Turing equivalent to the PSPACE-jump (resp. EXP-jump) of any set. It is also shown that if PH collapses to p k , then the p k-jump is not invertible. In particular, if NP = co-NP, then the NP-jump is not invertible, witnessed, in particular, by G SAT, where G is any 1-generic set. These results run contrary to the Friedberg Completeness Criterion Fri57] in recursion theory, which says that every (recursive) Turing degree above 0 0 is the Turing jump of another degree. The sets used in the paper to witness C-jump noninvertibility are all of the form G C, where G is 1-generic and C is some C-complete set. Other facts regarding 1-generics G and GSAT are also explored in this paper; in particular, G always lies in NP A ?P A for some A p tt G, but if A p ?tt G SAT p T SAT A for some A, then G p T ASAT, which in turn implies either G p T A or P 6 = NP.